Research ArticleMultistage Spectral Relaxation Method for Solvingthe Hyperchaotic Complex Systems

Hassan Saberi Nik1 and Paulo Rebelo2

1 Department of Mathematics Islamic Azad University Mashhad Branch Mashhad Iran2Departamento de Matematica Universidade da Beira Interior 6201-001 Covilha Portugal

Correspondence should be addressed to Hassan Saberi Nik saberi hssnyahoocom

Received 4 August 2014 Accepted 12 September 2014 Published 16 October 2014

Academic Editor Fazlollah Soleymani

Copyright copy 2014 H Saberi Nik and P Rebelo This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We present a pseudospectral method application for solving the hyperchaotic complex systems The proposed method called themultistage spectral relaxation method (MSRM) is based on a technique of extending Gauss-Seidel type relaxation ideas to systemsof nonlinear differential equations and using the Chebyshev pseudospectral methods to solve the resulting system on a sequence ofmultiple intervals In this new application the MSRM is used to solve famous hyperchaotic complex systems such as hyperchaoticcomplex Lorenz system and the complex permanent magnet synchronous motor We compare this approach to the Runge-Kuttabased ode45 solver to show that the MSRM gives accurate results

1 Introduction

Chaos theory studies the behaviour of dynamical systems thatare highly sensitive to initial conditions and have complexand highly unpredictable profiles [1 2] Chaotic systems canbe observed in a wide variety of applications In 1982 thecomplex Lorenz equations were proposed by Fowler et al [3]which extended nonlinear systems into complex space Afterthat some research works in this field have been achieved[4ndash9] With in-depth study of complex nonlinear systemsa variety of physical phenomena could be described by thechaotic or hyperchaotic complex systems for instance thedetuned laser systems and the amplitudes of electromagneticfields

The nature of complex chaotic systems precludes thepossibility of obtaining closed form analytical solutions ofthe underlying governing equations Thus approximate-analytical methods which are implemented on a sequence ofmultiple intervals to increase their radius of convergence areoften used to solve IVPsmodelling chaotic systems Examplesof multistage methods that have been developed recently tosolve IVPs for chaotic and nonchaotic systems include the

multistage homotopy analysis method [10] piecewise homo-topy perturbation methods [11 12] multistage variationaliteration method [13] and multistage differential transfor-mation method [14] Other multistage methods which usenumerical integration techniques have also been proposedsuch as the piecewise spectral homotopy analysis method[15ndash17] which uses a spectral collocation method to performthe integration process Accurate solutions of highly chaoticand hyperchaotic systems require resolution over manysmall intervals Thus seeking analytical solutions over thenumerous intervals may be impractical or computationallyexpensive if the solution is sought over very long intervals

In this paper we propose a piecewise or multistagespectral relaxation method (MSRM) for solving the hyper-chaotic complex systems as an accurate and robust alternativeto recent multistage methods The proposed MSRM wasdeveloped using the Gauss-Seidel idea of decoupling systemsof equations and using Chebyshev pseudospectral methodsto solve the resulting decoupled system on a sequence ofmultiple intervalsThe spectral relaxationmethod (SRM)wasrecently proposed in [18 19]

The rest of the paper is organized as follows In Section 2we give a brief description of the proposedMSRM algorithm

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 943293 10 pageshttpdxdoiorg1011552014943293

2 The Scientific World Journal

In Section 3 we present the numerical implementation of theMSRM on two examples of hyperchaotic complex systemsFinally the conclusion is given in Section 4

2 Multistage Spectral Relaxation Method

In this section we give a brief description of the numericalmethod of solution used to solve the nonlinear hyperchaoticcomplex We employ the multistage spectral relaxationmethod (MSRM) proposed in [19] The MSRM algorithm isbased on a Gauss-Seidel type of relaxation that decouplesand linearises the system and the use of spectral collocationmethod to solve the linearised equations in a sequentialmanner For compactness we express the system of 119898

nonlinear first order differential equations in the form

119903(119905) =

119898

sum

119896=1

120572119903119896119909119896(119905)

+ 119891119903[1199091(119905) 1199092(119905) 119909

119903minus1(119905)

119909119903+1

(119905) 119909119898(119905)]

(1)

subject to the initial conditions

119909119903(0) = 119909

lowast

119903 119903 = 1 2 119898 (2)

where 119909119903are the unknown variables and 119909

lowast

119903are the corre-

sponding initial conditions 120572119903119896

are known constant inputparameters and 119891

119903is the nonlinear component of the 119903th

equation and the dot denotes differentiation with respect totime 119905

The scheme computes the solution of (1) in a sequenceof equal subintervals that makes the entire interval Wedefine the interval of integration as Ω = [0 119879] and divideit into a sequence of nonoverlapping subintervals Ω

119894=

[119905119894minus1

119905119894] (119894 = 1 2 3 119891) where 119905

0= 0 and 119905

119891= 119879

We denote the solution of (1) in the first subinterval [1199050 1199051]

as 1199091

119903(119905) and the solutions in the subsequent subintervals

[119905119894minus1

119905119894] (119894 = 2 3 119891) as 119909119894

119903(119905) For obtaining the solution

in the first interval [1199050 1199051] (2) is used as the initial condition

By using the continuity condition between neighbouringsubintervals the obtained solution in the interval [119905

0 1199051] is

used to obtain the initial condition for the next subinterval[1199051 1199052] This is applied over the 119891 successive subintervals

that is the obtained solution for each subinterval [119905119894minus1

119905119894] is

used to obtain the initial condition for the next subinterval[119905119894 119905119894+1

] (119894 = 1 2 119891minus1)Thus in each interval [119905119894minus1

119905119894]we

must solve

119894

119903= 120572119903119903119909119894

119903+ (1 minus 120575

119903119904)

119898

sum

119896=1

120572119903119896119909119894

119896

+ 119891119903[119909119894

1 119909

119894

119903minus1 119909119894

119903+1 119909

119894

119899]

(3)

subject to

119909119894

119903(119905119894minus1

) = 119909119894minus1

119903(119905119894minus1

) (4)

where 120575119903119904is the Kronecker delta As mentioned earlier the

main idea behind the MSRM scheme is decoupling the

system of nonlinear IVPs using the Gauss-Seidel idea ofdecoupling systems of algebraic equations The proposedMSRM iteration scheme for the solution in the interval Ω

119894=

[119905119894minus1

119905119894] is given as

119894

1119904+1minus 12057211

119909119894

1119904+1= 12057212

119909119894

2119904+ 12057213

119909119894

3119904

+ sdot sdot sdot + 1205721119899

119909119894

119899119904+ 1198911[119909119894

1119904 119909

119894

119899119904]

119894

2119904+1minus 12057222

119909119894

2119904+1= 12057221

119909119894

1119904+1+ 12057223

119909119894

3119904

+ sdot sdot sdot + 1205722119899

119909119894

119899119904

+ 1198912[119909119894

1119904+1 119909119894

2119904 119909

119894

119899119904]

119894

119898119904+1minus 120572119898119898

119909119894

119898119904+1= 1205721198981

119909119894

1119904+1+ sdot sdot sdot + 120572

119898119898minus1119909119894

119898minus1119904+1

+ 119891119898[119909119894

1119904+1 119909

119894

119898minus1119904+1 119909119894

119898119904]

(5)

subject to the initial conditions

119909119894

119903119904+1(119905119894minus1

) = 119909119894minus1

119903(119905119894minus1

) 119903 = 1 2 119898 (6)

where 119909119903119904

is the estimate of the solution after 119904 iterations Asuitable initial guess to start the iteration scheme (5) is onethat satisfies the initial condition (6) A convenient choiceof initial guess that was found to work in the numericalexperiments considered in this work is

119909119894

1199030(119905) =

119909lowast

119903if 119894 = 1

119909119894minus1

119903(119905119894minus1

) if 2 le 119894 le 119891

(7)

The Chebyshev spectral method is used to solve (5) oneach interval [119905

119894minus1 119905119894] First the region [119905

119894minus1 119905119894] is transformed

to the interval [minus1 1] onwhich the spectralmethod is definedby using the linear transformation

119905 =

(119905119894minus 119905119894minus1

) 120591

2

+

(119905119894+ 119905119894minus1

)

2

(8)

in each interval [119905119894minus1

119905119894] for 119894 = 1 119891 We then discretize

the interval [119905119894minus1

119905119894] using the Chebyshev-Gauss-Lobatto

collocation points [20]

120591119894

119895= cos(

120587119895

119873

) 119895 = 1 2 119873 (9)

which are the extrema of the 119873th order Chebyshev polyno-mial

119879119873(120591) = cos (119873cosminus1120591) (10)

The Chebyshev spectral collocation method is based onthe idea of introducing a differentiation matrix 119863 which isused to approximate the derivatives of the unknown variables119909119894

119903119904+1(119905) at the collocation points as thematrix vector product

119889119909119894

119903119904+1

119889119905

1003816100381610038161003816100381610038161003816100381610038161003816119905=119905119895

=

119873

sum

119896=0

D119895119896119909119894

119903119904+1= DX119894

119903119904+1 119895 = 1 2 119873

(11)

