Research Article Dynamics of Generalized Tachyon Field in ...downloads.hindawi.com/journals/ahep/2015/283273.pdf · Research Article Dynamics of Generalized Tachyon Field in Teleparallel

Research ArticleDynamics of Generalized Tachyon Field in Teleparallel Gravity

Behnaz Fazlpour1 and Ali Banijamali2

1Department of Physics Islamic Azad University Babol Branch Babol Iran2Department of Basic Sciences Babol University of Technology Babol 47148-71167 Iran

Correspondence should be addressed to Behnaz Fazlpour bfazlpourumzacir

Received 11 October 2014 Accepted 15 January 2015

Academic Editor Chao-Qiang Geng

Copyright copy 2015 B Fazlpour and A Banijamali This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

We study dynamics of generalized tachyon scalar field in the framework of teleparallel gravity This model is an extension oftachyonic teleparallel dark energy model which has been proposed by Banijamali and Fazlpour (2012) In contrast with tachyonicteleparallel dark energymodel that has no scaling attractors here we find some scaling attractors whichmeans that the cosmologicalcoincidence problem can be alleviated Scaling attractors are presented for both interacting and noninteracting dark energy anddark matter cases

1 Introduction

The usual proposal to explain the late-time acceleratedexpansion of our universe is an unknown energy componentdubbed as dark energyThenatural choice andmost attractivecandidate for dark energy is the cosmological constant butit is not well accepted because of the cosmological constantproblem [1 2] as well as the age problem [3] Thus manydynamical dark energy models as alternative possibilitieshave been proposed Quintessence phantom k-essencequintom and tachyon field are the most familiar dark energymodels in the literature (for reviews on dark energy modelssee [4 5]) The tachyon field arising in the context of stringtheory [4 5] and its application in cosmology both as a sourceof early inflation and late-time cosmic acceleration have beenextensively studied [6ndash9]

The so-called ldquoteleparallel equivalent of general relativityrdquoor teleparallel gravity was first constructed by Einstein [10ndash13] In this formulation one uses the curvature-less Weitzen-bock connection instead of the torsion-less Levi-Civita con-nection The relevant Lagrangian in teleparallel gravity is thetorsion scalar 119879 which is constructed by contraction of thetorsion tensorWe recall that the Einstein-Hilbert Lagrangian119877 is constructed by contraction of the curvature tensor Sinceteleparallel gravitywith torsion scalar as Lagrangian density iscompletely equivalent to a matter-dominated universe in the

framework of general relativity it cannot be acceleratedThusone should generalize teleparallel gravity either by replacing119879 with an arbitrary function the so-called 119891(119879) gravity[14ndash16] or by adding dark energy into teleparallel gravityallowing also a nonminimal coupling between dark energyand gravity Note that both approaches are inspired by thesimilarmodifications of general relativity that is119891(119877) gravity[17 18] and nonminimally coupled dark energymodels in theframework of general relativity [19ndash26]

Recently Geng et al [27 28] have included a nonminimalcoupling between quintessence and gravity in the context ofteleparallel gravity This theory has been called ldquoteleparalleldark energyrdquo and its dynamics was studied in [29ndash31] Tachy-onic teleparallel dark energy is a generalization of teleparalleldark energy by inserting a noncanonical scalar field insteadof quintessence in the action [32] Phase-space analysis of thismodel has been investigated in [33] On the other hand thereis no physical argument to exclude the interaction betweendark energy and dark matter The interaction between thesecompletely different components of our universe has the sameimportant consequences such as addressing the coincidenceproblem [34] There are also observational evidences of theinteraction in dark sector [35ndash40] In [35ndash38] Bertolami et alhave shown that this interaction might imply a violation ofthe equivalence principle By using optical X-ray and weaklensing data from the relaxed galaxy clusters Abdalla et al

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 283273 12 pageshttpdxdoiorg1011552015283273

2 Advances in High Energy Physics

[39 40] have found the signature of interaction between darkenergy and dark matter An interacting scenario providessolution to a number of cosmological problems in bothcanonical scalar field models [41] and nonminimally coupledscalar field models such as Brans-Dick field [42 43]

In this paper we consider generalized tachyon field asresponsible for dark energy in the framework of teleparallelgravity We will be interested in performing a dynamicalanalysis of such a model in FRW space time In such astudy we investigate our model for both interacting andnoninteracting cases We consider the nonminimal couplingfunction of the form 119891(120601) prop 120601

2 and choose 119899 = 2 in (2)The basic equations are presented in Section 2 In Section 3the evolution equations are translated in the language of theautonomous dynamical system by suitable transformation ofthe basic variables Section 31 deals with phase-space analysisas well as the cosmological implications of the equilibriumpoints of the model in noninteracting dark energy darkmatter case In Section 32 an interaction between darkenergy and dark matter has been considered and criticalpoints and their behavior have been extracted Section 4 isdevoted to a short summary of our results

2 Basic Equations

Our model is described by the following action as a general-ization of tachyon teleparallel dark energy model [32]

119878 = int1198894119909119890 [L

119879+L120593+L119898]

L119879=

119879

21205812

(1)

L120593= 120585119891 (120593) 119879 minus 119881 (120593) (1 minus 2119883)

119899 (2)

where 119890 = det(119890119894120583) = radicminus119892 (119890119894

120583are the orthonormal comp-

onents of the tetrad) while11987921205812 is the Lagrangian of telepar-allelism with 119879 as the torsion scalar (for an introductoryreview of teleparallelism see [12]) L

120593shows nonminimal

coupling of generalized tachyon field 120593 with gravity in theframework of teleparallel gravity and 119883 = (12)120597

120583120593120597120583120593 The

second part in L120593is the Lagrangian density of the general-

ized tachyon field which has been studied in [44 45] 119891(120593)is the nonminimal coupling function 120585 is a dimensionlessconstant measuring the nonminimal coupling and L

119898is

the matter Lagrangian For 119899 = 12 our model reduced totachyonic teleparallel dark energy discussed in [32] Here weconsider the case 119899 = 2 for two reasons The first is that forarbitrary 119899 our equations will be very complicated and onecannot solve themanalytically and the second is that for 119899 = 2wewill obtain interesting physical results as we will see below

Furthermore due to complexity of tachyon dynamics[46] has proposed an approach based on the redefinition ofthe tachyon field as follows

120593 997888rarr 120601 = int119889120593radic119881 (120593) lArrrArr 120597120593 =120597120601

radic119881 (120601)

(3)

In order to obtain a closed autonomous system and performthe phase-space analysis of the model we apply (3) in (2) for119899 = 2 that leads to the following action

119878 = int1198894119909119890 [

119879

21205812+ 120585119891 (120601) 119879 minus 119881 (120601)(1 minus

2119883

119881 (120601))

2

+L119898]

(4)

In a spatially flat FRW space-time

1198891199042= 1198891199052minus 1198862(119905) (119889119903

2+ 1199032119889Ω2) (5)

and a vierbein choice of the form 119890119894

120583= diag(1 119886 119886 119886) the

corresponding Friedmann equations are given by

1198672=1

3(120588120601+ 120588119898)

