Numerical Chaotic Behavior of the Fractional Rikitake System€¦ · Numerical Chaotic Behavior of the Fractional Rikitake System ... dynamics of the Rikitake system at inﬁnity,

ISSN 1 746-7233, England, UKWorld Journal of Modelling and Simulation

Vol. 9 (2013) No. 2, pp. 120-129

Numerical Chaotic Behavior of the Fractional Rikitake System

Mohammad Javidi1∗ , Nemat Nyamoradi2

Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran

(Received April 5 2012, Revised September 18 2012, Accepted April 14 2013)

Abstract. In this paper, a numerical solution for the system described by a generalized fractional Rikitakesystem is presented. The first step in the proposed procedure is represent the fractional order Rikitake systemas an equivalent system of ordinary differential equations. In the second step, we solved the system obtainedin the first step by using the well known fourth order Runge-Kutta method. We derived some chaotic behaviorof the fractional Rikitake system.

Keywords: fractional Rikitake system, ordinary differential equations, dynamic of system

1 Introduction

This paper concern the numerical solution of fractional Rikitake sysyem which have the following formdαxdtα = −µx+ zy,dβydtβ

= −µy + (z − a)x,dγzdtγ = 1− xy,

(1)

subject to the initial conditionsx(δ) = δ1, y(δ) = δ2, z(δ) = δ3,

where α, β, γ, δ1, δ2 and δ3 are real numbers.The Rikitake system [10, 12] been widely investigated in the last years. This system is a model which

attempts to explain the irregular polarity switching of the Earth’s magnetic field. But intervals among suchgeomagnetic polarity reversals are highly irregular. Thus while their average is about 7.105 years, there areintervals as long as 3.107 years without polarity change, but with large deviations of the poles from actualpositions. The mechanism of generating the geomagnetic field is explained by the dynamo theory.

Many authors worked on this systems. Tudoran and Grban[11] proposed some relevant dynamical prop-erties of the Rikitake dynamo system from the Poisson geometry point of view. Also the introduced someHamiltonian realizations of a particular case of the Rikitake system and study the system from the Poisson ge-ometry point of view. Machuca et al[6] introduced synchronization problem via nonlinear observer design. Anew exponential polynomial observer for a class of nonlinear oscillators is proposed, which is robust againstoutput noises. In [14], the authors presents a new 3-D autonomous chaotic system, which is topologicallynonequivalent to the original Lorenz and all Lorenz-like systems. Of particular interest is that the chaoticsystem can generate double-scroll chaotic attractors in a very wide parameter domain with only two stableequilibria. Also the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parame-ters is investigated. The Rikitake system[15] as nonlinear dynamical systems in geomagnetism can be studiedbased on the KCC-theory and the unified field theory. Especially, the behavior of the magnetic field of theRikitake system is represented in the electrical system projected from the electromechanical unified system.∗ Corresponding author. E-mail address: mo [email protected].

Published by World Academic Press, World Academic Union

World Journal of Modelling and Simulation, Vol. 9 (2013) No. 2, pp. 120-129 121

By using the Poincare compactification for polynomial vector fields in R3, Llibre and Messias[8] studied thedynamics of the Rikitake system at infinity, and show that there are orbits which escape to, or come from,infinity, instead of going towards the attractor. Moreover we study, for particular values of the parameters, theflow over two invariant planes, and describe the global flow of the system when it has two independent firstintegrals and thus is completely integrable. The global analysis performed, allows us to give a numerical de-scription of the creation of Rikitake attractor. In [5], the attractors synthesis algorithm for a class of dissipativedynamical systems with hyperbolic equilibria proposed to model numerically any attractor of the Rikitake sys-tem. The author of [9] analyze a particular case of the Rikitake two-dynamo system from the stability theorypoint of view. More exactly, a special type of dissipative system (the so-called metriplectic system) in such away that each Lyapunov stable equilibrium state of the unperturbed system generates a small one dimensionalattracting neighborhood for the dissipative system. present an analytical study of the stability of the equi-libria of the Rikitake system. Braga[4] proved that the two non-hyperbolic equilibria of the Rikitake systemare unstable for all positive values of the parameters. The authors of [1] proposed a master/slave scheme inconjunction with an adaptive control technique for synchronization and identification of the uncertain chaoticRikitake system, where only a partial knowledge of the state is available.

