Numerical Chaotic Behavior of the Fractional Rikitake System

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1 ISSN 7-7, England, UK World Journal of Modelling and Simulaion Vol. 9 ) No., pp. -9 Numerical Chaoic Behavior of he Fracional Rikiake Ssem Mohammad Javidi, Nema Namoradi Deparmen of Mahemaics, Facul of Sciences, Rai Universi, 79 Kermanshah, Iran Received April, Revised Sepember, Acceped April ) Absrac. In his paper, a numerical soluion for he ssem described b a generalied fracional Rikiake ssem is presened. he firs sep in he proposed procedure is represen he fracional order Rikiake ssem as an equivalen ssem of ordinar differenial equaions. In he second sep, we solved he ssem obained in he firs sep b using he well known fourh order Runge-Kua mehod. We derived some chaoic behavior of he fracional Rikiake ssem. Kewords: fracional Rikiake ssem, ordinar differenial equaions, dnamic of ssem Inroducion his paper concern he numerical soluion of fracional Rikiake ssem which have he following form d α d = µ +, α d β = µ + a), ) d β =, subjec o he iniial condiions d γ d γ δ) = δ, δ) = δ, δ) = δ, where α, β, γ, δ, δ and δ are real numbers. he Rikiake ssem [, ] been widel invesigaed in he las ears. his ssem is a model which aemps o eplain he irregular polari swiching of he Earh s magneic field. Bu inervals among such geomagneic polari reversals are highl irregular. hus while heir average is abou 7. ears, here are inervals as long as.7 ears wihou polari change, bu wih large deviaions of he poles from acual posiions. he mechanism of generaing he geomagneic field is eplained b he dnamo heor. Man auhors worked on his ssems. udoran and Grban [] proposed some relevan dnamical properies of he Rikiake dnamo ssem from he Poisson geomer poin of view. Also he inroduced some Hamilonian realiaions of a paricular case of he Rikiake ssem and sud he ssem from he Poisson geomer poin of view. Machuca e al [] inroduced snchroniaion problem via nonlinear observer design. A new eponenial polnomial observer for a class of nonlinear oscillaors is proposed, which is robus agains oupu noises. In [], he auhors presens a new -D auonomous chaoic ssem, which is opologicall nonequivalen o he original Loren and all Loren-like ssems. Of paricular ineres is ha he chaoic ssem can generae double-scroll chaoic aracors in a ver wide parameer domain wih onl wo sable equilibria. Also he eisence of singularl degenerae heeroclinic ccles for a suiable choice of he parameers is invesigaed. he Rikiake ssem [] as nonlinear dnamical ssems in geomagneism can be sudied based on he KCC-heor and he unified field heor. Especiall, he behavior of he magneic field of he Rikiake ssem is represened in he elecrical ssem projeced from he elecromechanical unified ssem. Corresponding auhor. address: mo javidi@ahoo.com. Published b World Academic Press, World Academic Union