= minus1

2(120588120601+ 119875120601+ 120588119898+ 119875119898)

(6)

where 119867 = 119886119886 is the Hubble parameter and a dot standsfor the derivative with respect to the cosmic time 119905 In theseequations 120588

119898and 119875

119898are the matter energy density and

pressure respectivelyThe effective energy density and pressure of generalized

tachyon dark energy read

120588120601= 119881 (120601) + 2 120601

2minus 3

1206014

119881 (120601)minus 6120585119867

2119891 (120601)

119875120601= minus119881 (120601) + 2120585 (3119867

2+ 2) 119891 (120601) + 10120585119867119891

120601120601

+ 1206012(2 minus

1206012

119881 (120601))

(7)

where 119891120601= 119889119891119889120601

The equation of motion of the scalar field can be obtainedby variation of the action (4) with respect to 120601

120601 + 3120583minus2]2119867 120601 +

1

4]2 (1 +

3 1206014

1198812 (120601))119881120601

+ 6120585]21198672119891120601= minus

]21198764 120601

(8)

with 119876 a general interaction coupling term between darkenergy and dark matter 120583 = 1radic1 minus 2119883119881 and ] =

1radic1 minus 6119883119881 In (7) and (8) we have used the useful relation

119879 = minus61198672 (9)

which simply arises from the calculation of torsion scalar forthe FRW metric (5) The scalar field evolution (8) expressesthe continuity equation for the field and matter as follows

120588120601+ 3119867(1 + 120596

120601) 120588120601= minus119876

120588119898+ 3119867 (1 + 120596

119898) 120588119898= 119876

(10)

Advances in High Energy Physics 3

Table 1 Location of the critical points and the corresponding values of the dark energy density parameter Ω120601and equation of state 120596

120601and

the condition required for an accelerating universe for119876 = 0 Here 1205821= 120572(4120572120585+radic1612057221205852 minus 21205822120585)120582

2 and 1205822= 120572(4120572120585minusradic1612057221205852 minus 21205822120585)120582

2

Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198601

0 2radic12058211205821205821

212057212058541205821(1 minus

1205822

812058512057221205821)

1612058521205724

(12058221205822

1+ 21205851205722) (1205822120582

1minus 81205851205722)

1205822gt1205851205722(41205821minus 1)

1205822

1

(1 + radic1 minus321205821

(1 minus 41205821)2)

or

1205822lt1205851205722(41205821minus 1)

1205822

1

(1 minus radic1 minus321205821

(1 minus 41205821)2)

1198602

0 2radic12058221205821205822

212057212058541205822(1 minus

1205822

812058512057221205822)

1612058521205724

(12058221205822

2+ 21205851205722) (1205822120582

2minus 81205851205722)

1205822gt1205851205722(41205822minus 1)

1205822

2

(1 + radic1 minus321205822

(1 minus 41205822)2)

or

1205822lt1205851205722(41205822minus 1)

1205822

2


(1 minus 41205822)2)

Table 2 Stability and existence conditions of the critical points ofthe model for 119876 = 0

Label Stability Existence

1198601

Saddle pointif 120572 gt 0 and 120585120582 gt 0 and

stable pointif 120585 gt 0 120572 lt 0 and 120582 lt 0

For all 120585 lt 0or

120585 ge 120582281205722

1198602

Saddle pointif 120572 lt 0 and 120585120582 lt 0 and

stable pointif 120585 gt 0 120572 gt 0 and 120582 gt 0


120585 ge 120582281205722

where 120596120601= 119875120601120588120601is the equation of state parameter of dark

energy which is attributed to the scalar field 120601The barotropicindex is defined by 120574 equiv 1 + 120596

119898with 0 lt 120574 lt 2

Although dynamics of tachyonic teleparallel dark energyhas been studied in [33] no scaling attractors has been foundHere we are going to perform a phase-space analysis ofgeneralized tachyonic teleparallel dark energy and as we willsee below that some interesting scaling attractors appear insuch theory

3 Cosmological Dynamics

In order to perform phase-space and stability analysis of themodel we introduce the following auxiliary variables

119909 equiv

120601

radic119881 119910 equiv

radic119881

radic3119867 119906 equiv radic119891 (11)

The auxiliary variables allow us to straightforwardly obtainthe density parameter of dark energy and dark matter

Ω120601equiv

120588120601

31198672= 120583minus21199102(1 + 3119909

2) minus 2120585119906

2 (12)

Ω119898equiv120588119898

31198672= 1 minus Ω

120601 (13)

while the equation of state of the field reads

120596120601equiv

119875120601

120588120601

=

minus120583minus41199102+ 2120585119906 [(5radic33) 120572119909119910 + 119906 (1 minus (23) 119904)]

120583minus21199102 (1 + 31199092) minus 21205851199062

(14)

where 120572 equiv 119891120601radic119891 and

119904 = minus

1198672= (2120585119906

2+ 1)minus1

sdot [5radic3120572120585119906119909119910 + 6120583minus211990921199102

minus3

2120574120583minus21199102(1 + 3119909

2)] +

3120574

2

(15)

From now we concentrate on exponential scalar field poten-tial of the form 119881 = 119881

0119890minus119896120582120601 and the nonminimal coupling

function of the form 119891(120601) prop 1206012 These choices lead to

constant 120582 and 120572 respectivelyAnother quantities with great physical significance

namely the total equation of state parameter and the decel-eration parameter are given by

120596tot equiv119875120601+ 119875119898

120588120601+ 120588119898

= 120583minus21199102(41199092minus 120574 (1 + 3119909

2))

+ 2120585119906 [5radic3

3120572119909119910 + 119906(120574 minus

2

3119904)]

+ 120574 minus 1

(16)

119902 equiv minus1 minus

1198672=1

2+3

2120596tot

=3

2120583minus21199102(41199092minus 120574 (1 + 3119909

2))

+ 120585119906 [5radic3120572119909119910 + 119906 (3120574 minus 2119904)] +3120574

2minus 1

(17)


110

109

108

107

106

105

104

103

102

101

y(t)

0 005 010 015 020

x(t)

(a)

0 005 010 015 020

x(t)

003

002

004

005

006

007

008

u(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

003002 004 005 006 007 008

u(t)

(c)

Figure 1 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 and 120572 = minus2for 119876 = 0 For these values of the parameters point 119860

1is a stable attractor of the model

Using auxiliary variables (11) the evolution equations (6)and (8) can be recast as a dynamical system of ordinarydifferential equations

1199091015840=radic3

2[1205821199092119910 +

1

2120582]2 (1 + 31199094) 119910

minus 4120572120585]2119906119910minus1 minus 2radic3120583minus2]2119909] minus 119876

1199101015840= (minus

radic3

2120582119909119910 + 119904)119910

1199061015840=radic3120572119909119910

2

(18)

where 119876 = 119876 120601119867radic119881(120601) 120582 equiv minus119881120601119896119881 and prime in (18)

denotes differentiation with respect to the so-called e-foldingtime119873 = ln 119886


110

109

108

107

106

105

104

103

102

101

y(t)

minus005 0 005 010 015 020

x(t)