Vincent[13], applid the active control to synchronize two identical Rikitake attractors; which describesthe chaotic behavior exhibited by the reversal of dipole polarity. In this paper, we used an efficient numer-ical method based on a transformation of the original fraction Rikitake system into a system of OrdinaryDifferential Equations (ODEs).

2 Preliminaries

Definition 1. The Riemann-Liouville fractional integral operator of order α > 0, of function f ∈ L1(R+) isdefined as

Iαt0f(t) =

1Γ (α)

∫ t

t0

(t− s)α−1f(s)ds,

where Γ (·) is the Euler gamma function.

Definition 2. The Riemann-Liouville fractional derivative of order α > 0, n − 1 < α < n, n ∈ N is definedas

Dαt0f(t) =

1Γ (n− α)

( ddt

)n∫ t

t0

(t− s)n−α−1f(s)ds,

where the function f(t) have absolutely continuous derivatives up to order (n− 1).

The initial value problem related to Definition 2 is{Dαx(t) = f(t, x(t)),x(t)|t=0+ = x0,

(2)

where 0 < α < 1 and Dα = Dα0 .

In [7], the following results about the existence and uniqueness of solutions for Eq. (2) are further pre-sented.

Theorem 1. Assume that < : [0, T ∗]× [x0−δ, x0 +δ] with some T ∗ > 0 and some δ > 0, and let the function

f : < → R be continuous. Furthermore, define T := min{T ∗,

( δΓ (α+1)||f ||∞

) 1α

}, then there exists a function

x : [0, T ] → R solving the initial value problem Eq. (2). Notice that ||f ||∞ is the norm of function f .

Theorem 2. Assume that < : [0, T ∗]× [x0−δ, x0 +δ] with some T ∗ > 0 and some δ > 0, and let the functionf : < → R be bounded on < and fulfill a Lipschitz condition with respect to the second variable, i.e.

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122 M. Javidi & N. Nymoradi: Numerical Chaotic Behavior of the Fractional

|f(t, x)− f(t, y)| ≤ L|x− y|

with some constant number L > 0 independent of t, x, y. Then denoting T as Theorem 1, there exists at mostone function x : [0, T ] → R solving the initial value problem Eq. (2).

Furthermore, the above definition in one dimension can naturally be generalized to the case of multipledimensions. That is, let X(t) = (x1(t), x2(t), · · · , xn(t))T ∈ Rn and α = (α1, α2, · · · , αn)T ∈ Rn, 0 <αi < 1, i = 1, 2, · · · , n. The n-dimension FODE is described as follows:

DαX(t) =1

Γ (1− α)

∫ t

0

X ′(u)(t− u)α

du , F (t,X(t)), (3)

where

1Γ (1− α)

∫ t

0

X ′(u)(t− u)α

du = (Dα1x1(t), Dα2x2(t), · · · , Dαnxn(t))T

and

F (t,X(t)) =

f1(t, x1(t), x2(t), · · · , xn(t))f2(t, x1(t), x2(t), · · · , xn(t))

...fn(t, x1(t), x2(t), · · · , xn(t))

.

The results of Theorems 1, 2 can be easily generalizes to the initial value problem of the vector-valuefunctions Eq. (3).

3 Expansion formulas for the fractional derivatives

In order to solve Eq. 1, we shall use a numerical method introduced by Atanackovic and Stankovic [2]to solve the single linear FDE. Also the same authors [3] developed the method to solve the nonlinear FDE. Inthis paper we developed the same method to solve the fractional . Now we explain the method.