The Scientific World Journal 3

Table 1 Numerical comparison between MSRM and ode45 for the hyperchaotic complex Lorenz system

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

2 minus291138 minus291138 2173155 2173155 minus324491 minus3244914 minus363001 minus363001 652144 652144 minus630884 minus6308846 280571 280571 minus277638 minus277638 minus237099 minus2370998 001134 001134 209585 209585 minus014880 minus01488010 minus080219 minus080219 1648559 1648560 minus006690 minus006690

where D = 2119863(119905119894minus 119905119894minus1

) and X119894119903119904+1

= [119909119894

119903119904+1(120591119894

0) 119909119894

119903119904+1(120591119894

1)

119909119894

119903119904+1(120591119894

119873)] are the vector functions at the collocation

points 120591119894119895

Applying the Chebyshev spectral collocation method in(5) gives

A119903X119894119903119904+1

= B119894119903 X119894

119903119904+1(120591119894minus1

119873) = X119894minus1

119903(120591119894minus1

119873)

119903 = 1 2 119898

(12)

withA119903= D minus 120572

119903119903I

B1198941= 12057212X1198942119904

+ sdot sdot sdot + 1205721119899X119894119899119904

+ 1198911[X1198941119904 X119894

119898119904]

B1198942= 12057221X1198941119904+1

+ 12057223X1198943119904

+ sdot sdot sdot + 1205722119898

X119894119898119904

+ 1198912[X1198941119904+1

X1198942119904 X119894

119898119904]

B119894119898

= 1205721198981

X1198941119904+1

+ 1205721198982

X1198942119904+1

+ sdot sdot sdot + 120572119898119898minus1

X119894119898minus1119904+1

+ 119891119898[X1198941119904+1

X119894119898minus1119904+1

X119894119898119904

]

(13)

where I is an identity matrix of order 119873 + 1 Thus startingfrom the initial approximation (7) the recurrence formula

X119894119903119904+1

= Aminus1119903B119894119903 119903 = 1 2 119898 (14)

can be used to obtain the solution 119909119894

119903(119905) in the interval

[119905119894minus1

119905119894] The solution approximating 119909

119903(119905) in the entire

interval [1199050 119905119865] is given by

119909119903(119905) =

1199091

119903(119905) 119905 isin [119905

0 1199051]

1199092

119903(119905) 119905 isin [119905

1 1199052]

119909119865

119903(119905) 119905 isin [119905

119891minus1 119905119891]

(15)

3 Numerical Examples

In this section we consider two examples which demonstratethe efficiency and accuracy of the proposed method Inparticular we use the MSRM algorithm as an appropriatetool for solving nonlinear IVPs we apply the method to twocomplex nonlinear chaotic systems

Example 1 The hyperchaotic complex Lorenz system can bedescribed as

1= 1198861(1199112minus 1199111) + 119895119911

4

2= 11988621199111minus 1199112minus 11991111199113+ 1198951199114

3=

1

2

(11991111199112+ 11991111199112) minus 11988631199113

4=

1

2

(11991111199112+ 11991111199112) minus 11988641199114

(16)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 1199114

= 1199096

119895 = radicminus1 1199111and 119911

2are the conjugates of 119911

1and 119911

2 When

the parameters are chosen as 1198861= 15 119886

2= 36 119886

3= 45 and

1198864= 12 the system (16) is hyperchaotic [21]Replacing the complex variables in system (16) with real

and imaginary number variables one can get an equivalentsystem as follows

1= 1198861(1199093minus 1199091)

2= 1198861(1199094minus 1199092) + 1199096

3= 11988621199091minus 1199093minus 11990911199095

4= 11988621199092minus 1199094minus 11990921199095+ 1199096

5= 11990911199093+ 11990921199094minus 11988631199095

6= 11990911199093+ 11990921199094minus 11988641199096

(17)

For (17) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus1198861 120572

13= 1198861 120572

22= minus1198861

12057224

= 1198861 120572

26= 1

12057231

= 1198862 120572

33= minus1 120572

42= 1198862

12057244

= minus1 12057246

= 1

12057255

= minus1198863 120572

66= minus1198864 119891

3= minus11990911199095

1198914= minus11990921199095 119891

5= 1198916= 11990911199093+ 11990921199094

(18)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 6

Through numerical experimentation it was determinedthat 119873 = 6 collocation points and 5 iterations of the MSRMscheme at each interval were sufficient to give accurate resultsin each [119905

119894minus1 119905119894] interval Tables 1 and 2 show a comparison

of the solutions of the hyperchaotic complex Lorenz system

4 The Scientific World Journal

Table 2 Numerical comparison between MSRM and ode45 for the hyperchaotic complex Lorenz system

119905

1199094(119905) 119909

5(119905) 119909

6(119905)

MSRM ode45 MSRM ode45 MSRM ode45

2 2396851 2396851 4432071 4432071 2654682 26546824 1130830 1130830 1468007 1468007 325221 3252216 465208 465208 3934559 3934559 1299055 12990558 minus499685 minus499685 3379560 3379560 802232 80223210 198179 198179 5059739 5059740 2448234 2448234

MSRMode45

0 2 4 6 8 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

t

x1(t)

MSRMode45

0 2 4 6 8 10t

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x2(t)

Figure 1 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

computed by the MSRM and ode45 In Figures 1 2 and 3the MSRM graphical results are also compared with ode45and good agreement is observed The MRSM phase portraitsin Figures 4 and 5 were also found to be exactly the same asthose computed using ode45 This shows that the proposedMSRM is a valid tool for solving the hyperchaotic complexLorenz system

Example 2 State equations of a permanent magnet syn-chronous motor system in a field-oriented rotor can bedescribed as follows [22 23]

119889119894119889

119889119905

=

minus1198771119894119889+ 120596119871119902119894119902+ 119906119889

119871119889

119889119894119902

119889119905

=

1198771119894119902+ 120596119871119889119894119902+ 119906119902minus 120596Ψ119903

119871119902

119889120596

119889119905

=

119899119902Ψ119903119894119889+ 119899119901(119871119889minus 119871119902) 119894119889119894119902minus 119879119871minus 120573120596

119869

(19)

where 119894119889 119894119902 and 120596 are the state variables which represent

currents and motor angular frequency respectively 119906119889and

119906119902are the direct-axis stator and quadrature-axis stator voltage

components respectively 119869 is the polar moment of inertia119879119871is the external load torque 120573 is the viscous damping

coefficient 1198771is the stator winding resistance 119871

119889and 119871

119902

are the direct-axis stator inductors and quadrature-axis statorinductors respectivelyΨ

119903is the permanent magnet flux and

119899119901is the number of pole-pairs the parameters 119871

119889 119871119902 119869 119879119871

1198771 Ψ119903 120573 are all positive

When the air gap is even and the motor has no load orpower outage the dimensionless equations of a permanentmagnet synchronous motor system can be depicted as

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3= 11991111199112minus 1199113

(20)

where 119886 119887 are both positive parameters If the current in thesystem (19) is plural and the variables 119911

1 1199112in the system (20)

are complex numbers by changing cross coupled terms 1199111

and 1199112to conjugate form Wang and Zhang got a complex

The Scientific World Journal 5

MSRMode45

MSRMode45

0 2 4 6 8 10minus30

minus20

0

10

20

30

t

minus10

0 2 4 6 8 10minus30

minus20

minus10

0

10

20

30

40

t

x3(t)

x4(t)

Figure 2 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

MSRMode45

x5(t)

0 2 4 6 8 100

10

20

30

40

50

60

t

MSRMode45

x6(t)

0 2 4 6 8 100

5

10

15

20

25

30

35

40

t

Figure 3 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

permanent magnet synchronous motor system as follows[24]

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3=

1

2

(11991111199112+ 11991111199112) minus 1199113

(21)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 119895 = radicminus1

1199111and 119911

2are the conjugates of 119911

1and 119911

2 Replacing the

complex variables in system (21) with real and imaginary

number variables Wang and Zhang got an equivalent systemas follows (see [24])

1= 119886 (119909

3minus 1199091)

2= 119886 (119909

4minus 1199092)

3= 1198871199091minus 1199093minus 11990911199095

4= 1198871199092minus 1199094minus 11990921199095

5= 11990911199093+ 11990921199094minus 1199095

(22)

where 119886 119887 are positive parameters determining the chaoticbehaviors and bifurcations of system (22) When the param-eters satisfy 1 le 119886 le 11 10 le 119887 le 20 there is one positive

6 The Scientific World Journal

MSRMode45

x2(t)

x1(t)

minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

MSRMode45

x4(t)

x3(t)minus30 minus20 minus10 0 10 20 30

minus30

minus20

minus10

0

10

20

30

40

Figure 4 Phase portraits of the hyperchaotic complex Lorenz system

x6(t)

x5(t)0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

x3(t)

x2 (t) x1

(t)

MSRMode45

MSRMode45

minus20

020

40

minus40

minus20

020

40minus30

minus20

minus10

0

10

20

30

Figure 5 Phase portraits of the hyperchaotic complex Lorenz system

Lyapunov exponent two Lyapunov exponents of zero andtwo negative Lyapunov exponents for system (22) whichmeans system (22) is chaotic [24] The values of parametersand initial values are 119886 = 11 119887 = 20 and 119909