(a)

007

009

008

011

010

013

012

015

014

u(t)

minus005 0 005 010 015 020

x(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

007 008 010 012 014009 011 013 015

u(t)

(c)

Figure 2 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 and 120572 = 2 for119876 = 0 For these values of the parameters point 119860


The next step is the introduction of interaction term 119876

to obtain an autonomous system out of (18) The fixed points(119909119888 119910119888 119906119888) for which 1199091015840 = 1199101015840 = 1199061015840 = 0 depend on the choice

of the interaction term 119876 and two general possibilities willbe treated in the sequel The stability of the system at a fixedpoint can be obtained from the analysis of the determinantand trace of the perturbation matrix 119872 Such a matrix canbe constructed by substituting linear perturbations 119909 rarr

119909119888+ 120575119909 119910 rarr 119910

119888+ 120575119910 and 119906 rarr 119906

119888+ 120575119906 about the

critical point (119909119888 119910119888 119906119888) into the autonomous system (18)The

3 times 3 matrix119872 of the linearized perturbation equations ofthe autonomous system is shown in the appendix Thereforefor each critical point we examine the sign of the real partof the eigenvalues of 119872 According to the usual dynamicalsystem analysis if the eigenvalues are real and have oppositesigns the corresponding critical point is a saddle point Afixed point is unstable if the eigenvalues are positive and itis stable for negative real part of the eigenvalues

In the following subsections we will study the dynamicsof generalized tachyon field with different interaction term


109

107105

103

101 y(t)

x(t)

u(t)

008

007

006

005

004

003

002

0005

010015

020

(a)

110

107105 y

(t)

u(t)

minus005 0 005 010 015 020

x(t)

015

014

013

012

011

010

009

008

(b)

Figure 3 Three-dimensional phase-space trajectories of the model for 119876 = 0 with stable attractors 1198601(a) and 119860

2(b) The values of the

parameters are those mentioned in Figures 1 and 2 respectively

119876 Although one can work with a general 120574 (120574 = 1 and 43correspond to dust matter and radiation resp) without lossof generality we assume 120574 = 1 for simplicity

31 The Case for 119876 = 0 The first case 119876 = 0 clearlymeans that there is no interaction between dark energy andbackground matter In this case there are two critical pointspresented in Table 1 From (12) and (14) one can obtain thecorresponding values of density parameter Ω

120601and equation

of state of dark energy120596120601at each point Also using (17)we can

find the condition required for acceleration (119902 lt 0) at eachpoint These parameters and conditions have been shownin Table 1 The stability and existence conditions of criticalpoints 119860

1and 119860

2are presented in Table 2 We mention that

the corresponding eigenvalues of perturbation matrix 119872 atcritical points 119860

1and 119860

2are considerably involved and here

we do not present their explicit expressions but we can findsign of them numerically

Critical Point 1198601 This critical point is a scaling attractor if

120585 gt 0 120572 lt 0 and 120582 lt 0 Therefore cosmological coincidenceproblem can be alleviated at this point 119860

1is a saddle point

for 120572 gt 0 and 120585120582 gt 0

Critical Point 1198602 1198602can also be a scaling attractor of the

model or a saddle point under the same conditions as for1198601

In Figure 1 we have chosen the values of the parameters120585 120582 and 120572 such that 119860

1became a stable attractor of the

model Plots in Figure 1 show the phase-space trajectories on119909-119910 119909-119906 and 119906-119910 planes from (a) to (c) respectively Thesame plots are shown in Figure 2 for critical point 119860

2 Note

that the values of the parameter have been chosen in the waythat 119860

2became a stable point of the model In Figure 3 the

corresponding 3-dimensional phase-space trajectories of themodel have been presented One can see that 119860

1and 119860

2are

stable attractors of themodel in (a) and (b) plots respectively

32The Case for119876 = 120573120581120588119898120601 This deals with the most famil-

iar interaction term extensively considered in the literature(see eg [31 47ndash51]) Here 119876 in terms of auxiliary variablesis 119876 = radic3120573119910minus1Ω

119898 Inserting such an interaction term in (18)

and setting the left hand sides of the equations to zero leadto the critical points 119861

1 1198612 1198613 and 119861

4presented in Table 3

In the same table we have provided the corresponding valuesofΩ120601and 120596

120601as well as the condition needed for accelerating

universe at each fixed pointsThe stability and existence conditions for each point are

presented in Table 4 Since the corresponding eigenvalues ofthe fixed points are complicated we do not give them herebut one can obtain their signs numerically and then examinethe stability properties of the critical points

Critical Point 1198611This point exists for 120582 gt 0 and 120585 ge 120582281205722

However it is an unstable saddle point

Critical Point 1198612The critical point 119861

2exists for 120582 gt 0 and 120585 lt

0 or 120585 ge 120582281205722 This point is a scaling attractor of the modelif 120572 gt 0 and 120585 gt 0 Figure 4 shows clearly such a behavior ofthe model for suitable choices of 120585 120582 and 120572

Critical Point 1198613This point exists for negative values of 120582 and

120585 or when 120585 ge 120582281205722 Also it is a stable point if 120572 lt 0

and 120585 gt 0 and a saddle point if 120572 gt 0 and 120585 lt 0 Thevalues of parameters have been chosen in Figure 5 such that


110

109

108

107

106

105

104

103

102

101

y(t)

0 01 02

x(t)

minus01

(a)

0 01 02

x(t)

minus01

024

022

020

018

016

014

012

010

008

u(t)

(b)

110107105103101 102 104 106 108 109

y(t)

024

022

020

018

016

014

012

010

008

u(t)

(c)

Figure 4 From (a) to (c) the projections of the phase-space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = 06 120572 = 2 and120573 = 15 for 119876 = 120573120581120588

119898120601 For these values of the parameters point 119861


1198613became a attractor of the model as it is clear from phase-

space trajectories

Critical Point 1198614Thepoint119861

4exists for120582 lt 0 and 120585 ge 120582281205722

It is a stable point if 120572 lt 0 and 120585 gt 0 In Figure 6 values of theparameters 120585 and 120572 are those satisfying these constraints andso 1198614becomes a attractor point for phase-plane trajectories

The corresponding 3-dimensional phase-space trajectories ofthe model for attractor points 119861

2(a) 119861

3(b) and 119861

4(c) are

plotted in Figure 7

4 Conclusion

A model of dark energy with nonminimal coupling ofquintessence scalar field with gravity in the framework ofteleparallel gravity was called teleparallel dark energy [27] Ifone replaces quintessence by tachyon field in such a modelthen tachyonic teleparallel dark energy will be constructed[32]

Moreover although dark energy and dark matter scaledifferently with the expansion of our universe according to


110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)

Figure 5 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus2 and120573 = 15 for 119876 = 120573120581120588



the observations [52ndash54] we are living in an epoch in whichdark energy and dark matter densities are comparable andthis is the well-known cosmological coincidence problem[30] This problem can be alleviated in most dark energymodels via the method of scaling solutions in which thedensity parameters of dark energy and dark matter are bothnonvanishing over there