It is well known that for an analytic function f(t) the Caputo fractional derivative Dαf(t), 0 < α < 1defined as

Dαf(t) =1

Γ (1− α)

∫ t

0(t− τ)−αf (1)(τ)dτ, (4)

where f (1)(τ) denote the first derivative of f(τ). From Eq. 4, we obtain

Dαf(t) =1

Γ (2− α)

[f (1)(0)tα−1

+∫ t

0(t− τ)1−αf (2)(τ)dτ

]. (5)

By using the binomial formula, we have

(t− τ)1−α = t1−α

(1− τ

t

)1−α

= t1−α∞∑

p=0

(1− αp

)(−1)p

(τ

t

)p

(6)

= t1−α∞∑

p=0

Γ (p− 1 + α)Γ (α− 1)p!

(τ

t

)p

,

∣∣∣∣τt∣∣∣∣ < 1.

By substitution of Eq. (6) into Eq. (5), we have

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Dαf(t) =1

Γ (2− α)

[f (1)(0)tα−1

+1

tα−1

∫ t

0

∞∑p=0

Γ (p− 1 + α)Γ (α− 1)p!

(τ

t)pf (2)(τ)dτ

]. (7)

We can rewrite Eq. (7) as the following form

Dαf(t) =1

Γ (2− α)

[f (1)(t)tα−1

+1

tα−1

∞∑p=1

Γ (p− 1 + α)Γ (α− 1)p!tp

∫ t

0τpf (2)(τ)dτ

]. (8)

Using integration by part, we obtain∫ t

0τpf (2)(τ)dτ = tpf (1)(t)− p

∫ t

0τp−1f (1)(τ)dτ

= tpf (1)(t)− ptp−1f(t) + p(p− 1)∫ t

0τp−2f(τ)dτ, p ≥ 2. (9)

By substitution of Eq. (9) into Eq. (8), we obtain

Dαf(t) =1

Γ (2− α)

{f (1)(t)tα−1

[1 +

∞∑p=1

Γ (p− 1 + α)Γ (α− 1)p!

]

−[α− 1tα

f(t) +∞∑

p=2

Γ (p− 1 + α)Γ (α− 1)(p− 1)!

(f(t)tα

+Vp(f)(t)tp−1+α

)]}. (10)

where

Vp(f)(t) = −(p− 1)∫ t

0τp−2f(τ)dτ, p = 2, 3, · · · , (11)

with the following properties

ddtVp(f) = −(p− 1)tp−2f(t), p = 2, 3, · · · . (12)

We approximate Dαf(t) by using M terms in sums appearing in Eq. (10) as follows

Dαf(t) ' 1Γ (2− α)

{f (1)(t)tα−1

[1 +

M∑p=1

Γ (p− 1 + α)Γ (α− 1)p!

]

−[α− 1tα

f(t) +M∑

p=2

Γ (p− 1 + α)Γ (α− 1)(p− 1)!

(f(t)tα

+Vp(f)(t)tp−1+α

)]}. (13)

By setting T1(t) = x(t), TM+1(t) = y(t), T2M+1(t) = z(t) and Tp(t) = Vp(x)(t),TM+p(t) =Vp(y)(t), T2M+p(t) = Vp(z)(t), p = 2, 3, · · · , we can rewrite Rikitake system Eq. (1) as the following formordinary differential equations

T ′1(t)tα−1

Π(p, α,M)−[α− 1tα

T1(t) +M∑

p=2

Ξ(p, α)(T1(t)tα

+Tp(t)tp−1+α

)]= ψ(α)(−µT1(t) + T2M+1(t)TM+1(t)),

T ′M+1(t)tβ−1

Π(p, β,M)−[β − 1tβ

TM+1(t) +M∑

p=2

Ξ(p, β)(TM+1(t)

tβ+TM+p(t)tp−1+β

)]= ψ(β)(−µTM+1(t) + (T2M+1(t)− a)T1(t)),

T ′2M+1(t)tγ−1

Π(p, γ,M)−[γ − 1tγ

T2M+1(t) +M∑

p=2

Ξ(p, γ)(T2M+1(t)

tγ+TM+p(t)tp−1+γ

)]= ψ(γ)(1− TM+1(t)T1(t)), (14)



where

Tp(t) = −(p− 1)∫ t0 τ

p−2T1(t)(τ)dτ, p = 2, 3, · · · ,M,

TM+p(t) = −(p− 1)∫ t0 τ

p−2TM+1(t)(τ)dτ, p = 2, 3, · · · ,M,

T2M+p(t) = −(p− 1)∫ t0 τ

p−2T2M+1(t)(τ)dτ, p = 2, 3, · · · ,M,

Π(p, γ,M) = 1 +M∑

p=1

Γ (p−1+γ)Γ (γ−1)p! ,

Ξ(p, γ) = Γ (p−1+γ)Γ (γ−1)(p−1)! ,

ψ(γ) = 1Γ (2−γ) .