2 World Journal of Modelling and Simulaion, Vol. 9 ) No., pp. -9 B using he Poincaré compacificaion for polnomial vecor fields in R, Llibre and Messias [] sudied he dnamics of he Rikiake ssem a infini, and show ha here are orbis which escape o, or come from, infini, insead of going owards he aracor. Moreover we sud, for paricular values of he parameers, he flow over wo invarian planes, and describe he global flow of he ssem when i has wo independen firs inegrals and hus is compleel inegrable. he global analsis performed, allows us o give a numerical descripion of he creaion of Rikiake aracor. In [], he aracors snhesis algorihm for a class of dissipaive dnamical ssems wih hperbolic equilibria proposed o model numericall an aracor of he Rikiake ssem. he auhor of [9] anale a paricular case of he Rikiake wo-dnamo ssem from he sabili heor poin of view. More eacl, a special pe of dissipaive ssem he so-called meriplecic ssem) in such a wa ha each Lapunov sable equilibrium sae of he unperurbed ssem generaes a small one dimensional aracing neighborhood for he dissipaive ssem. presen an analical sud of he sabili of he equilibria of he Rikiake ssem. Braga [] proved ha he wo non-hperbolic equilibria of he Rikiake ssem are unsable for all posiive values of he parameers. he auhors of [] proposed a maser/slave scheme in conjuncion wih an adapive conrol echnique for snchroniaion and idenificaion of he uncerain chaoic Rikiake ssem, where onl a parial knowledge of he sae is available. Vincen [], applid he acive conrol o snchronie wo idenical Rikiake aracors; which describes he chaoic behavior ehibied b he reversal of dipole polari. In his paper, we used an efficien numerical mehod based on a ransformaion of he original fracion Rikiake ssem ino a ssem of Ordinar Differenial Equaions ODEs). Preliminaries Definiion. he Riemann-Liouville fracional inegral operaor of order α >, of funcion f L R + ) is defined as where Γ ) is he Euler gamma funcion. I α f) = s) α fs)ds, Γ α) Definiion. he Riemann-Liouville fracional derivaive of order α >, n < α < n, n N is defined as D α f) = d ) n s) Γ n α) n α fs)ds, d where he funcion f) have absoluel coninuous derivaives up o order n ). he iniial value problem relaed o Definiion is D α ) = f, )), ) = + =, ) where < α < and D α = D α. In [7], he following resuls abou he eisence and uniqueness of soluions for Eq. ) are furher presened. heorem. Assume ha R : [, ] [ δ, + δ] wih some > and some δ >, and le he funcion f : R R be coninuous. Furhermore, define := min, δγ α+)) } α f, hen here eiss a funcion : [, ] R solving he iniial value problem Eq. ). Noice ha f is he norm of funcion f. heorem. Assume ha R : [, ] [ δ, + δ] wih some > and some δ >, and le he funcion f : R R be bounded on R and fulfill a Lipschi condiion wih respec o he second variable, i.e. WJMS for subscripion: info@wjms.org.uk

3 M. Javidi & N. Nmoradi: Numerical Chaoic Behavior of he Fracional f, ) f, ) L wih some consan number L > independen of,,. hen denoing as heorem, here eiss a mos one funcion : [, ] R solving he iniial value problem Eq. ). Furhermore, he above definiion in one dimension can naurall be generalied o he case of muliple dimensions. ha is, le X) = ), ),, n )) R n and α = α, α,, α n ) R n, < α i <, i =,,, n. he n-dimension FODE is described as follows: D α X u) X) = du F, X)), ) Γ α) u) α where X u) Γ α) u) α du = Dα ), D α ),, D αn n )) and F, X)) = f, ), ),, n )) f, ), ),, n )). f n, ), ),, n )). he resuls of heorems, can be easil generalies o he iniial value problem of he vecor-value funcions Eq. ). Epansion formulas for he fracional derivaives In order o solve Eq., we shall use a numerical mehod inroduced b Aanackovic and Sankovic [] o solve he single linear FDE. Also he same auhors [] developed he mehod o solve he nonlinear FDE. In his paper we developed he same mehod o solve he fracional. Now we eplain he mehod. I is well known ha for an analic funcion f) he Capuo fracional derivaive D α f), < α < defined as D α f) = Γ α) where f ) τ) denoe he firs derivaive of fτ). From Eq., we obain D α f) = B using he binomial formula, we have [ f ) ) Γ α) α + τ) α = α τ ) α = α p= = α p= B subsiuion of Eq. ) ino Eq. ), we have α p τ) α f ) τ)dτ, ) Γ p + α) Γ α )p! τ) α f ) τ)dτ ]. ) ) ) p τ ) p ) ) τ p, τ <. WJMS for conribuion: submi@wjms.org.uk