In this paper we investigated the phase-space analysisof generalized tachyon cosmology in the framework ofteleparallel gravity Our model was described by action (2)

which generalizes tachyonic teleparallel dark energy modelproposed in [32] We found some scaling attractors in ourmodel for the case 119899 = 2 and the coupling function119891(120601) prop 120601

2 These scaling attractors are 1198601and 119860

2when

there is no interaction between dark energy and dark matter1198612 1198613 and 119861

4are scaling attractors in the case that dark

energy interacts with dark matter through the interactingterm 119876 = 120573120581120588

119898120601 As it is shown in [29] original teleparallel

dark energy model also has a late-time attractor in whichdark energy behaves like a cosmological constant and so


y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)

Figure 6 From (a) to (c) the projections of the phase space trajectories on the 119909-119910 119909-119906 and 119906-119910 planes with 120585 = 05 120582 = minus06 120572 = minus05and 120573 = 15 for 119876 = 120573120581120588



it provides a natural way for the stabilization of the darkenergy equation of state to the cosmological constant valuewithout the need for parameter tuning Also the authors in[55] by using power-law potentials and nonminimal couplingfunctions have shown that teleparallel gravity has no stablefuture solutions for positive nonminimal coupling and thescalar field results always in oscillationsOur results show thatgeneralized tachyon field represents interesting cosmologicalbehavior in comparison with ordinary tachyon fields in the

framework of teleparallel gravity because there is no scalingattractor in the latter model Note that however in [33]Otalora has considered a dynamically changing coefficient120572 = 119891

120601radic119891 and found a field matter dominated solution

in which it has some portions of the dark energy density inthe matter dominated era So generalized tachyon field givesus the hope that cosmological coincidence problem can bealleviated without fine-tunings One can study our model fordifferent kinds of potential and other famous interaction termbetween dark energy and dark matter


u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)

Figure 7Three-dimensional phase-space trajectories of the model for119876 = 120573120581120588119898120601with stable attractors 119861

2(a) 119861

3(b) and 119861

4(c)The values

of the parameters are those mentioned in Figures 4 5 and 6 respectively

Table 3 Location of the critical points and the corresponding values of the dark energy density parameterΩ120601and equation of state120596

120601and the

condition required for an accelerating universe for119876 = 120573120581120588119898120601 Here 120579

1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579

2= (4120572120585minusradic1612057221205852 minus 21205822120585)2120582120585

Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611

Saddle pointif 120572 gt 0 and 120585 gt 0

120582 gt 0 and 120585 ge 120582281205722

1198612


stable pointif 120572 gt 0 and 120585 gt 0

120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613

Saddle pointif 120572 gt 0 and 120585 lt 0 and

stable pointif 120572 lt 0 and 120585 gt 0

120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614

Stable pointif 120572 lt 0 and 120585 gt 0

120582 lt 0 and 120585 ge 120582281205722

Appendix

Perturbation Matrix Elements

The elements of 3 times 3 matrix 119872 of the linearized per-turbation equations for the real and physically meaningfulcritical points (119909

119888 119910119888 119906119888) of the autonomous system (18) read

as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888

minus 4radic3120572120585119906119888119909119888]2119888119910minus1

119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)

minusradic3120582119910

2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)

where in the case of 119876 = 0 we have M11= M12= M13= 0

and in the case of 119876 = 120573120581120588119898120601 we have

M11= 4radic3120573]minus2

119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)

Examining the eigenvalues of the matrix119872 for each criticalpoint one determines its stability conditions

Conflict of Interests

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

References

[1] S Weinberg ldquoThe cosmological constant problemrdquo Reviews ofModern Physics vol 61 no 1 pp 1ndash23 1989

[2] S M Carroll ldquoThe cosmological constantrdquo Living Reviews inRelativity vol 4 article 1 2001

[3] R-J Yang and S N Zhang ldquoThe age problem in the ΛCDMmodelrdquo Monthly Notices of the Royal Astronomical Society vol407 no 3 pp 1835ndash1841 2010

[4] Y-F Cai E N Saridakis M R Setare and J-Q Xia ldquoQuintomcosmology theoretical implications and observationsrdquo PhysicsReports vol 493 no 1 pp 1ndash60 2010

[5] K Bamba S Capozziello S Nojiri and S D Odintsov ldquoDarkenergy cosmology the equivalent description via differenttheoretical models and cosmography testsrdquo Astrophysics andSpace Science vol 342 no 1 pp 155ndash228 2012

[6] A Sen ldquoRemarks on tachyon driven cosmologyrdquo PhysicaScripta vol 2005 no 117 article 70 2005

[7] GWGibbons ldquoCosmological evolution of the rolling tachyonrdquoPhysics Letters B vol 537 no 1-2 pp 1ndash4 2002

[8] M R Garousi ldquoTachyon couplings on non-BPS D-branes andDiracndashBornndashInfeld actionrdquo Nuclear Physics B vol 584 no 1-2pp 284ndash299 2000

[9] A Sen ldquoField theory of tachyon matterrdquoModern Physics LettersA vol 17 no 27 pp 1797ndash1804 2002

[10] A Unzicker and T Case ldquoTranslation of Einsteinrsquos attempt ofa unified field theory with teleparallelismrdquo httparxivorgabsphysics0503046

[11] K Hayashi and T Shirafuji ldquoNew general relativityrdquo PhysicalReview D vol 19 pp 3524ndash3553 1979 Addendum in PhysicalReview D vol 24 pp 3312-3314 1982

[12] R Aldrovandi and J G Pereira Teleparallel Gravity An Intro-duction Springer Dordrecht Netherlands 2013

[13] J W Maluf ldquoThe teleparallel equivalent of general relativityrdquoAnnalen der Physik vol 525 no 5 pp 339ndash357 2013

[14] R Ferraro andF Fiorini ldquoModified teleparallel gravity inflationwithout an inflatonrdquo Physical Review D Particles Fields Gravi-tation and Cosmology vol 75 no 8 Article ID 084031 2007

[15] G R Bengochea and R Ferraro ldquoDark torsion as the cosmicspeed-uprdquo Physical Review D Particles Fields Gravitation andCosmology vol 79 Article ID 124019 2009

[16] E V Linder ldquoEinsteins other gravity and the acceleration of theUniverserdquo Physical Review D vol 81 Article ID 127301 2010


[17] A de Felice and S Tsujikawa ldquof (R) theoriesrdquo Living Reviews inRelativity vol 13 no 3 pp 1ndash161 2010

[18] S Nojiri and S D Odintsov ldquoUnified cosmic history inmodified gravity from F(R) theory to Lorentz non-invariantmodelsrdquo Physics Reports vol 505 no 2ndash4 pp 59ndash144 2011

[19] B L Spokoiny ldquoInflation and generation of perturbations inbroken-symmetric theory of gravityrdquo Physics Letters B vol 147no 1ndash3 pp 39ndash43 1984

[20] F Perrotta C Baccigalupi and S Matarrese ldquoExtendedquintessencerdquo Physical Review D vol 61 Article ID 0235071999

[21] V Faraoni ldquoInflation and quintessence with nonminimal cou-plingrdquo Physical Review D vol 62 no 2 Article ID 023504 15pages 2000