1(0) = 1 119909

2(0) = 2

1199093(0) = 3 119909

4(0) = 4 119909

5(0) = 5

For (21) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus119886 12057213

= 119886 12057222

= minus119886

12057224

= 119886 12057231

= 119887 12057233

= minus1

12057242

= 119887 12057244

= minus1 12057255

= minus1

1198913= minus11990911199095 119891

4= minus11990921199095 119891

5= 11990911199093+ 11990921199094

(23)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 5

The results obtained were compared to those from theMATLAB inbuilt solver ode45 The ode45 solver integratesa system of ordinary differential equations using explicit4th and 5th Runge-Kutta formula Tables 3 and 4 show

The Scientific World Journal 7

Table 3 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

3 minus385711 minus385711 minus566683 minus566683 minus520445 minus52044510 minus033729 minus033729 minus049554 minus049554 minus049104 minus04910417 012630 012631 018555 018557 015550 01555124 005091 005105 007480 007501 019500 01951831 minus255034 minus254878 minus374694 minus374465 minus079819 minus07932638 minus393154 minus373551 minus577619 minus548818 minus533693 minus520595

Table 4 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199094(119905) 119909

5(119905)

MSRM ode45 MSRM ode45

3 minus764635 minus764635 1505932 150593210 minus072144 minus072143 1073663 107366317 022846 022848 1425582 142558324 028649 028675 1933844 193392131 minus117270 minus116545 2534856 253573938 minus784098 minus764855 1498250 1403140

x1(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

t

x2(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

Figure 6 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

a comparison of the solutions of the complex permanentmagnet synchronous motor computed by the MSRM andode45 In Figures 6 7 and 8 the MSRM graphical results arealso compared with ode45 and good agreement is observedThe MRSM phase portraits in Figures 9 and 10 were alsofound to be exactly the same as those computed usingode45 This shows that the proposed MSRM is a valid toolfor solving the complex permanent magnet synchronousmotor

4 Conclusion

In this paper we have applied a spectral method called themultistage spectral relaxation method (MSRM) for the solu-tions of hyperchaotic complex systemsThe proposedMSRMwas developed using the Gauss-Seidel idea of decouplingsystems of equations and using Chebyshev pseudospectralmethods to solve the resulting decoupled system on asequence of multiple intervals The proposed MSRM was

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

In Section 3 we present the numerical implementation of theMSRM on two examples of hyperchaotic complex systemsFinally the conclusion is given in Section 4

2 Multistage Spectral Relaxation Method

In this section we give a brief description of the numericalmethod of solution used to solve the nonlinear hyperchaoticcomplex We employ the multistage spectral relaxationmethod (MSRM) proposed in [19] The MSRM algorithm isbased on a Gauss-Seidel type of relaxation that decouplesand linearises the system and the use of spectral collocationmethod to solve the linearised equations in a sequentialmanner For compactness we express the system of 119898

nonlinear first order differential equations in the form

119903(119905) =

119898

sum

119896=1

120572119903119896119909119896(119905)

+ 119891119903[1199091(119905) 1199092(119905) 119909

119903minus1(119905)

119909119903+1

(119905) 119909119898(119905)]

(1)

subject to the initial conditions

119909119903(0) = 119909

lowast

119903 119903 = 1 2 119898 (2)

where 119909119903are the unknown variables and 119909

lowast

119903are the corre-

sponding initial conditions 120572119903119896

are known constant inputparameters and 119891

119903is the nonlinear component of the 119903th

equation and the dot denotes differentiation with respect totime 119905

The scheme computes the solution of (1) in a sequenceof equal subintervals that makes the entire interval Wedefine the interval of integration as Ω = [0 119879] and divideit into a sequence of nonoverlapping subintervals Ω

119894=

[119905119894minus1

119905119894] (119894 = 1 2 3 119891) where 119905

0= 0 and 119905

119891= 119879

We denote the solution of (1) in the first subinterval [1199050 1199051]

as 1199091

119903(119905) and the solutions in the subsequent subintervals

[119905119894minus1

119905119894] (119894 = 2 3 119891) as 119909119894

119903(119905) For obtaining the solution

in the first interval [1199050 1199051] (2) is used as the initial condition

By using the continuity condition between neighbouringsubintervals the obtained solution in the interval [119905

0 1199051] is

used to obtain the initial condition for the next subinterval[1199051 1199052] This is applied over the 119891 successive subintervals

that is the obtained solution for each subinterval [119905119894minus1

119905119894] is

used to obtain the initial condition for the next subinterval[119905119894 119905119894+1

] (119894 = 1 2 119891minus1)Thus in each interval [119905119894minus1

119905119894]we

must solve

119894

119903= 120572119903119903119909119894

119903+ (1 minus 120575

119903119904)

119898

sum

119896=1

120572119903119896119909119894

119896

+ 119891119903[119909119894

1 119909

119894

119903minus1 119909119894

119903+1 119909

119894

119899]

(3)

subject to

119909119894

119903(119905119894minus1

) = 119909119894minus1

119903(119905119894minus1

) (4)

where 120575119903119904is the Kronecker delta As mentioned earlier the

main idea behind the MSRM scheme is decoupling the

system of nonlinear IVPs using the Gauss-Seidel idea ofdecoupling systems of algebraic equations The proposedMSRM iteration scheme for the solution in the interval Ω

119894=

[119905119894minus1

119905119894] is given as

119894

1119904+1minus 12057211

119909119894

1119904+1= 12057212

119909119894

2119904+ 12057213

119909119894

3119904

+ sdot sdot sdot + 1205721119899

119909119894

119899119904+ 1198911[119909119894

1119904 119909

119894

119899119904]

119894

2119904+1minus 12057222

119909119894

2119904+1= 12057221

119909119894

1119904+1+ 12057223

119909119894

3119904

+ sdot sdot sdot + 1205722119899

119909119894

119899119904

+ 1198912[119909119894

1119904+1 119909119894

2119904 119909

119894

119899119904]

119894

119898119904+1minus 120572119898119898

119909119894

119898119904+1= 1205721198981

119909119894

1119904+1+ sdot sdot sdot + 120572

119898119898minus1119909119894

119898minus1119904+1

+ 119891119898[119909119894

1119904+1 119909

119894

119898minus1119904+1 119909119894

119898119904]

(5)

subject to the initial conditions

119909119894

119903119904+1(119905119894minus1

) = 119909119894minus1

119903(119905119894minus1

) 119903 = 1 2 119898 (6)

where 119909119903119904

is the estimate of the solution after 119904 iterations Asuitable initial guess to start the iteration scheme (5) is onethat satisfies the initial condition (6) A convenient choiceof initial guess that was found to work in the numericalexperiments considered in this work is

119909119894

1199030(119905) =

119909lowast

119903if 119894 = 1

119909119894minus1

119903(119905119894minus1

) if 2 le 119894 le 119891

(7)

The Chebyshev spectral method is used to solve (5) oneach interval [119905

119894minus1 119905119894] First the region [119905

119894minus1 119905119894] is transformed

to the interval [minus1 1] onwhich the spectralmethod is definedby using the linear transformation

119905 =

(119905119894minus 119905119894minus1

) 120591

2

+

(119905119894+ 119905119894minus1

)

2

(8)

in each interval [119905119894minus1

119905119894] for 119894 = 1 119891 We then discretize

the interval [119905119894minus1

119905119894] using the Chebyshev-Gauss-Lobatto

collocation points [20]

120591119894

119895= cos(

120587119895

119873

) 119895 = 1 2 119873 (9)

which are the extrema of the 119873th order Chebyshev polyno-mial

119879119873(120591) = cos (119873cosminus1120591) (10)

The Chebyshev spectral collocation method is based onthe idea of introducing a differentiation matrix 119863 which isused to approximate the derivatives of the unknown variables119909119894

119903119904+1(119905) at the collocation points as thematrix vector product

119889119909119894

119903119904+1

119889119905

1003816100381610038161003816100381610038161003816100381610038161003816119905=119905119895

=

119873

sum

119896=0

D119895119896119909119894

119903119904+1= DX119894

119903119904+1 119895 = 1 2 119873

(11)

The Scientific World Journal 3

Table 1 Numerical comparison between MSRM and ode45 for the hyperchaotic complex Lorenz system

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

2 minus291138 minus291138 2173155 2173155 minus324491 minus3244914 minus363001 minus363001 652144 652144 minus630884 minus6308846 280571 280571 minus277638 minus277638 minus237099 minus2370998 001134 001134 209585 209585 minus014880 minus01488010 minus080219 minus080219 1648559 1648560 minus006690 minus006690

where D = 2119863(119905119894minus 119905119894minus1

) and X119894119903119904+1

= [119909119894

119903119904+1(120591119894

0) 119909119894

119903119904+1(120591119894

1)

119909119894

119903119904+1(120591119894

119873)] are the vector functions at the collocation

points 120591119894119895

Applying the Chebyshev spectral collocation method in(5) gives

A119903X119894119903119904+1

= B119894119903 X119894

119903119904+1(120591119894minus1

119873) = X119894minus1

119903(120591119894minus1

119873)