4 World Journal of Modelling and Simulaion, Vol. 9 ) No., pp. -9 where D α f) = [ f ) ) Γ α) α + α We can rewrie Eq. 7) as he following form [ D α f ) ) f) = Γ α) α + α Using inegraion b par, we obain τ p f ) τ)dτ = p f ) ) p p= p= τ p f ) τ)dτ Γ p + α) τ ] Γ α )p! )p f ) τ)dτ. 7) Γ p + α) Γ α )p! p = p f ) ) p p f) + pp ) B subsiuion of Eq. 9) ino Eq. ), we obain D α f ) [ ) ] Γ p + α) f) = Γ α) α + Γ α )p! p= [ α α f) + Γ p + α) f) Γ α )p )! α wih he following properies p= V p f)) = p ) ] τ p f ) τ)dτ. ) τ p fτ)dτ, p. 9) )]} + V pf)) p +α. ) τ p fτ)dτ, p =,,, ) d d V pf) = p ) p f), p =,,. ) We approimae D α f) b using M erms in sums appearing in Eq. ) as follows D α f) Γ α) f ) ) α [ + p= ] Γ p + α) Γ α )p! [ α M α f) + Γ p + α) f) Γ α )p )! α p= )]} + V pf)) p +α. ) B seing ) = ), M+ ) = ), M+ ) = ) and p ) = V p )), M+p ) = V p )), M+p ) = V p )), p =,,, we can rewrie Rikiake ssem Eq. ) as he following form ordinar differenial equaions ) [ α Πp, α, M) α α ) + p= ) Ξp, α) α + )] p) p +α = ψα) µ ) + M+ ) M+ )), M+ ) [ β M+ ) β Πp, β, M) β M+ ) + Ξp, β) β + )] M+p) p +β = ψβ) µ M+ ) + M+ ) a) )), M+ ) [ γ M+ ) γ Πp, γ, M) γ M+ ) + Ξp, γ) γ p= p= + )] M+p) p +γ = ψγ) M+ ) )), ) WJMS for subscripion: info@wjms.org.uk

5 M. Javidi & N. Nmoradi: Numerical Chaoic Behavior of he Fracional where p ) = p ) τ p )τ)dτ, p =,,, M, M+p ) = p ) τ p M+ )τ)dτ, p =,,, M, M+p ) = p ) τ p M+ )τ)dτ, p =,,, M, Πp, γ, M) = + M Ξp, γ) = ψγ) = Γ γ). p= Γ p +γ) Γ γ )p )!, Γ p +γ) Γ γ )p!, ) Now we can rewrie Eq. ) and Eq. ) as he following form ) = α Πp,α,M) α α ) + M Ξp, α) ) α p= + p) p +α ) + ψα) µ ) + M+ ) M+ )) p) = p ) ), p =,,, M, M+ ) = β β Πp,β,M) β M+ ) + M Ξp, β) ) p= β } + ψβ) µ M+ ) + M+ ) a) )) wih he following iniial condiions M+p ) = p ) M+), p =,,, M, M+ ) = γ γ Πp,γ,M) γ M+ ) + M Ξp, γ) } p= + ψγ) M+ ) )) M+p ) = p ) M+), p =,,, M, δ) = δ, p δ) =, p =,,, M, M+ δ) = δ, M+p δ) =, p =,,, M, M+ δ) = δ, M+p δ) =, p =,,, M, we can rewrie he Eq. ) and Eq. 7) as he following form ) ). M ) ) γ S ) S S,M S ) S S,M =... S M, ) S M, S M,M + ψα) M+ ) M+ ) α Πp,α,M). β Πp,β,M) ψβ) M+) a) )) γ Πp,γ,M) ψγ) M+) )) } ) + M+p) p +β ) + M+p) p +γ ) ). M ), ) 7) ) where WJMS for conribuion: submi@wjms.org.uk

6 World Journal of Modelling and Simulaion, Vol. 9 ) No., pp. -9 S ) = α Πp,α,M) + M Ξp, α) µψα), α p= S p ) = α Ξp,α) Πp,α,M), p =,,, M, p= p ), j =, p =,,, M, S pj ) =, < j M, p =,,, M, j =,,, M, 9) S M+, ) = S M+,p ) = S M+p,M+j ) = β Πp,β,M) + M β Πp,β,M) p= p= Ξp, β) β µψβ), Ξp,β) p +β, p =,,, M, p ), j =, p =,,, M,, < j M, p =,,, M j =,,, M, ) S M+, ) = S M+,p ) = S M+p,M+j ) = γ Πp,γ,M) + M γ Πp,γ,M) p= p= Ξp, γ) γ µψγ), Ξp,γ) p +γ, p =,,, M, p ), j =, p =,,, M,, < j M, p =,,, M j =,,, M. ) In he ne secion we solve he ssem of ordinar differenial Eq. ) wih he iniial condiions Eq. 7) b using fourh order. ) ) ) Fig.. Chaoic aracor of Rikiake ssem a = [.,.9, Figure : Chaoic aracor of Rikiake ssem a Q.], = [ , =[:; :9; :] ; P.99], µ =, M = and a = =[:99::99; :99] ;μ= ;M = and a = Numerical Simulaion WJMS for subscripion: info@wjms.org.uk No we consider he numerical soluion of Rikiake ssem b using he well known Runge KuaRK) mehod of order fourh. In Figure,we displa he numerical soluion of Rikiake ssem. he parameers in he have been se o μ =;M = and a Q =. he iniial condiions [););)] = =[:; :9; :] ; P =[ff; fi; fl] =[:99::99; :99] have been used. In numerical soluion has