[22] E Elizalde S Nojiri and S D Odintsov ldquoLate-time cosmologyin a (phantom) scalar-tensor theory dark energy and thecosmic speed-uprdquo Physical Review D vol 70 Article ID 0435392004

[23] OHrycyna andM Szydlowski ldquoNon-minimally coupled scalarfield cosmology on the phase planerdquo Journal of Cosmology andAstroparticle Physics vol 2009 article 026 2009

[24] O Hrycyna and M Szydlowski ldquoExtended Quintessence withnon-minimally coupled phantom scalar fieldrdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 76 ArticleID 123510 2007

[25] R C de Souza andGMKremer ldquoConstraining non-minimallycoupled tachyon fields by the Noether symmetryrdquo Classical andQuantum Gravity vol 26 no 13 Article ID 135008 2009

[26] A A Sen and N C Devi ldquoCosmology with non-minimallycoupled 119896-fieldrdquo General Relativity and Gravitation vol 42 no4 pp 821ndash838 2010

[27] C-Q Geng C-C Lee E N Saridakis and Y-P Wu ldquolsquoTelepar-allelrsquo dark energyrdquoPhysics Letters SectionBNuclear ElementaryParticle and High-Energy Physics vol 704 no 5 pp 384ndash3872011

[28] C-Q Geng C-C Lee and E N Saridakis ldquoObservationalconstraints on teleparallel dark energyrdquo Journal of Cosmologyand Astroparticle Physics vol 2012 no 1 article 002 2012

[29] C Xu E N Saridakis and G Leon ldquoPhase-space analysis ofteleparallel dark energyrdquo Journal of Cosmology and AstroparticlePhysics vol 2012 no 7 article 5 2012

[30] H Wei ldquoDynamics of teleparallel dark energyrdquo Physics LettersB vol 712 no 4-5 pp 430ndash436 2012

[31] G Otalora ldquoScaling attractors in interacting teleparallel darkenergyrdquo Journal of Cosmology and Astroparticle Physics vol2013 no 7 article 44 2013

[32] A Banijamali and B Fazlpour ldquoTachyonic teleparallel darkenergyrdquo Astrophysics and Space Science vol 342 no 1 pp 229ndash235 2012

[33] G Otalora ldquoCosmological dynamics of tachyonic teleparalleldark energyrdquo Physical Review D vol 88 no 6 Article ID063505 8 pages 2013

[34] SH Pereira S S A Pinho and JMH da Silva ldquoSome remarkson the attractor behaviour in ELKO cosmologyrdquo Journal ofCosmology and Astroparticle Physics vol 2014 article 020 2014

[35] O Bertolami F Gil Pedro and M Le Delliou ldquoDark energy-dark matter interaction and putative violation of the equiva-lence principle from the Abell cluster A586rdquo Physics Letters Bvol 654 no 5-6 pp 165ndash169 2007

[36] O Bertolami F Gil Pedro and M le Delliou ldquoThe Abellcluster A586 and the detection of violation of the equivalence

principlerdquo General Relativity and Gravitation vol 41 no 12 pp2839ndash2846 2009

[37] M Le Delliou O Bertolami and F Gil Pedro ldquoDark energy-dark matter interaction from the abell cluster A586 and viola-tion of the equivalence principlerdquo AIP Conference Proceedingsvol 957 p 421 2007

[38] O Bertolami F G Pedro and M L Delliou ldquoDark energy-dark matter interaction from the abell cluster A586rdquo EuropeanAstronomical Society Publications Series vol 30 pp 161ndash1672008

[39] E Abdalla L R Abramo L Sodre and B Wang ldquoSignature ofthe interaction between dark energy and dark matter in galaxyclustersrdquo Physics Letters B vol 673 pp 107ndash110 2009

[40] E Abdalla L R Abramo and J C C de Souza ldquoSignatureof the interaction between dark energy and dark matter inobservationsrdquo Physical Review D vol 82 Article ID 0235082010

[41] N Banerjee and S Das ldquoAcceleration of the universe with a sim-ple trigonometric potentialrdquoGeneral Relativity and Gravitationvol 37 no 10 pp 1695ndash1703 2005

[42] S Das and N Banerjee ldquoAn interacting scalar field and therecent cosmic accelerationrdquo General Relativity and Gravitationvol 38 no 5 pp 785ndash794 2006

[43] S Das and N Banerjee ldquoBrans-Dicke scalar field as achameleonrdquo Physical Review D vol 78 Article ID 043512 2008

[44] S Unnikrishnan ldquoCan cosmological observations uniquelydetermine the nature of dark energyrdquo Physical Review D vol78 Article ID 063007 2008

[45] R Yang and J Qi ldquoDynamics of generalized tachyon fieldrdquoTheEuropean Physical Journal C vol 72 article 2095 2012

[46] I Quiros T Gonzalez D Gonzalez Y Napoles R Garcıa-Salcedo andCMoreno ldquoA study of tachyondynamics for broadclasses of potentialsrdquoClassical andQuantumGravity vol 27 no21 Article ID 215021 2010

[47] E J Copeland A R Liddle and D Wands ldquoExponentialpotentials and cosmological scaling solutionsrdquo Physical ReviewD vol 57 no 8 pp 4686ndash4690 1998

[48] L Amendola ldquoScaling solutions in general nonminimal cou-pling theoriesrdquo Physical Review D vol 60 Article ID 0435011999

[49] Z K Guo R G Cai and Y Z Zhang ldquoCosmological evolutionof interacting phantom energy with dark matterrdquo Journal ofCosmology andAstroparticle Physics vol 2005 article 002 2005

[50] H Wei ldquoCosmological evolution of quintessence and phantomwith a new type of interaction in dark sectorrdquoNuclear Physics Bvol 845 no 3 pp 381ndash392 2011

[51] L P Chimento A S Jakubi D Pavon and W ZimdahlldquoInteracting quintessence solution to the coincidence problemrdquoPhysical Review D vol 67 no 8 Article ID 083513 2003

[52] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal ofModern Physics D GravitationAstrophysics Cosmology vol 15 no 11 pp 1753ndash1935 2006

[53] J A Frieman M S Turner and D Huterer ldquoDark energy andthe accelerating universerdquo Annual Review of Astronomy andAstrophysics vol 46 pp 385ndash432 2008

[54] S Tsujikawa ldquoDark energy investigation and modelingrdquo inA Challenge for Modern Cosmology vol 370 of Astrophysicsand Space Science Library pp 331ndash402 Springer DordrechtNetherlands 2011

[55] M Skugoreva E Saridakis and A Toporensky ldquoDynamicalfeatures of scalar-torsion theoriesrdquo to appear in Physical ReviewD

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


[39 40] have found the signature of interaction between darkenergy and dark matter An interacting scenario providessolution to a number of cosmological problems in bothcanonical scalar field models [41] and nonminimally coupledscalar field models such as Brans-Dick field [42 43]

In this paper we consider generalized tachyon field asresponsible for dark energy in the framework of teleparallelgravity We will be interested in performing a dynamicalanalysis of such a model in FRW space time In such astudy we investigate our model for both interacting andnoninteracting cases We consider the nonminimal couplingfunction of the form 119891(120601) prop 120601