119903 = 1 2 119898

(12)

withA119903= D minus 120572

119903119903I

B1198941= 12057212X1198942119904

+ sdot sdot sdot + 1205721119899X119894119899119904

+ 1198911[X1198941119904 X119894

119898119904]

B1198942= 12057221X1198941119904+1

+ 12057223X1198943119904

+ sdot sdot sdot + 1205722119898

X119894119898119904

+ 1198912[X1198941119904+1

X1198942119904 X119894

119898119904]

B119894119898

= 1205721198981

X1198941119904+1

+ 1205721198982

X1198942119904+1

+ sdot sdot sdot + 120572119898119898minus1

X119894119898minus1119904+1

+ 119891119898[X1198941119904+1

X119894119898minus1119904+1

X119894119898119904

]

(13)

where I is an identity matrix of order 119873 + 1 Thus startingfrom the initial approximation (7) the recurrence formula

X119894119903119904+1

= Aminus1119903B119894119903 119903 = 1 2 119898 (14)

can be used to obtain the solution 119909119894

119903(119905) in the interval

[119905119894minus1

119905119894] The solution approximating 119909

119903(119905) in the entire

interval [1199050 119905119865] is given by

119909119903(119905) =

1199091

119903(119905) 119905 isin [119905

0 1199051]

1199092

119903(119905) 119905 isin [119905

1 1199052]

119909119865

119903(119905) 119905 isin [119905

119891minus1 119905119891]

(15)

3 Numerical Examples

In this section we consider two examples which demonstratethe efficiency and accuracy of the proposed method Inparticular we use the MSRM algorithm as an appropriatetool for solving nonlinear IVPs we apply the method to twocomplex nonlinear chaotic systems

Example 1 The hyperchaotic complex Lorenz system can bedescribed as

1= 1198861(1199112minus 1199111) + 119895119911

4

2= 11988621199111minus 1199112minus 11991111199113+ 1198951199114

3=

1

2

(11991111199112+ 11991111199112) minus 11988631199113

4=

1

2

(11991111199112+ 11991111199112) minus 11988641199114

(16)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 1199114

= 1199096

119895 = radicminus1 1199111and 119911

2are the conjugates of 119911

1and 119911

2 When

the parameters are chosen as 1198861= 15 119886

2= 36 119886

3= 45 and

1198864= 12 the system (16) is hyperchaotic [21]Replacing the complex variables in system (16) with real

and imaginary number variables one can get an equivalentsystem as follows

1= 1198861(1199093minus 1199091)

2= 1198861(1199094minus 1199092) + 1199096

3= 11988621199091minus 1199093minus 11990911199095

4= 11988621199092minus 1199094minus 11990921199095+ 1199096

5= 11990911199093+ 11990921199094minus 11988631199095

6= 11990911199093+ 11990921199094minus 11988641199096

(17)

For (17) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus1198861 120572

13= 1198861 120572

22= minus1198861

12057224

= 1198861 120572

26= 1

12057231

= 1198862 120572

33= minus1 120572

42= 1198862

12057244

= minus1 12057246

= 1

12057255

= minus1198863 120572

66= minus1198864 119891

3= minus11990911199095

1198914= minus11990921199095 119891

5= 1198916= 11990911199093+ 11990921199094

(18)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 6

Through numerical experimentation it was determinedthat 119873 = 6 collocation points and 5 iterations of the MSRMscheme at each interval were sufficient to give accurate resultsin each [119905

119894minus1 119905119894] interval Tables 1 and 2 show a comparison

of the solutions of the hyperchaotic complex Lorenz system

4 The Scientific World Journal

Table 2 Numerical comparison between MSRM and ode45 for the hyperchaotic complex Lorenz system

119905

1199094(119905) 119909

5(119905) 119909

6(119905)

MSRM ode45 MSRM ode45 MSRM ode45

2 2396851 2396851 4432071 4432071 2654682 26546824 1130830 1130830 1468007 1468007 325221 3252216 465208 465208 3934559 3934559 1299055 12990558 minus499685 minus499685 3379560 3379560 802232 80223210 198179 198179 5059739 5059740 2448234 2448234

MSRMode45

0 2 4 6 8 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

t

x1(t)

MSRMode45

0 2 4 6 8 10t

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x2(t)

Figure 1 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

computed by the MSRM and ode45 In Figures 1 2 and 3the MSRM graphical results are also compared with ode45and good agreement is observed The MRSM phase portraitsin Figures 4 and 5 were also found to be exactly the same asthose computed using ode45 This shows that the proposedMSRM is a valid tool for solving the hyperchaotic complexLorenz system

Example 2 State equations of a permanent magnet syn-chronous motor system in a field-oriented rotor can bedescribed as follows [22 23]

119889119894119889

119889119905

=

minus1198771119894119889+ 120596119871119902119894119902+ 119906119889

119871119889

119889119894119902

119889119905

=

1198771119894119902+ 120596119871119889119894119902+ 119906119902minus 120596Ψ119903

119871119902

119889120596

119889119905

=

119899119902Ψ119903119894119889+ 119899119901(119871119889minus 119871119902) 119894119889119894119902minus 119879119871minus 120573120596

119869

(19)

where 119894119889 119894119902 and 120596 are the state variables which represent

currents and motor angular frequency respectively 119906119889and

119906119902are the direct-axis stator and quadrature-axis stator voltage

components respectively 119869 is the polar moment of inertia119879119871is the external load torque 120573 is the viscous damping

coefficient 1198771is the stator winding resistance 119871

119889and 119871

119902

are the direct-axis stator inductors and quadrature-axis statorinductors respectivelyΨ

119903is the permanent magnet flux and

119899119901is the number of pole-pairs the parameters 119871

119889 119871119902 119869 119879119871

1198771 Ψ119903 120573 are all positive

When the air gap is even and the motor has no load orpower outage the dimensionless equations of a permanentmagnet synchronous motor system can be depicted as

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3= 11991111199112minus 1199113

(20)

where 119886 119887 are both positive parameters If the current in thesystem (19) is plural and the variables 119911

1 1199112in the system (20)

are complex numbers by changing cross coupled terms 1199111

and 1199112to conjugate form Wang and Zhang got a complex

The Scientific World Journal 5

MSRMode45

MSRMode45

0 2 4 6 8 10minus30

minus20

0

10

20

30

t

minus10

0 2 4 6 8 10minus30

minus20

minus10

0

10

20

30

40

t

x3(t)

x4(t)

Figure 2 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

MSRMode45

x5(t)

0 2 4 6 8 100

10

20

30

40

50

60

t

MSRMode45

x6(t)

0 2 4 6 8 100

5

10

15

20

25

30

35

40

t

Figure 3 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

permanent magnet synchronous motor system as follows[24]

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3=

1

2

(11991111199112+ 11991111199112) minus 1199113

(21)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 119895 = radicminus1

1199111and 119911

2are the conjugates of 119911

1and 119911

2 Replacing the

complex variables in system (21) with real and imaginary

number variables Wang and Zhang got an equivalent systemas follows (see [24])

1= 119886 (119909

3minus 1199091)

2= 119886 (119909

4minus 1199092)

3= 1198871199091minus 1199093minus 11990911199095

4= 1198871199092minus 1199094minus 11990921199095

5= 11990911199093+ 11990921199094minus 1199095

(22)

where 119886 119887 are positive parameters determining the chaoticbehaviors and bifurcations of system (22) When the param-eters satisfy 1 le 119886 le 11 10 le 119887 le 20 there is one positive

6 The Scientific World Journal

MSRMode45

x2(t)

x1(t)

minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

MSRMode45

x4(t)

x3(t)minus30 minus20 minus10 0 10 20 30

minus30

minus20

minus10

0

10

20

30

40

Figure 4 Phase portraits of the hyperchaotic complex Lorenz system

x6(t)

x5(t)0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

x3(t)

x2 (t) x1

(t)

MSRMode45

MSRMode45

minus20

020

40

minus40

minus20

020

40minus30

minus20

minus10

0

10

20

30

Figure 5 Phase portraits of the hyperchaotic complex Lorenz system

Lyapunov exponent two Lyapunov exponents of zero andtwo negative Lyapunov exponents for system (22) whichmeans system (22) is chaotic [24] The values of parametersand initial values are 119886 = 11 119887 = 20 and 119909

1(0) = 1 119909

2(0) = 2

1199093(0) = 3 119909

4(0) = 4 119909

5(0) = 5

For (21) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus119886 12057213

= 119886 12057222

= minus119886

12057224

= 119886 12057231

= 119887 12057233

= minus1

12057242

= 119887 12057244

= minus1 12057255

= minus1

1198913= minus11990911199095 119891

4= minus11990921199095 119891

5= 11990911199093+ 11990921199094

(23)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 5

The results obtained were compared to those from theMATLAB inbuilt solver ode45 The ode45 solver integratesa system of ordinary differential equations using explicit4th and 5th Runge-Kutta formula Tables 3 and 4 show