7 M. Javidi & N. Nmoradi: Numerical Chaoic Behavior of he Fracional Numerical simulaion No we consider he numerical soluion of Rikiake ssem b using he well known Runge-Kua RK) mehod of order fourh. In Fig., we displa he numerical soluion of Rikiake ssem. he parameers have been se o µ =, M = and a =. he iniial condiions [δ), δ), δ)] = = [.,.9,.], = [α, β, γ] = [ ,.99] have been used. he numerical soluion has been approimaed from = o = and he numerical soluions have been saved and ploed a inervals of., i.e. a imes =,.,.,,. In all numerical runs, he soluion has been approimaed a =. = δ. ) ) ) Fig.. Chaoic aracor of Rikiake ssem a = [.,., Figure : Chaoic aracor of Rikiake ssem a Q.], = [.9..9, = [ :; :; :] ; P.9], µ =., M = and a =.9 = [:9::9; :9] ; μ =:;M = and a =:9 7 9 ) 7 ) ) Fig.. Chaoic aracor of Rikiake ssem a = [,.,.] Figure : Chaoic aracor of Rikiake ssem a Q, = [ , = [; :; :] ; P.97], µ =., M = and a = = [:97::97; :97] ; μ = :;M = and a = WJMS for conribuion: submi@wjms.org.uk

8 World Journal of Modelling and Simulaion, Vol. 9 ) No., pp In Fig., we displa he numerical soluion of Rikiake ssem. he parameers have been se o µ =., M = and a =.9. he iniial condiions = [.,.,.], = [.9..9,.9] have been used. he numerical soluion has been approimaed from = o = and he numerical soluions have been saved and ploed a inervals of., i.e. a imes =,.,.,,. In all numerical runs, he soluion has been approimaed a =.. In Fig., we displa he numerical soluion of Rikiake ssem. he parameers in he have been se o µ =., M = and a =. he iniial condiions = [,.,.], = [ ,.97] have been used. he numerical soluion has been approimaed from = o = and he numerical soluions have been saved and ploed a inervals of., i.e. a imes =,.,.,,. In all numerical runs, he soluion has been approimaed a =.. ) ) ) Fig.. Chaoic aracor of Rikiake ssem a = [.,., Figure : Chaoic aracor of Rikiake ssem a Q.], = [.9..9, =[:; :; :] ; P.9], µ =., M = and =[:9::9; :9] ;μ= a = :;M = and a = imes = ; :; :; ;. In all numerical runs, he soluion has been approimaed a = :. In Figure, we displa he numerical soluion of Rikiake ssem wih μ = :;M = ; P =[:9::9; :9] ; =:; = and a =. he iniial condiions ) = :;) = :;) = : have been used. In Figure, we displa he numerical soluion of Rikiake ssem wih μ =;M = ; P = [:99::99; :99] ; = :; = and a =. he iniial condiions ) = :;) = :9;) = : have been used. In Fig., we displa he numerical soluion of Rikiake ssem. he parameers in he have been se o µ =., M = and a =. he iniial condiions = [.,.,.], = [.9..9,.9] have been used. he numerical soluion has been approimaed from = o = and he numerical soluions have been saved and ploed a inervals of., i.e. a imes =,.,.,,. In all numerical runs, he soluion has been approimaed a =.. In Fig., we displa he numerical soluion of Rikiake ssem wih µ =., M =, = [.9..9,.9] Conclusions, =., = and a =. he iniial condiions ) =., ) =., ) =. have been used. In his paper,we applied an epansion formula for fracional derivaives given b ). I conains In Fig., we displa he numerical soluion of Rikiake ssem wih µ =, M =, = ineger derivaives up o he finie order k and ime momens of k-h derivaive, given b ). [ , Firsl,.99] b, using = Eq.., ) = we conver and a he = fracional. he iniial differenial condiions equaion ) o= a ssem., ) of ordinar =.9, ) =. have beendifferenial used. equaions of ineger order. We use fourh order Runge-Kua formula for he numerical inegraion of he ssem of ODEs. we solved he ssem obained in he firs sep b using he well known fourh order Runge-Kua mehod. We derived some chaoic behavior of he fracional Rikiake ssem. Conclusions In his References paper, we applied an epansion formula for fracional derivaives given b Eq. ). I conains ineger derivaives up o he finie order k and ime momens of k-h derivaive, given b Eq. ). Firsl, b using Eq. ) [] Ramona we conver A. andhe udoran. fracional On asmpoicall differenial equaion sabiliing ohe a ssem rikiakeof wo-disk ordinar dnamo differenial dnamics. equaions of Nonlinear Analsis: Real World Applicaions,):,. WJMS for subscripion: info@wjms.org.uk