2 and choose 119899 = 2 in (2)The basic equations are presented in Section 2 In Section 3the evolution equations are translated in the language of theautonomous dynamical system by suitable transformation ofthe basic variables Section 31 deals with phase-space analysisas well as the cosmological implications of the equilibriumpoints of the model in noninteracting dark energy darkmatter case In Section 32 an interaction between darkenergy and dark matter has been considered and criticalpoints and their behavior have been extracted Section 4 isdevoted to a short summary of our results

2 Basic Equations

Our model is described by the following action as a general-ization of tachyon teleparallel dark energy model [32]

119878 = int1198894119909119890 [L

119879+L120593+L119898]

L119879=

119879

21205812

(1)

L120593= 120585119891 (120593) 119879 minus 119881 (120593) (1 minus 2119883)

119899 (2)

where 119890 = det(119890119894120583) = radicminus119892 (119890119894

120583are the orthonormal comp-

onents of the tetrad) while11987921205812 is the Lagrangian of telepar-allelism with 119879 as the torsion scalar (for an introductoryreview of teleparallelism see [12]) L

120593shows nonminimal

coupling of generalized tachyon field 120593 with gravity in theframework of teleparallel gravity and 119883 = (12)120597

120583120593120597120583120593 The

second part in L120593is the Lagrangian density of the general-

ized tachyon field which has been studied in [44 45] 119891(120593)is the nonminimal coupling function 120585 is a dimensionlessconstant measuring the nonminimal coupling and L

119898is

the matter Lagrangian For 119899 = 12 our model reduced totachyonic teleparallel dark energy discussed in [32] Here weconsider the case 119899 = 2 for two reasons The first is that forarbitrary 119899 our equations will be very complicated and onecannot solve themanalytically and the second is that for 119899 = 2wewill obtain interesting physical results as we will see below

Furthermore due to complexity of tachyon dynamics[46] has proposed an approach based on the redefinition ofthe tachyon field as follows

120593 997888rarr 120601 = int119889120593radic119881 (120593) lArrrArr 120597120593 =120597120601

radic119881 (120601)

(3)

In order to obtain a closed autonomous system and performthe phase-space analysis of the model we apply (3) in (2) for119899 = 2 that leads to the following action

119878 = int1198894119909119890 [

119879

21205812+ 120585119891 (120601) 119879 minus 119881 (120601)(1 minus

2119883

119881 (120601))

2

+L119898]

(4)

In a spatially flat FRW space-time

1198891199042= 1198891199052minus 1198862(119905) (119889119903

2+ 1199032119889Ω2) (5)

and a vierbein choice of the form 119890119894

120583= diag(1 119886 119886 119886) the

corresponding Friedmann equations are given by

1198672=1

3(120588120601+ 120588119898)

= minus1

2(120588120601+ 119875120601+ 120588119898+ 119875119898)

(6)

where 119867 = 119886119886 is the Hubble parameter and a dot standsfor the derivative with respect to the cosmic time 119905 In theseequations 120588

119898and 119875

119898are the matter energy density and

pressure respectivelyThe effective energy density and pressure of generalized

tachyon dark energy read

120588120601= 119881 (120601) + 2 120601

2minus 3

1206014

119881 (120601)minus 6120585119867

2119891 (120601)

119875120601= minus119881 (120601) + 2120585 (3119867

2+ 2) 119891 (120601) + 10120585119867119891

120601120601

+ 1206012(2 minus

1206012

119881 (120601))

(7)

where 119891120601= 119889119891119889120601

The equation of motion of the scalar field can be obtainedby variation of the action (4) with respect to 120601

120601 + 3120583minus2]2119867 120601 +

1

4]2 (1 +

3 1206014

1198812 (120601))119881120601

+ 6120585]21198672119891120601= minus

]21198764 120601

(8)

with 119876 a general interaction coupling term between darkenergy and dark matter 120583 = 1radic1 minus 2119883119881 and ] =

1radic1 minus 6119883119881 In (7) and (8) we have used the useful relation

119879 = minus61198672 (9)

which simply arises from the calculation of torsion scalar forthe FRW metric (5) The scalar field evolution (8) expressesthe continuity equation for the field and matter as follows

120588120601+ 3119867(1 + 120596

120601) 120588120601= minus119876

120588119898+ 3119867 (1 + 120596

119898) 120588119898= 119876

(10)



120601and



2

Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198601

0 2radic12058211205821205821

212057212058541205821(1 minus

1205822

812058512057221205821)

1612058521205724

(12058221205822

1+ 21205851205722) (1205822120582

1minus 81205851205722)

1205822gt1205851205722(41205821minus 1)

1205822

1

(1 + radic1 minus321205821

(1 minus 41205821)2)

or

1205822lt1205851205722(41205821minus 1)

1205822

1


(1 minus 41205821)2)

1198602

0 2radic12058221205821205822

212057212058541205822(1 minus

1205822

812058512057221205822)

1612058521205724

(12058221205822

2+ 21205851205722) (1205822120582

2minus 81205851205722)

1205822gt1205851205722(41205822minus 1)

1205822

2

(1 + radic1 minus321205822

(1 minus 41205822)2)

or

1205822lt1205851205722(41205822minus 1)

1205822

2


(1 minus 41205822)2)



1198601




120585 ge 120582281205722

1198602




120585 ge 120582281205722



119898with 0 lt 120574 lt 2




119909 equiv

120601


radic119881



Ω120601equiv

120588120601

31198672= 120583minus21199102(1 + 3119909

2) minus 2120585119906

2 (12)

Ω119898equiv120588119898

31198672= 1 minus Ω

120601 (13)


120596120601equiv

119875120601

120588120601

=


120583minus21199102 (1 + 31199092) minus 21205851199062

(14)


119904 = minus

1198672= (2120585119906

2+ 1)minus1

sdot [5radic3120572120585119906119909119910 + 6120583minus211990921199102

minus3

2120574120583minus21199102(1 + 3119909

2)] +

3120574

2

(15)






120596tot equiv119875120601+ 119875119898

120588120601+ 120588119898

= 120583minus21199102(41199092minus 120574 (1 + 3119909

2))

+ 2120585119906 [5radic3

3120572119909119910 + 119906(120574 minus

2

3119904)]

+ 120574 minus 1

(16)


1198672=1

2+3

2120596tot

=3

2120583minus21199102(41199092minus 120574 (1 + 3119909

2))

+ 120585119906 [5radic3120572119909119910 + 119906 (3120574 minus 2119904)] +3120574

2minus 1

(17)


110

109

108

107

106

105

104

103

102

101

y(t)

0 005 010 015 020

x(t)

(a)

0 005 010 015 020

x(t)

003

002

004

005

006

007

008

u(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

003002 004 005 006 007 008

u(t)

(c)




1199091015840=radic3

2[1205821199092119910 +

1

2120582]2 (1 + 31199094) 119910


1199101015840= (minus

radic3

2120582119909119910 + 119904)119910

1199061015840=radic3120572119909119910

2

(18)