(15)

Now we can rewrite Eq. (14) and Eq. (15) as the following form

T ′1(t) = tα−1

Π(p,α,M)

{α−1tα T1(t) +

M∑p=2

Ξ(p, α)(

T1(t)tα

+ Tp(t)tp−1+α

)+ ψ(α)(−µT1(t) + T2∗M+1(t)TM+1(t))

}T ′p(t) = −(p− 1)T1(t), p = 2, 3, · · · ,M,

T ′M+1(t) = tβ−1

Π(p,β,M)

{β−1tβTM+1(t) +

M∑p=2

Ξ(p, β)(

T1(t)tβ

+ TM+p(t)

tp−1+β

)+ ψ(β)(−µTM+1(t) + (T2M+1(t)− a)T1(t))

}T ′M+p(t) = −(p− 1)TM+1(t), p = 2, 3, · · · ,M,

T ′2M+1(t) = tγ−1

Π(p,γ,M)

{γ−1tγ T2M+1(t) +

M∑p=2

Ξ(p, γ)(

T1(t)tγ + T2M+p(t)

tp−1+γ

)+ ψ(γ)(1− TM+1(t)T1(t))

}T ′2M+p(t) = −(p− 1)T2M+1(t), p = 2, 3, · · · ,M,

(16)

with the following initial conditions

T1(δ) = δ1,Tp(δ) = 0, p = 2, 3, · · · ,M,TM+1(δ) = δ2,TM+p(δ) = 0, p = 2, 3, · · · ,M,T2M+1(δ) = δ2,T2M+p(δ) = 0, p = 2, 3, · · · ,M,

(17)

we can rewrite the Eq. (16) and Eq. (17) as the following formT1(t)T2(t)...

T3M (t)

′

=

S11(t) S12 · · · S1,3M

S21(t) S22 · · · S2,3M...... · · ·

...S3M,1(t) S3M,2 · · · S3M,3M

T1(t)T2(t)...

T3M (t)

+

ψ(α)T2M+1(t)TM+1(t) tα−1

Π(p,α,M)...

tβ−1

Π(p,β,M)ψ(β)(T2M+1(t)− a)T1(t))tγ−1

Π(p,γ,M)ψ(γ)(1− TM+1(t)T1(t))

,(18)

where



S11(t) = α−1tΠ(p,α,M) +

M∑p=2

Ξ(p, α) 1tα − µψ(α),

S1p(t) = tα−1

Π(p,α,M)

M∑p=2

Ξ(p,α)tp−1+α , p = 2, 3, · · · ,M,

Spj(t) =

{−(p− 1), j = 1, p = 2, 3, · · · ,M,

0, 1 < j ≤M, p = 2, 3, · · · ,M, j = 2, 3, · · · ,M,

(19)

SM+1,1(t) = β−1tΠ(p,β,M) +

M∑p=2

Ξ(p, β) 1tβ− µψ(β),

SM+1,p(t) = tβ−1

Π(p,β,M)

M∑p=2

Ξ(p,β)tp−1+β , p = 2, 3, · · · ,M,

SM+p,M+j(t) =

{−(p− 1), j = 1, p = 2, 3, · · · ,M,

0, 1 < j ≤M, p = 2, 3, · · · ,M j = 2, 3, · · · ,M,

(20)

S2M+1,1(t) = γ−1tΠ(p,γ,M) +

M∑p=2

Ξ(p, γ) 1tγ − µψ(γ),

S2M+1,p(t) = tγ−1

Π(p,γ,M)

M∑p=2

Ξ(p,γ)tp−1+γ , p = 2, 3, · · · ,M,

S2M+p,2M+j(t) =

{−(p− 1), j = 1, p = 2, 3, · · · ,M,

0, 1 < j ≤M, p = 2, 3, · · · ,M j = 2, 3, · · · ,M.