The Scientific World Journal 7

Table 3 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

3 minus385711 minus385711 minus566683 minus566683 minus520445 minus52044510 minus033729 minus033729 minus049554 minus049554 minus049104 minus04910417 012630 012631 018555 018557 015550 01555124 005091 005105 007480 007501 019500 01951831 minus255034 minus254878 minus374694 minus374465 minus079819 minus07932638 minus393154 minus373551 minus577619 minus548818 minus533693 minus520595

Table 4 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199094(119905) 119909

5(119905)

MSRM ode45 MSRM ode45

3 minus764635 minus764635 1505932 150593210 minus072144 minus072143 1073663 107366317 022846 022848 1425582 142558324 028649 028675 1933844 193392131 minus117270 minus116545 2534856 253573938 minus784098 minus764855 1498250 1403140

x1(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

t

x2(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

Figure 6 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

a comparison of the solutions of the complex permanentmagnet synchronous motor computed by the MSRM andode45 In Figures 6 7 and 8 the MSRM graphical results arealso compared with ode45 and good agreement is observedThe MRSM phase portraits in Figures 9 and 10 were alsofound to be exactly the same as those computed usingode45 This shows that the proposed MSRM is a valid toolfor solving the complex permanent magnet synchronousmotor

4 Conclusion

In this paper we have applied a spectral method called themultistage spectral relaxation method (MSRM) for the solu-tions of hyperchaotic complex systemsThe proposedMSRMwas developed using the Gauss-Seidel idea of decouplingsystems of equations and using Chebyshev pseudospectralmethods to solve the resulting decoupled system on asequence of multiple intervals The proposed MSRM was

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

Table 1 Numerical comparison between MSRM and ode45 for the hyperchaotic complex Lorenz system

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

2 minus291138 minus291138 2173155 2173155 minus324491 minus3244914 minus363001 minus363001 652144 652144 minus630884 minus6308846 280571 280571 minus277638 minus277638 minus237099 minus2370998 001134 001134 209585 209585 minus014880 minus01488010 minus080219 minus080219 1648559 1648560 minus006690 minus006690

where D = 2119863(119905119894minus 119905119894minus1

) and X119894119903119904+1

= [119909119894

119903119904+1(120591119894

0) 119909119894

119903119904+1(120591119894

1)

119909119894

119903119904+1(120591119894

119873)] are the vector functions at the collocation

points 120591119894119895

Applying the Chebyshev spectral collocation method in(5) gives

A119903X119894119903119904+1

= B119894119903 X119894

119903119904+1(120591119894minus1

119873) = X119894minus1

119903(120591119894minus1

119873)

119903 = 1 2 119898

(12)

withA119903= D minus 120572

119903119903I

B1198941= 12057212X1198942119904

+ sdot sdot sdot + 1205721119899X119894119899119904

+ 1198911[X1198941119904 X119894

119898119904]

B1198942= 12057221X1198941119904+1

+ 12057223X1198943119904

+ sdot sdot sdot + 1205722119898

X119894119898119904

+ 1198912[X1198941119904+1

X1198942119904 X119894

119898119904]

B119894119898

= 1205721198981

X1198941119904+1

+ 1205721198982

X1198942119904+1

+ sdot sdot sdot + 120572119898119898minus1

X119894119898minus1119904+1

+ 119891119898[X1198941119904+1

X119894119898minus1119904+1

X119894119898119904

]

(13)

where I is an identity matrix of order 119873 + 1 Thus startingfrom the initial approximation (7) the recurrence formula

X119894119903119904+1

= Aminus1119903B119894119903 119903 = 1 2 119898 (14)

can be used to obtain the solution 119909119894

119903(119905) in the interval

[119905119894minus1

119905119894] The solution approximating 119909

119903(119905) in the entire

interval [1199050 119905119865] is given by

119909119903(119905) =

1199091

119903(119905) 119905 isin [119905

0 1199051]

1199092

119903(119905) 119905 isin [119905

1 1199052]

119909119865

119903(119905) 119905 isin [119905

119891minus1 119905119891]

(15)

3 Numerical Examples

In this section we consider two examples which demonstratethe efficiency and accuracy of the proposed method Inparticular we use the MSRM algorithm as an appropriatetool for solving nonlinear IVPs we apply the method to twocomplex nonlinear chaotic systems

Example 1 The hyperchaotic complex Lorenz system can bedescribed as

1= 1198861(1199112minus 1199111) + 119895119911

4

2= 11988621199111minus 1199112minus 11991111199113+ 1198951199114

3=

1

2

(11991111199112+ 11991111199112) minus 11988631199113

4=

1

2

(11991111199112+ 11991111199112) minus 11988641199114

(16)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 1199114

= 1199096

119895 = radicminus1 1199111and 119911

2are the conjugates of 119911

1and 119911

2 When

the parameters are chosen as 1198861= 15 119886

2= 36 119886

3= 45 and

1198864= 12 the system (16) is hyperchaotic [21]Replacing the complex variables in system (16) with real

and imaginary number variables one can get an equivalentsystem as follows

1= 1198861(1199093minus 1199091)

2= 1198861(1199094minus 1199092) + 1199096

3= 11988621199091minus 1199093minus 11990911199095

4= 11988621199092minus 1199094minus 11990921199095+ 1199096

5= 11990911199093+ 11990921199094minus 11988631199095

6= 11990911199093+ 11990921199094minus 11988641199096

(17)

For (17) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus1198861 120572

13= 1198861 120572

22= minus1198861

12057224

= 1198861 120572

26= 1

12057231

= 1198862 120572

33= minus1 120572

42= 1198862

12057244

= minus1 12057246

= 1

12057255

= minus1198863 120572

66= minus1198864 119891

3= minus11990911199095

1198914= minus11990921199095 119891

5= 1198916= 11990911199093+ 11990921199094

(18)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 6

Through numerical experimentation it was determinedthat 119873 = 6 collocation points and 5 iterations of the MSRMscheme at each interval were sufficient to give accurate resultsin each [119905

119894minus1 119905119894] interval Tables 1 and 2 show a comparison

of the solutions of the hyperchaotic complex Lorenz system

4 The Scientific World Journal

Table 2 Numerical comparison between MSRM and ode45 for the hyperchaotic complex Lorenz system

119905

1199094(119905) 119909

5(119905) 119909

6(119905)

MSRM ode45 MSRM ode45 MSRM ode45

2 2396851 2396851 4432071 4432071 2654682 26546824 1130830 1130830 1468007 1468007 325221 3252216 465208 465208 3934559 3934559 1299055 12990558 minus499685 minus499685 3379560 3379560 802232 80223210 198179 198179 5059739 5059740 2448234 2448234

MSRMode45

0 2 4 6 8 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

t

x1(t)

MSRMode45

0 2 4 6 8 10t

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x2(t)

Figure 1 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

computed by the MSRM and ode45 In Figures 1 2 and 3the MSRM graphical results are also compared with ode45and good agreement is observed The MRSM phase portraitsin Figures 4 and 5 were also found to be exactly the same asthose computed using ode45 This shows that the proposedMSRM is a valid tool for solving the hyperchaotic complexLorenz system

Example 2 State equations of a permanent magnet syn-chronous motor system in a field-oriented rotor can bedescribed as follows [22 23]

119889119894119889

119889119905

=

minus1198771119894119889+ 120596119871119902119894119902+ 119906119889

119871119889

119889119894119902

119889119905

=

1198771119894119902+ 120596119871119889119894119902+ 119906119902minus 120596Ψ119903

119871119902

119889120596

119889119905

=

119899119902Ψ119903119894119889+ 119899119901(119871119889minus 119871119902) 119894119889119894119902minus 119879119871minus 120573120596

119869

(19)

where 119894119889 119894119902 and 120596 are the state variables which represent

currents and motor angular frequency respectively 119906119889and

119906119902are the direct-axis stator and quadrature-axis stator voltage

components respectively 119869 is the polar moment of inertia119879119871is the external load torque 120573 is the viscous damping

coefficient 1198771is the stator winding resistance 119871

119889and 119871

119902

are the direct-axis stator inductors and quadrature-axis statorinductors respectivelyΨ

119903is the permanent magnet flux and

119899119901is the number of pole-pairs the parameters 119871

119889 119871119902 119869 119879119871

1198771 Ψ119903 120573 are all positive

When the air gap is even and the motor has no load orpower outage the dimensionless equations of a permanentmagnet synchronous motor system can be depicted as

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3= 11991111199112minus 1199113

(20)

where 119886 119887 are both positive parameters If the current in thesystem (19) is plural and the variables 119911

1 1199112in the system (20)

are complex numbers by changing cross coupled terms 1199111

and 1199112to conjugate form Wang and Zhang got a complex

The Scientific World Journal 5

MSRMode45

MSRMode45

0 2 4 6 8 10minus30

minus20

0

10

20

30

t

minus10

0 2 4 6 8 10minus30

minus20

minus10

0

10

20

30

40

t

x3(t)

x4(t)

Figure 2 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

MSRMode45

x5(t)

0 2 4 6 8 100

10

20

30

40

50

60

t

MSRMode45

x6(t)