110

109

108

107

106

105

104

103

102

101

y(t)

minus005 0 005 010 015 020

x(t)

(a)

007

009

008

011

010

013

012

015

014

u(t)

minus005 0 005 010 015 020

x(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

007 008 010 012 014009 011 013 015

u(t)

(c)






119909119888+ 120575119909 119910 rarr 119910

119888+ 120575119910 and 119906 rarr 119906

119888+ 120575119906 about the





109

107105

103

101 y(t)

x(t)

u(t)

008

007

006

005

004

003

002

0005

010015

020

(a)

110

107105 y

(t)

u(t)

minus005 0 005 010 015 020

x(t)

015

014

013

012

011

010

009

008

(b)






120601and equation



1and 119860



1and 119860





1is a saddle point

for 120572 gt 0 and 120585120582 gt 0






2 Note




1and 119860

2are






1 1198612 1198613 and 119861















110

109

108

107

106

105

104

103

102

101

y(t)

0 01 02

x(t)

minus01

(a)

0 01 02

x(t)

minus01

024

022

020

018

016

014

012

010

008

u(t)

(b)

110107105103101 102 104 106 108 109

y(t)

024

022

020

018

016

014

012

010

008

u(t)

(c)





space trajectories





2(a) 119861

3(b) and 119861

4(c) are

plotted in Figure 7

4 Conclusion




110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)








2when







y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120601and



2

Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198601

0 2radic12058211205821205821

212057212058541205821(1 minus

1205822

812058512057221205821)

1612058521205724

(12058221205822

1+ 21205851205722) (1205822120582

1minus 81205851205722)

1205822gt1205851205722(41205821minus 1)

1205822

1

(1 + radic1 minus321205821

(1 minus 41205821)2)

or

1205822lt1205851205722(41205821minus 1)

1205822

1


(1 minus 41205821)2)

1198602

0 2radic12058221205821205822

212057212058541205822(1 minus

1205822

812058512057221205822)

1612058521205724

(12058221205822

2+ 21205851205722) (1205822120582

2minus 81205851205722)

1205822gt1205851205722(41205822minus 1)

1205822

2

(1 + radic1 minus321205822

(1 minus 41205822)2)

or

1205822lt1205851205722(41205822minus 1)

1205822

2


(1 minus 41205822)2)



1198601




120585 ge 120582281205722

1198602




120585 ge 120582281205722



119898with 0 lt 120574 lt 2




119909 equiv

120601


radic119881



Ω120601equiv

120588120601

31198672= 120583minus21199102(1 + 3119909

2) minus 2120585119906

2 (12)

Ω119898equiv120588119898

31198672= 1 minus Ω

120601 (13)


120596120601equiv

119875120601

120588120601

=


120583minus21199102 (1 + 31199092) minus 21205851199062

(14)


119904 = minus

1198672= (2120585119906

2+ 1)minus1

sdot [5radic3120572120585119906119909119910 + 6120583minus211990921199102

minus3

2120574120583minus21199102(1 + 3119909

2)] +

3120574

2

(15)






120596tot equiv119875120601+ 119875119898

120588120601+ 120588119898

= 120583minus21199102(41199092minus 120574 (1 + 3119909

2))

+ 2120585119906 [5radic3

3120572119909119910 + 119906(120574 minus

2

3119904)]

+ 120574 minus 1

(16)


1198672=1

2+3

2120596tot

=3

2120583minus21199102(41199092minus 120574 (1 + 3119909

2))

+ 120585119906 [5radic3120572119909119910 + 119906 (3120574 minus 2119904)] +3120574

2minus 1

(17)


110

109

108

107

106

105

104

103

102

101

y(t)

0 005 010 015 020

x(t)

(a)

0 005 010 015 020

x(t)

003

002

004

005

006

007

008

u(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

003002 004 005 006 007 008

u(t)

(c)




1199091015840=radic3

2[1205821199092119910 +

1

2120582]2 (1 + 31199094) 119910


1199101015840= (minus

radic3

2120582119909119910 + 119904)119910

1199061015840=radic3120572119909119910

2

(18)




110

109

108

107

106

105

104

103

102

101

y(t)

minus005 0 005 010 015 020

x(t)

(a)

007

009

008

011

010

013

012

015

014

u(t)

minus005 0 005 010 015 020

x(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

007 008 010 012 014009 011 013 015

u(t)

(c)






119909119888+ 120575119909 119910 rarr 119910

119888+ 120575119910 and 119906 rarr 119906

119888+ 120575119906 about the





109

107105

103

101 y(t)

x(t)

u(t)

008

007

006

005

004

003

002

0005

010015

020

(a)

110

107105 y

(t)

u(t)

minus005 0 005 010 015 020

x(t)

015

014

013

012

011

010

009

008

(b)






120601and equation



1and 119860



1and 119860





1is a saddle point

for 120572 gt 0 and 120585120582 gt 0






2 Note




1and 119860

2are






1 1198612 1198613 and 119861















110

109

108

107

106

105

104

103

102

101

y(t)

0 01 02

x(t)

minus01

(a)

0 01 02

x(t)

minus01

024

022

020

018

016

014

012

010

008

u(t)

(b)

110107105103101 102 104 106 108 109

y(t)

024

022

020

018

016

014

012

010

008

u(t)

(c)





space trajectories





2(a) 119861

3(b) and 119861

4(c) are

plotted in Figure 7

4 Conclusion




110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)








2when







y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




110

109

108

107

106

105

104

103

102

101

y(t)

0 005 010 015 020

x(t)

(a)

0 005 010 015 020

x(t)

003

002

004

005

006

007

008

u(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

003002 004 005 006 007 008

u(t)

(c)




1199091015840=radic3

2[1205821199092119910 +

1

2120582]2 (1 + 31199094) 119910


1199101015840= (minus

radic3

2120582119909119910 + 119904)119910

1199061015840=radic3120572119909119910

2

(18)




110

109

108

107

106

105

104

103

102

101

y(t)

minus005 0 005 010 015 020

x(t)

(a)

007

009

008

011

010

013

012

015

014

u(t)

minus005 0 005 010 015 020

x(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

007 008 010 012 014009 011 013 015

u(t)

(c)






119909119888+ 120575119909 119910 rarr 119910

119888+ 120575119910 and 119906 rarr 119906

119888+ 120575119906 about the





109

107105

103

101 y(t)

x(t)

u(t)

008

007

006

005

004

003

002

0005

010015

020

(a)

110

107105 y

(t)

u(t)

minus005 0 005 010 015 020

x(t)

015

014

013

012

011

010

009

008

(b)






120601and equation



1and 119860



1and 119860





1is a saddle point

for 120572 gt 0 and 120585120582 gt 0






2 Note




1and 119860

2are






1 1198612 1198613 and 119861















110

109

108

107

106

105

104

103

102

101

y(t)

0 01 02

x(t)

minus01

(a)

0 01 02

x(t)

minus01

024

022

020

018

016

014

012

010

008

u(t)

(b)

110107105103101 102 104 106 108 109

y(t)

024

022

020

018

016

014

012

010

008

u(t)