(21)

In the next section we solve the system of ordinary differential Eq. (18) with the initial conditions Eq. (17) byusing fourth order.

−4 −3 −2 −1 0 1 2 3 4 5−4

−3

−2

−1

0

1

2

3

4

x

y

−4 −3 −2 −1 0 1 2 3 4 5−2

−1

0

1

2

3

4

5

x

z

−2 −1 0 1 2 3 4 5−4

−3

−2

−1

0

1

2

3

4

z

y

−4−2

02

46

−2

0

2

4

6−4

−3

−2

−1

0

1

2

3

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x(t)z(t)

y(t)

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�

Fig. 1. Chaotic attractor of Rikitake system at∏

= [0.8, 0.9, 0.5]T ,∑

= [0.99.0.99, 0.99]T , µ = 1, M = 5 and a = 1



4 Numerical simulation

No we consider the numerical solution of Rikitake system by using the well known Runge-Kutta (RK)method of order fourth. In Fig. 1, we display the numerical solution of Rikitake system. The parameters havebeen set to µ = 1,M = 5 and a = 1. The initial conditions [x(δ), y(δ), z(δ)]T =

∏= [0.8, 0.9, 0.5]T ,

∑=

[α, β, γ]T = [0.99.0.99, 0.99]T have been used. The numerical solution has been approximated from t = 0to t = 200 and the numerical solutions have been saved and plotted at intervals of 0.01, i.e. at times t =0, 0.01, 0.02, · · · , 200. In all numerical runs, the solution has been approximated at ∆t = 0.01 = δ.

−12 −10 −8 −6 −4 −2 0 2 4−4

−3

−2

−1

0

1

2

3

x

y

−12 −10 −8 −6 −4 −2 0 2 4−3

−2

−1

0

1

2

3

4

5

x

z

−3 −2 −1 0 1 2 3 4 5−4

−3

−2

−1

0

1

2

3

z

y

−15

−10

−5

0

5

−4

−2

0

2

4

6−4

−3

−2

−1

0

1

2

3

x(t)z(t)

y(t)

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<


= [−1.2, 0.2, 0.5]T ,∑

= [0.98.0.98, 0.98]T , µ = 1.2, M = 5 anda = 0.9

−10 −5 0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4

5

x

y

−10 −5 0 5 10 15 20−1

0

1

2

3

4

5

6

7

8

x

z

−1 0 1 2 3 4 5 6 7 8−4

−3

−2

−1

0

1

2

3

4

5

z

y

−10−5

05

1015

20

−2

0

2

4

6

8−4

−3

−2

−1

0

1

2

3

4

5

x(t)z(t)

y(t)

��'�� "��

( /,� ��E� ��70 ��

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,�


= [1, 0.2, 0.5]T ,∑

= [0.97.0.97, 0.97]T , µ = 0.8, M = 5 and a = 4



In Fig. 2, we display the numerical solution of Rikitake system. The parameters have been set to µ =1.2,M = 5 and a = 0.9. The initial conditions

∏= [−1.2, 0.2, 0.5]T ,

∑= [0.98.0.98, 0.98]T have been

used. The numerical solution has been approximated from t = 0 to t = 200 and the numerical solutions havebeen saved and plotted at intervals of 0.01, i.e. at times t = 0, 0.01, 0.02, · · · , 200. In all numerical runs, thesolution has been approximated at ∆t = 0.01.