0 2 4 6 8 100

5

10

15

20

25

30

35

40

t

Figure 3 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

permanent magnet synchronous motor system as follows[24]

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3=

1

2

(11991111199112+ 11991111199112) minus 1199113

(21)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 119895 = radicminus1

1199111and 119911

2are the conjugates of 119911

1and 119911

2 Replacing the

complex variables in system (21) with real and imaginary

number variables Wang and Zhang got an equivalent systemas follows (see [24])

1= 119886 (119909

3minus 1199091)

2= 119886 (119909

4minus 1199092)

3= 1198871199091minus 1199093minus 11990911199095

4= 1198871199092minus 1199094minus 11990921199095

5= 11990911199093+ 11990921199094minus 1199095

(22)

where 119886 119887 are positive parameters determining the chaoticbehaviors and bifurcations of system (22) When the param-eters satisfy 1 le 119886 le 11 10 le 119887 le 20 there is one positive

6 The Scientific World Journal

MSRMode45

x2(t)

x1(t)

minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

MSRMode45

x4(t)

x3(t)minus30 minus20 minus10 0 10 20 30

minus30

minus20

minus10

0

10

20

30

40

Figure 4 Phase portraits of the hyperchaotic complex Lorenz system

x6(t)

x5(t)0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

x3(t)

x2 (t) x1

(t)

MSRMode45

MSRMode45

minus20

020

40

minus40

minus20

020

40minus30

minus20

minus10

0

10

20

30

Figure 5 Phase portraits of the hyperchaotic complex Lorenz system

Lyapunov exponent two Lyapunov exponents of zero andtwo negative Lyapunov exponents for system (22) whichmeans system (22) is chaotic [24] The values of parametersand initial values are 119886 = 11 119887 = 20 and 119909

1(0) = 1 119909

2(0) = 2

1199093(0) = 3 119909

4(0) = 4 119909

5(0) = 5

For (21) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus119886 12057213

= 119886 12057222

= minus119886

12057224

= 119886 12057231

= 119887 12057233

= minus1

12057242

= 119887 12057244

= minus1 12057255

= minus1

1198913= minus11990911199095 119891

4= minus11990921199095 119891

5= 11990911199093+ 11990921199094

(23)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 5

The results obtained were compared to those from theMATLAB inbuilt solver ode45 The ode45 solver integratesa system of ordinary differential equations using explicit4th and 5th Runge-Kutta formula Tables 3 and 4 show

The Scientific World Journal 7

Table 3 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

3 minus385711 minus385711 minus566683 minus566683 minus520445 minus52044510 minus033729 minus033729 minus049554 minus049554 minus049104 minus04910417 012630 012631 018555 018557 015550 01555124 005091 005105 007480 007501 019500 01951831 minus255034 minus254878 minus374694 minus374465 minus079819 minus07932638 minus393154 minus373551 minus577619 minus548818 minus533693 minus520595

Table 4 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199094(119905) 119909

5(119905)

MSRM ode45 MSRM ode45

3 minus764635 minus764635 1505932 150593210 minus072144 minus072143 1073663 107366317 022846 022848 1425582 142558324 028649 028675 1933844 193392131 minus117270 minus116545 2534856 253573938 minus784098 minus764855 1498250 1403140

x1(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

t

x2(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

Figure 6 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

a comparison of the solutions of the complex permanentmagnet synchronous motor computed by the MSRM andode45 In Figures 6 7 and 8 the MSRM graphical results arealso compared with ode45 and good agreement is observedThe MRSM phase portraits in Figures 9 and 10 were alsofound to be exactly the same as those computed usingode45 This shows that the proposed MSRM is a valid toolfor solving the complex permanent magnet synchronousmotor

4 Conclusion

In this paper we have applied a spectral method called themultistage spectral relaxation method (MSRM) for the solu-tions of hyperchaotic complex systemsThe proposedMSRMwas developed using the Gauss-Seidel idea of decouplingsystems of equations and using Chebyshev pseudospectralmethods to solve the resulting decoupled system on asequence of multiple intervals The proposed MSRM was

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

Table 2 Numerical comparison between MSRM and ode45 for the hyperchaotic complex Lorenz system

119905

1199094(119905) 119909

5(119905) 119909

6(119905)

MSRM ode45 MSRM ode45 MSRM ode45

2 2396851 2396851 4432071 4432071 2654682 26546824 1130830 1130830 1468007 1468007 325221 3252216 465208 465208 3934559 3934559 1299055 12990558 minus499685 minus499685 3379560 3379560 802232 80223210 198179 198179 5059739 5059740 2448234 2448234

MSRMode45

0 2 4 6 8 10minus20

minus15

minus10

minus5

0

5

10

15

20

25

t

x1(t)

MSRMode45

0 2 4 6 8 10t

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x2(t)

Figure 1 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

computed by the MSRM and ode45 In Figures 1 2 and 3the MSRM graphical results are also compared with ode45and good agreement is observed The MRSM phase portraitsin Figures 4 and 5 were also found to be exactly the same asthose computed using ode45 This shows that the proposedMSRM is a valid tool for solving the hyperchaotic complexLorenz system

Example 2 State equations of a permanent magnet syn-chronous motor system in a field-oriented rotor can bedescribed as follows [22 23]

119889119894119889

119889119905

=

minus1198771119894119889+ 120596119871119902119894119902+ 119906119889

119871119889

119889119894119902

119889119905

=

1198771119894119902+ 120596119871119889119894119902+ 119906119902minus 120596Ψ119903

119871119902

119889120596

119889119905

=

119899119902Ψ119903119894119889+ 119899119901(119871119889minus 119871119902) 119894119889119894119902minus 119879119871minus 120573120596

119869

(19)

where 119894119889 119894119902 and 120596 are the state variables which represent

currents and motor angular frequency respectively 119906119889and

119906119902are the direct-axis stator and quadrature-axis stator voltage

components respectively 119869 is the polar moment of inertia119879119871is the external load torque 120573 is the viscous damping

coefficient 1198771is the stator winding resistance 119871

119889and 119871

119902

are the direct-axis stator inductors and quadrature-axis statorinductors respectivelyΨ

119903is the permanent magnet flux and

119899119901is the number of pole-pairs the parameters 119871

119889 119871119902 119869 119879119871

1198771 Ψ119903 120573 are all positive

When the air gap is even and the motor has no load orpower outage the dimensionless equations of a permanentmagnet synchronous motor system can be depicted as

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3= 11991111199112minus 1199113

(20)

where 119886 119887 are both positive parameters If the current in thesystem (19) is plural and the variables 119911

1 1199112in the system (20)

are complex numbers by changing cross coupled terms 1199111

and 1199112to conjugate form Wang and Zhang got a complex

The Scientific World Journal 5

MSRMode45

MSRMode45

0 2 4 6 8 10minus30

minus20

0

10

20

30

t

minus10

0 2 4 6 8 10minus30

minus20

minus10

0

10

20

30

40

t

x3(t)

x4(t)

Figure 2 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

MSRMode45

x5(t)

0 2 4 6 8 100

10

20

30

40

50

60

t

MSRMode45

x6(t)

0 2 4 6 8 100

5

10

15

20

25

30

35

40

t

Figure 3 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

permanent magnet synchronous motor system as follows[24]

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3=

1

2

(11991111199112+ 11991111199112) minus 1199113

(21)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 119895 = radicminus1

1199111and 119911

2are the conjugates of 119911

1and 119911

2 Replacing the

complex variables in system (21) with real and imaginary

number variables Wang and Zhang got an equivalent systemas follows (see [24])

1= 119886 (119909

3minus 1199091)

2= 119886 (119909

4minus 1199092)

3= 1198871199091minus 1199093minus 11990911199095

4= 1198871199092minus 1199094minus 11990921199095

5= 11990911199093+ 11990921199094minus 1199095

(22)

where 119886 119887 are positive parameters determining the chaoticbehaviors and bifurcations of system (22) When the param-eters satisfy 1 le 119886 le 11 10 le 119887 le 20 there is one positive

6 The Scientific World Journal

MSRMode45

x2(t)

x1(t)

minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

MSRMode45

x4(t)

x3(t)minus30 minus20 minus10 0 10 20 30

minus30

minus20

minus10

0

10

20

30

40

Figure 4 Phase portraits of the hyperchaotic complex Lorenz system

x6(t)

x5(t)0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

x3(t)

x2 (t) x1

(t)

MSRMode45

MSRMode45

minus20

020

40

minus40

minus20

020

40minus30

minus20

minus10

0

10

20

30

Figure 5 Phase portraits of the hyperchaotic complex Lorenz system

Lyapunov exponent two Lyapunov exponents of zero andtwo negative Lyapunov exponents for system (22) whichmeans system (22) is chaotic [24] The values of parametersand initial values are 119886 = 11 119887 = 20 and 119909