(c)





space trajectories





2(a) 119861

3(b) and 119861

4(c) are

plotted in Figure 7

4 Conclusion




110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)








2when







y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




110

109

108

107

106

105

104

103

102

101

y(t)

minus005 0 005 010 015 020

x(t)

(a)

007

009

008

011

010

013

012

015

014

u(t)

minus005 0 005 010 015 020

x(t)

(b)

110

109

108

107

106

105

104

103

102

101

y(t)

007 008 010 012 014009 011 013 015

u(t)

(c)






119909119888+ 120575119909 119910 rarr 119910

119888+ 120575119910 and 119906 rarr 119906

119888+ 120575119906 about the





109

107105

103

101 y(t)

x(t)

u(t)

008

007

006

005

004

003

002

0005

010015

020

(a)

110

107105 y

(t)

u(t)

minus005 0 005 010 015 020

x(t)

015

014

013

012

011

010

009

008

(b)






120601and equation



1and 119860



1and 119860





1is a saddle point

for 120572 gt 0 and 120585120582 gt 0






2 Note




1and 119860

2are






1 1198612 1198613 and 119861















110

109

108

107

106

105

104

103

102

101

y(t)

0 01 02

x(t)

minus01

(a)

0 01 02

x(t)

minus01

024

022

020

018

016

014

012

010

008

u(t)

(b)

110107105103101 102 104 106 108 109

y(t)

024

022

020

018

016

014

012

010

008

u(t)

(c)





space trajectories





2(a) 119861

3(b) and 119861

4(c) are

plotted in Figure 7

4 Conclusion




110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)








2when







y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




109

107105

103

101 y(t)

x(t)

u(t)

008

007

006

005

004

003

002

0005

010015

020

(a)

110

107105 y

(t)

u(t)

minus005 0 005 010 015 020

x(t)

015

014

013

012

011

010

009

008

(b)






120601and equation



1and 119860



1and 119860





1is a saddle point

for 120572 gt 0 and 120585120582 gt 0






2 Note




1and 119860

2are






1 1198612 1198613 and 119861















110

109

108

107

106

105

104

103

102

101

y(t)

0 01 02

x(t)

minus01

(a)

0 01 02

x(t)

minus01

024

022

020

018

016

014

012

010

008

u(t)

(b)

110107105103101 102 104 106 108 109

y(t)

024

022

020

018

016

014

012

010

008

u(t)

(c)





space trajectories





2(a) 119861

3(b) and 119861

4(c) are

plotted in Figure 7

4 Conclusion




110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)








2when







y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




110

109

108

107

106

105

104

103

102

101

y(t)

0 01 02

x(t)

minus01

(a)

0 01 02

x(t)

minus01

024

022

020

018

016

014

012

010

008

u(t)

(b)

110107105103101 102 104 106 108 109

y(t)

024

022

020

018

016

014

012

010

008

u(t)

(c)





space trajectories





2(a) 119861

3(b) and 119861

4(c) are

plotted in Figure 7

4 Conclusion




110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)








2when







y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




110

108

106

104

102

100

y(t)

minus005 0 005 010 015 020

x(t)

(a)u(t)

minus005 0 005 010 015 020

x(t)

minus002

minus004

008

006

004

002

0

(b)

110108106104102100

y(t)

u(t)

minus002

minus004

008

006

004

002

0

(c)








2when







y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




y(t)

x(t)

323

322

321

320

319

318

317

minus0004 minus0002 0 000600040002

(a)u(t)

x(t)

3025

3020

3015

3010

3005

minus0001 0001 0003 0005 0007

(b)

y(t)

u(t)

3025

3020

3015

3010

3170 3175 3180 3185 3190

3005

(c)









u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




u(t)

minus01 0 01 02

x(t)

110

107

105104103

101

y(t)

024

022

020

018

016

014

012

010

008

(a)u(t)

minus002

minus004

008

006

004

002

0

minus005 0 005 010 015 020

x(t)

110

108

106

104

102

100

y(t)

(b)

y(t)

3170

3175

31803185

3190

minus00020

00020004

0006

x(t)

u(t)

3025

3020

3015

3010

3005

(c)


2(a) 119861

3(b) and 119861

4(c)The values



120601and the


1= (4120572120585+radic1612057221205852 minus 21205822120585)2120582120585 and 120579


Label (119909119888 119910119888 119906119888) Ω

120601120596120601

Acceleration

1198611

02radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198612

02radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572(120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)

1198613

0 minus2radic2120582120572120585120579

1

120582 1205791

21205851205791(4120572

120582minus 1205791)

2 (1205851205793

1+ 2120572120582)

(1205791minus 4120572120582) (1 + 2120585120579

2

1)

120582 gt8120572 (120585120579

2

1minus 1)

1205791(81205851205792

1+ 1)

1198614

0 minus2radic2120582120572120585120579

2

120582 1205792

21205851205792(4120572

120582minus 1205792)

2 (1205851205793

2+ 2120572120582)

(1205792minus 4120572120582) (1 + 2120585120579

2

2)

120582 gt8120572 (120585120579

2

2minus 1)

1205792(81205851205792

2+ 1)



119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119898120601


1198611


120582 gt 0 and 120585 ge 120582281205722

1198612



120582 gt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198613



120582 lt 0 andfor all 120585 lt 0

or120585 ge 120582

281205722

1198614


120582 lt 0 and 120585 ge 120582281205722

Appendix




as follows

11987211= 3]2119888(radic3

2120582119909119888119910119888(21199092

119888+ ]2119888(1 + 3119909

4

119888)) minus 6120583

minus2

119888]21198881199092

119888


119888) + radic3120582119909

119888119910119888minus 3 +M

11

11987212=radic3

4(120582 (2119909

2

119888+ ]2119888(1 + 3119909

4

119888)) + 8120572120585119906

119888]2119888119910minus2

119888) +M

12

11987213= minus2radic3120572120585]2

119888119910minus1

119888+M13

11987221=

21199102

119888(radic3120572120585119906

119888+ 3119909119888119910119888]minus2119888)

(21205851199062119888+ 1)


2

119888

2

11987222=

2119910119888(minus (94) 120583

minus4

119888119910119888+ 5radic3120572120585119909

119888119906119888)

(21205851199062119888+ 1)

minus radic3120582119909119888119910119888+3

2

11987223=

61205851199061198881199102

119888(minus (10radic33) 120572120585119906

119888119909119888+ 120583minus4

1198881199102

119888)

(21205851199062119888+ 1)2

+5radic3120572120585119909

1198881199102

119888

21205851199062119888+ 1

11987231=radic3120572119910

119888

2

11987232=radic3120572119909

119888

2

11987233= 0

(A1)




119888119909119888119910119888

M12= 2radic3120573120583

minus2

119888(1 + 3119909

2

119888)

M13= minus4radic3120573120585119906

119888119910minus1

119888

(A2)




References































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics

















































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



Documents

Research Article Dynamics of Generalized Tachyon Field in ...downloads.hindawi.com/journals/ahep/2015/283273.pdf · Research Article Dynamics of Generalized Tachyon Field in Teleparallel