In Fig. 3, we display the numerical solution of Rikitake system. The parameters in the have been set toµ = 0.8,M = 5 and a = 4. The initial conditions

∏= [1, 0.2, 0.5]T ,

∑= [0.97.0.97, 0.97]T have been


−10 0 10 20 30 40 50 60−6

−4

−2

0

2

4

6

8

x

y

−10 0 10 20 30 40 50 60−4

−2

0

2

4

6

8

10

x

z

−4 −2 0 2 4 6 8 10−6

−4

−2

0

2

4

6

8

z

y

−100

1020

3040

5060

−5

0

5

10−6

−4

−2

0

2

4

6

8

x(t)z(t)

y(t)

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( /��<D��<D� ��<D0 � � (,�7�� ( 7 �� ( �

�� ( �� ,� ��E� � � � � E�� =� � �� "� �� "� -�� ##��1�� N� ( ��,� =� ��'�� 7 %� �� #�� "� �� %��" � ( ,�7�� (7��

( /��<D��<D� ��<D0 �N� ( ��,� � ( ,�� ( �� !"� �� *�+ ( ,�� *�+ (��E� �*�+ ( ��7 "�&� -�� =� ��'�� D %� �� #�� "� �� %��" � ( ,�� ( 7�

�( /��<<��<<� ��<<0 �N� ( ��,� � ( ,�� ( 7� !"� ��

�*�+ ( �� *�+ ( ��<� �*�+ ( ��7 "�&� -��

� ��

=� �"� #�#�� %� �##�� 1#�� &��&� '�&�� -� *,�+� =� �� '�� &��&� �# �� "� 4�� $�" ��&��&� '�&�� -� *,,+�� -� � ��' 2�� *,�+ %� ��&�� "� �� '�� J� � � ��" �� '�$A�� "� ��'�� "� � �� @>2 � %� �&�� "� � �� -�� "� 4� � ��# -� � ��' �"�%� ��%� ��" �� '�$A�� "�� J� ��&�� "�� -�"�&�� "� ��

� � � � �

/,0 �� !�� @� � ��#�� -��;��' �"� �� %�$�� $�� ' �� (�� ,E*7+�E7�7 O E7,� E�,,�

,,


= [1.8, 0.2, 0.5]T ,∑

= [0.96.0.96, 0.96]T , µ = 1.5, M = 5 anda = 4

In Fig. 4, we display the numerical solution of Rikitake system. The parameters in the have been set toµ = 1.5,M = 5 and a = 4. The initial conditions

∏= [1.8, 0.2, 0.5]T ,

∑= [0.96.0.96, 0.96]T have been


In Fig. 5, we display the numerical solution of Rikitake system with µ = 1.5,M = 5,∑

=[0.96.0.96, 0.96]T ,∆t = 0.01, t = 100 and a = 4. The initial conditions x(0) = 1.8, y(0) = 0.2, z(0) = 0.5have been used.

In Fig. 6, we display the numerical solution of Rikitake system with µ = 1,M = 5,∑

=[0.99.0.99, 0.99]T ,∆t = 0.01, t = 100 and a = 5. The initial conditions x(0) = 0.8, y(0) = 0.9, z(0) = 0.5have been used.

5 Conclusions

In this paper, we applied an expansion formula for fractional derivatives given by Eq. (13). It containsinteger derivatives up to the finite order k and time moments of k-th derivative, given by Eq. (11). Firstly, byusing Eq. (13) we convert the fractional differential equation to a system of ordinary differential equations of