1(0) = 1 119909

2(0) = 2

1199093(0) = 3 119909

4(0) = 4 119909

5(0) = 5

For (21) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus119886 12057213

= 119886 12057222

= minus119886

12057224

= 119886 12057231

= 119887 12057233

= minus1

12057242

= 119887 12057244

= minus1 12057255

= minus1

1198913= minus11990911199095 119891

4= minus11990921199095 119891

5= 11990911199093+ 11990921199094

(23)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 5

The results obtained were compared to those from theMATLAB inbuilt solver ode45 The ode45 solver integratesa system of ordinary differential equations using explicit4th and 5th Runge-Kutta formula Tables 3 and 4 show

The Scientific World Journal 7

Table 3 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

3 minus385711 minus385711 minus566683 minus566683 minus520445 minus52044510 minus033729 minus033729 minus049554 minus049554 minus049104 minus04910417 012630 012631 018555 018557 015550 01555124 005091 005105 007480 007501 019500 01951831 minus255034 minus254878 minus374694 minus374465 minus079819 minus07932638 minus393154 minus373551 minus577619 minus548818 minus533693 minus520595

Table 4 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199094(119905) 119909

5(119905)

MSRM ode45 MSRM ode45

3 minus764635 minus764635 1505932 150593210 minus072144 minus072143 1073663 107366317 022846 022848 1425582 142558324 028649 028675 1933844 193392131 minus117270 minus116545 2534856 253573938 minus784098 minus764855 1498250 1403140

x1(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

t

x2(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

Figure 6 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

a comparison of the solutions of the complex permanentmagnet synchronous motor computed by the MSRM andode45 In Figures 6 7 and 8 the MSRM graphical results arealso compared with ode45 and good agreement is observedThe MRSM phase portraits in Figures 9 and 10 were alsofound to be exactly the same as those computed usingode45 This shows that the proposed MSRM is a valid toolfor solving the complex permanent magnet synchronousmotor

4 Conclusion

In this paper we have applied a spectral method called themultistage spectral relaxation method (MSRM) for the solu-tions of hyperchaotic complex systemsThe proposedMSRMwas developed using the Gauss-Seidel idea of decouplingsystems of equations and using Chebyshev pseudospectralmethods to solve the resulting decoupled system on asequence of multiple intervals The proposed MSRM was

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

MSRMode45

MSRMode45

0 2 4 6 8 10minus30

minus20

0

10

20

30

t

minus10

0 2 4 6 8 10minus30

minus20

minus10

0

10

20

30

40

t

x3(t)

x4(t)

Figure 2 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

MSRMode45

x5(t)

0 2 4 6 8 100

10

20

30

40

50

60

t

MSRMode45

x6(t)

0 2 4 6 8 100

5

10

15

20

25

30

35

40

t

Figure 3 Comparison between the MSRM and ode45 results for the hyperchaotic complex Lorenz system

permanent magnet synchronous motor system as follows[24]

1= 119886 (119911

2minus 1199111)

2= 1198871199111minus 1199112minus 11991111199113

3=

1

2

(11991111199112+ 11991111199112) minus 1199113

(21)

where 1199111

= 1199091+ 1198951199092 1199112

= 1199093+ 1198951199094 1199113

= 1199095 119895 = radicminus1

1199111and 119911

2are the conjugates of 119911

1and 119911

2 Replacing the

complex variables in system (21) with real and imaginary

number variables Wang and Zhang got an equivalent systemas follows (see [24])

1= 119886 (119909

3minus 1199091)

2= 119886 (119909

4minus 1199092)

3= 1198871199091minus 1199093minus 11990911199095

4= 1198871199092minus 1199094minus 11990921199095

5= 11990911199093+ 11990921199094minus 1199095

(22)

where 119886 119887 are positive parameters determining the chaoticbehaviors and bifurcations of system (22) When the param-eters satisfy 1 le 119886 le 11 10 le 119887 le 20 there is one positive

6 The Scientific World Journal

MSRMode45

x2(t)

x1(t)

minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

MSRMode45

x4(t)

x3(t)minus30 minus20 minus10 0 10 20 30

minus30

minus20

minus10

0

10

20

30

40

Figure 4 Phase portraits of the hyperchaotic complex Lorenz system

x6(t)

x5(t)0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

x3(t)

x2 (t) x1

(t)

MSRMode45

MSRMode45

minus20

020

40

minus40

minus20

020

40minus30

minus20

minus10

0

10

20

30

Figure 5 Phase portraits of the hyperchaotic complex Lorenz system

Lyapunov exponent two Lyapunov exponents of zero andtwo negative Lyapunov exponents for system (22) whichmeans system (22) is chaotic [24] The values of parametersand initial values are 119886 = 11 119887 = 20 and 119909

1(0) = 1 119909

2(0) = 2

1199093(0) = 3 119909

4(0) = 4 119909

5(0) = 5

For (21) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus119886 12057213

= 119886 12057222

= minus119886

12057224

= 119886 12057231

= 119887 12057233

= minus1

12057242

= 119887 12057244

= minus1 12057255

= minus1

1198913= minus11990911199095 119891

4= minus11990921199095 119891

5= 11990911199093+ 11990921199094

(23)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 5

The results obtained were compared to those from theMATLAB inbuilt solver ode45 The ode45 solver integratesa system of ordinary differential equations using explicit4th and 5th Runge-Kutta formula Tables 3 and 4 show

The Scientific World Journal 7

Table 3 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

3 minus385711 minus385711 minus566683 minus566683 minus520445 minus52044510 minus033729 minus033729 minus049554 minus049554 minus049104 minus04910417 012630 012631 018555 018557 015550 01555124 005091 005105 007480 007501 019500 01951831 minus255034 minus254878 minus374694 minus374465 minus079819 minus07932638 minus393154 minus373551 minus577619 minus548818 minus533693 minus520595

Table 4 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199094(119905) 119909

5(119905)

MSRM ode45 MSRM ode45

3 minus764635 minus764635 1505932 150593210 minus072144 minus072143 1073663 107366317 022846 022848 1425582 142558324 028649 028675 1933844 193392131 minus117270 minus116545 2534856 253573938 minus784098 minus764855 1498250 1403140

x1(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

t

x2(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

Figure 6 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

a comparison of the solutions of the complex permanentmagnet synchronous motor computed by the MSRM andode45 In Figures 6 7 and 8 the MSRM graphical results arealso compared with ode45 and good agreement is observedThe MRSM phase portraits in Figures 9 and 10 were alsofound to be exactly the same as those computed usingode45 This shows that the proposed MSRM is a valid toolfor solving the complex permanent magnet synchronousmotor

4 Conclusion

In this paper we have applied a spectral method called themultistage spectral relaxation method (MSRM) for the solu-tions of hyperchaotic complex systemsThe proposedMSRMwas developed using the Gauss-Seidel idea of decouplingsystems of equations and using Chebyshev pseudospectralmethods to solve the resulting decoupled system on asequence of multiple intervals The proposed MSRM was

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

MSRMode45

x2(t)

x1(t)

minus20 minus15 minus10 minus5 0 5 10 15 20 25minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

MSRMode45

x4(t)

x3(t)minus30 minus20 minus10 0 10 20 30

minus30

minus20

minus10

0

10

20

30

40

Figure 4 Phase portraits of the hyperchaotic complex Lorenz system

x6(t)

x5(t)0 10 20 30 40 50 60

0

5

10

15

20

25

30

35

40

x3(t)

x2 (t) x1

(t)

MSRMode45

MSRMode45

minus20

020

40

minus40

minus20

020

40minus30

minus20

minus10

0

10

20

30

Figure 5 Phase portraits of the hyperchaotic complex Lorenz system

Lyapunov exponent two Lyapunov exponents of zero andtwo negative Lyapunov exponents for system (22) whichmeans system (22) is chaotic [24] The values of parametersand initial values are 119886 = 11 119887 = 20 and 119909

1(0) = 1 119909

2(0) = 2

1199093(0) = 3 119909

4(0) = 4 119909

5(0) = 5

For (21) the parameters 120572119903119896

and 119891119903are defined as

12057211

= minus119886 12057213

= 119886 12057222

= minus119886

12057224

= 119886 12057231

= 119887 12057233

= minus1

12057242

= 119887 12057244

= minus1 12057255

= minus1

1198913= minus11990911199095 119891

4= minus11990921199095 119891

5= 11990911199093+ 11990921199094

(23)

with all other 120572119903119896

and 119891119903= 0 for 119903 119896 = 1 2 5

The results obtained were compared to those from theMATLAB inbuilt solver ode45 The ode45 solver integratesa system of ordinary differential equations using explicit4th and 5th Runge-Kutta formula Tables 3 and 4 show

The Scientific World Journal 7

Table 3 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

3 minus385711 minus385711 minus566683 minus566683 minus520445 minus52044510 minus033729 minus033729 minus049554 minus049554 minus049104 minus04910417 012630 012631 018555 018557 015550 01555124 005091 005105 007480 007501 019500 01951831 minus255034 minus254878 minus374694 minus374465 minus079819 minus07932638 minus393154 minus373551 minus577619 minus548818 minus533693 minus520595

Table 4 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199094(119905) 119909

5(119905)

MSRM ode45 MSRM ode45

3 minus764635 minus764635 1505932 150593210 minus072144 minus072143 1073663 107366317 022846 022848 1425582 142558324 028649 028675 1933844 193392131 minus117270 minus116545 2534856 253573938 minus784098 minus764855 1498250 1403140

x1(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

t

x2(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

Figure 6 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

a comparison of the solutions of the complex permanentmagnet synchronous motor computed by the MSRM andode45 In Figures 6 7 and 8 the MSRM graphical results arealso compared with ode45 and good agreement is observedThe MRSM phase portraits in Figures 9 and 10 were alsofound to be exactly the same as those computed usingode45 This shows that the proposed MSRM is a valid toolfor solving the complex permanent magnet synchronousmotor

4 Conclusion

In this paper we have applied a spectral method called themultistage spectral relaxation method (MSRM) for the solu-tions of hyperchaotic complex systemsThe proposedMSRMwas developed using the Gauss-Seidel idea of decouplingsystems of equations and using Chebyshev pseudospectralmethods to solve the resulting decoupled system on asequence of multiple intervals The proposed MSRM was

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

Table 3 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199091(119905) 119909

2(119905) 119909

3(119905)

MSRM ode45 MSRM ode45 MSRM ode45

3 minus385711 minus385711 minus566683 minus566683 minus520445 minus52044510 minus033729 minus033729 minus049554 minus049554 minus049104 minus04910417 012630 012631 018555 018557 015550 01555124 005091 005105 007480 007501 019500 01951831 minus255034 minus254878 minus374694 minus374465 minus079819 minus07932638 minus393154 minus373551 minus577619 minus548818 minus533693 minus520595

Table 4 Numerical comparison between MSRM and ode45 for the complex permanent magnet synchronous motor

119905

1199094(119905) 119909

5(119905)

MSRM ode45 MSRM ode45

3 minus764635 minus764635 1505932 150593210 minus072144 minus072143 1073663 107366317 022846 022848 1425582 142558324 028649 028675 1933844 193392131 minus117270 minus116545 2534856 253573938 minus784098 minus764855 1498250 1403140

x1(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus6

minus4

minus2

0

2

4

6

8

t

x2(t)

MSRMode45

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

Figure 6 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

a comparison of the solutions of the complex permanentmagnet synchronous motor computed by the MSRM andode45 In Figures 6 7 and 8 the MSRM graphical results arealso compared with ode45 and good agreement is observedThe MRSM phase portraits in Figures 9 and 10 were alsofound to be exactly the same as those computed usingode45 This shows that the proposed MSRM is a valid toolfor solving the complex permanent magnet synchronousmotor

4 Conclusion

In this paper we have applied a spectral method called themultistage spectral relaxation method (MSRM) for the solu-tions of hyperchaotic complex systemsThe proposedMSRMwas developed using the Gauss-Seidel idea of decouplingsystems of equations and using Chebyshev pseudospectralmethods to solve the resulting decoupled system on asequence of multiple intervals The proposed MSRM was

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

0 5 10 15 20 25 30 35 40minus8

minus6

minus4

minus2

0

2

4

6

8

10

t

0 5 10 15 20 25 30 35 40minus10

minus5

0

5

10

15

t

MSRMode45

MSRMode45

x3(t)

x4(t)

Figure 7 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

0 5 10 15 20 25 30 35 405

10

15

20

25

30

35

t

x5(t)

Figure 8 Comparison between the MSRM and ode45 results for the complex permanent magnet synchronous motor

MSRMode45

x4(t)

x1(t)minus6 minus4 minus2 0 2 4 6 8

minus10

minus5

0

5

10

15

MSRMode45

x5(t)

x2(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

Figure 9 Phase portraits of the complex permanent magnet synchronous motor

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

MSRMode45

x5(t)

x4(t)minus10 minus5 0 5 10 155

10

15

20

25

30

35

MSRMode45

x3(t)

x2 (t)

x1(t)minus10 minus5 0 5 10minus20

0

20minus10

minus5

0

5

10

Figure 10 Phase portraits of the complex permanent magnet synchronous motor

used to solve the hyperchaotic complex Lorenz systemand complex permanent magnet synchronous motor Theaccuracy and validity of the proposed method was testedagainst Matlab Runge-Kutta based inbuilt solvers and againstpreviously published results

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465ndash1466 1999

[2] Z Wei R Wang and A Liu ldquoA new finding of the existence ofhidden hyperchaotic attractors with no equilibriardquoMathematicsand Computers in Simulation vol 100 pp 13ndash23 2014

[3] A C Fowler J D Gibbon andM JMcGuinness ldquoThe complexLorenz equationsrdquo Physica D Nonlinear Phenomena vol 4 no2 pp 139ndash163 198182

[4] A G Vladimirov V Y Toronov andV L Derbov ldquoProperties ofthe phase space and bifurcations in the complex LorenzmodelrdquoTechnical Physics vol 43ndash48 pp 877ndash884 1998

[5] A Rauh L Hannibal and N B Abraham ldquoGlobal stabilityproperties of the complex Lorenz modelrdquo Physica D NonlinearPhenomena vol 99 no 1 pp 45ndash58 1996

[6] R L Lang ldquoA stochastic complex model with random imag-inary noiserdquo Nonlinear Dynamics vol 62 no 3 pp 561ndash5652010

[7] G M Mahmoud M E Ahmed and E E Mahmoud ldquoAnalysisof hyperchaotic complex Lorenz systemsrdquo International Journalof Modern Physics C vol 19 no 10 pp 1477ndash1494 2008

[8] G M Mahmoud E E Mahmoud and M E Ahmed ldquoOn thehyperchaotic complex Lu systemrdquoNonlinear Dynamics vol 58no 4 pp 725ndash738 2009

[9] G M Mahmoud M A Al-Kashif and S A Aly ldquoBasicproperties and chaotic synchronization of complex Lorenzsystemrdquo International Journal of Modern Physics C vol 18 no2 pp 253ndash265 2007

[10] A K Alomari M S Noorani and R Nazar ldquoExplicit seriessolutions of some linear and nonlinear Schrodinger equationsvia the homotopy analysis methodrdquo Communications in Non-linear Science and Numerical Simulation vol 14 no 4 pp 1196ndash1207 2009

[11] S Effati H Saberi Nik and M Shirazian ldquoAn improvement tothe homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equationrdquo IMA Journal ofMathematical Controland Information vol 30 no 4 pp 487ndash506 2013

[12] M S Chowdhury and I Hashim ldquoApplication of multistagehomotopy-perturbation method for the solutions of the ChensystemrdquoNonlinear Analysis RealWorld Applications vol 10 no1 pp 381ndash391 2009

[13] S M Goh M S Noorani and I Hashim ldquoOn solving thechaotic Chen system a new time marching design for thevariational iteration method using Adomianrsquos polynomialrdquoNumerical Algorithms vol 54 no 2 pp 245ndash260 2010

[14] A Gokdogan M Merdan and A Yildirim ldquoThe modifiedalgorithm for the differential transform method to solution ofGenesio systemsrdquo Communications in Nonlinear Science andNumerical Simulation vol 17 no 1 pp 45ndash51 2012

[15] H Saberi Nik S Effati S S Motsa and M Shirazian ldquoSpectralhomotopy analysis method and its convergence for solvinga class of nonlinear optimal control problemsrdquo NumericalAlgorithms vol 65 no 1 pp 171ndash194 2014

[16] H Saberi Nik S Effati S S Motsa and S Shateyi ldquoAnew piecewise-spectral homotopy analysis method for solvingchaotic systems of initial value problemsrdquo Mathematical Prob-lems in Engineering vol 2013 Article ID 583193 13 pages 2013

[17] S Effati H Saberi Nik and A Jajarmi ldquoHyperchaos controlof the hyperchaotic Chen system by optimal control designrdquoNonlinear Dynamics vol 73 no 1-2 pp 499ndash508 2013

[18] S S Motsa P G Dlamini and M Khumalo ldquoSolving hyper-chaotic systems using the spectral relaxation methodrdquo AbstractandAppliedAnalysis vol 2012 Article ID 203461 18 pages 2012

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 The Scientific World Journal

[19] S S Motsa P Dlamini and M Khumalo ldquoA new multistagespectral relaxation method for solving chaotic initial valuesystemsrdquoNonlinearDynamics vol 72 no 1-2 pp 265ndash283 2013

[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000[21] C Luo and XWang ldquoHybridmodified function projective syn-

chronization of two different dimensional complex nonlinearsystems with parameters identificationrdquo Journal of the FranklinInstitute vol 350 no 9 pp 2646ndash2663 2013

[22] J Yu B Chen H Yu and J Gao ldquoAdaptive fuzzy trackingcontrol for the chaotic permanent magnet synchronous motordrive system via backsteppingrdquo Nonlinear Analysis Real WorldApplications vol 12 no 1 pp 671ndash681 2011

[23] F Zhang C Mu X Wang I Ahmed and Y Shu ldquoSolutionbounds of a new complex PMSM systemrdquoNonlinear Dynamicsvol 74 no 4 pp 1041ndash1051 2013

[24] X Wang and H Zhang ldquoBackstepping-based lag synchro-nization of a complex permanent magnet synchronous motorsystemrdquo Chinese Physics B vol 22 no 4 Article ID 0489022013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of