0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

30

40

50

60

t

x

0 10 20 30 40 50 60 70 80 90 100−6

−4

−2

0

2

4

6

8

t

y

0 10 20 30 40 50 60 70 80 90 100−4

−2

0

2

4

6

8

10

t

x

��'�� 7� I��

( /,�� E� ��70 ��

( /��<D��<D� ��<D0 �N� (��,� � ( ,�� ( ,�7�� ( 7 �� ( �

0 10 20 30 40 50 60 70 80 90 100−6

−4

−2

0

2

4

6

t

x

0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t

y

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

t

z

��'�� D� I��

( /�� <� ��70 ��

( /��<<��<<� ��<<0 �N� (��,� � ( ,�� ( ,�� ( 7 �� ( 7

,E

Fig. 5. Numerical solution of Rikitake system at∏

= [1.8, 0.2, 0.5]T ,∑

= [0.96.0.96, 0.96]T ,∆t = 0.001, t = 100µ = 1.5, M = 5 and a = 4

0 10 20 30 40 50 60 70 80 90 100−10

0

10

20

30

40

50

60

t

x

0 10 20 30 40 50 60 70 80 90 100−6

−4

−2

0

2

4

6

8

t

y

0 10 20 30 40 50 60 70 80 90 100−4

−2

0

2

4

6

8

10

t

x

��'�� 7� I��

( /,�� E� ��70 ��

( /��<D��<D� ��<D0 �N� (��,� � ( ,�� ( ,�7�� ( 7 �� ( �

0 10 20 30 40 50 60 70 80 90 100−6

−4

−2

0

2

4

6

t

x

0 10 20 30 40 50 60 70 80 90 100−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t

y

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

t

z

��'�� D� I��

( /�� <� ��70 ��

( /��<<��<<� ��<<0 �N� (��,� � ( ,�� ( ,�� ( 7 �� ( 7

,E

Fig. 6. Numerical solution of Rikitake system at∏

= [0.8, 0.9, 0.5]T ,∑

= [0.99.0.99, 0.99]T ,∆t = 0.01, t = 100µ = 1, M = 5 and a = 5

integer order. We use fourth order Runge-Kutta formula for the numerical integration of the system of ODEs.we solved the system obtained in the first step by using the well known fourth order Runge-Kutta method. Wederived some chaotic behavior of the fractional Rikitake system.

References

[1] A. Carlos, M. Rafael, et al. Synchronization and parameter estimations of an uncertain rikitake system. PhysicsLetters A, 2010, 374(35): 3625–3628.

[2] T. Atanackovic, B. Stankovic. An expansion formula for fractional derivatives and its application. FractionalCalculus and Applied Analysis, 2004, 7(3): 365–378.

[3] T. Atanackovic, B. Stankovic. On a numerical scheme for solving differential equations of fractional order. Me-chanics Research Communications, 2008, 35(7): 429–438.

[4] D. Braga, F. Dias. On the stability of the equilibria of the rikitake system. Physics Letters A, 2010, 374(42):4316–4320.

[5] M. Danca, S. Codreanu. Modeling numerically the rikitake’s attractors by parameter switching. Journal of theFranklin Institute, 2011, (0).

[6] J. Mata-Machuca, R. Aguilar. An exponential polynomial observer for synchronization of chaotic systems. Com-munications in Nonlinear Science and Numerical Simulation, 2010, 15(12): 4114–4130.

[7] K. Diethelm, K. Diethelm, et al. Analysis of fractional differential equations. Journal of Mathematical Analysisand Applications, 2002, 265: 229–248.

[8] J. Llibre, M. Messias. Global dynamics of the rikitake system. Physica D: Nonlinear Phenomena, 2009, 238(3):241–252.

[9] R. Tudoran. On asymptotically stabilizing the rikitake two-disk dynamo dynamics. Nonlinear Analysis: Real WorldApplications, 2011, 12(5): 2505–2510.

[10] T. Rikitake. Oscillations of a system of disk dynamos. Mathematical Proceedings of the Cambridge PhilosophicalSociety, 1958, 54(01): 89–105.

[11] R. Tudoran, A. Grban. A hamiltonian look at the rikitake two-disk dynamo system. Nonlinear Analysis: RealWorld Applications, 2010, 11(4): 2888–2895.

[12] C. Valls. Rikitake system: analytic and darbouxian integrals. Proceedings of the Royal Society of Edinburgh,Section: A Mathematics, 2005, 135(06): 1309–1326.



[13] U. Vincent. Synchronization of rikitake chaotic attractor using active control. Physics Letters A, 2005, 343(1-3):133–138.

[14] Z. Wei, Q. Yang. Dynamical analysis of a new autonomous 3-d chaotic system only with stable equilibria. Nonlin-ear Analysis: Real World Applications, 2011, 12(1): 106–118.

[15] T. Yajima, H. Nagahama. Geometrical unified theory of rikitake system and kcc-theory. Nonlinear Analysis:Theory, Methods & Applications, 2009, 71(12): e203–e210.


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