9 M. Javidi & N. Nmoradi: Numerical Chaoic Behavior of he Fracional Figure : Numerical soluion of Rikiake ssem a Q =[:; :; :] ; P =[:9::9; :9] ; = :;= μ =:;M =anda = Fig.. Numerical soluion of Rikiake ssem a = [., Figure : Numerical soluion of Rikiake ssem a Q.,.], = [.9..9, =[:; :; :] ; P.9], =., = µ =., M = and a = =[:9::9; :9] ; = :;= μ =:;M =anda = Fig.. Numerical soluion of Rikiake ssem a = [.,.9,.], µ =, M = and a =. Figure : Numerical soluion of Rikiake ssem a Q =[:; :9; :] ; P =[:99::99; :99] ; = 7 :;= μ =;M =anda =. = [ ,.99], =., =. ineger order. We use fourh order Runge-Kua formula for he numerical inegraion of he ssem of ODEs.. we solved he ssem obained in he firs sep b using he well known fourh order Runge-Kua mehod. We derived some chaoic behavior of he fracional Rikiake ssem.. References Figure : Numerical soluion of Rikiake ssem a Q =[:; :9; :] ; P =[:99::99; :99] ; = [] A. Carlos, :;= M. Rafael, μ =;M e al. =anda Snchroniaion = and parameer esimaions of an uncerain rikiake ssem. Phsics Leers A,, 7):. []. Aanackovic, B. Sankovic. An epansion formula for fracional derivaives and is applicaion. Fracional Calculus and Applied Analsis,, 7): 7. []. Aanackovic, B. Sankovic. On a numerical scheme for solving differenial equaions of fracional order. Mechanics Research Communicaions,, 7): 9. [] D. Braga, F. Dias. On he sabili of he equilibria of he rikiake ssem. Phsics Leers A,, 7):. [] M. Danca, S. Codreanu. Modeling numericall he rikiake s aracors b parameer swiching. Journal of he Franklin Insiue,, ). [] J. Maa-Machuca, R. Aguilar. An eponenial polnomial observer for snchroniaion of chaoic ssems. Communicaions in Nonlinear Science and Numerical Simulaion,, ):. [7] K. Diehelm, K. Diehelm, e al. Analsis of fracional differenial equaions. Journal of Mahemaical Analsis and Applicaions,, : 9. [] J. Llibre, M. Messias. Global dnamics of he rikiake ssem. Phsica D: Nonlinear Phenomena, 9, ):. [9] R. udoran. On asmpoicall sabiliing he rikiake wo-disk dnamo dnamics. Nonlinear Analsis: Real World Applicaions,, ):. []. Rikiake. Oscillaions of a ssem of disk dnamos. Mahemaical Proceedings of he Cambridge Philosophical Socie, 9, ): 9. [] R. udoran, A. Grban. A hamilonian look a he rikiake wo-disk dnamo ssem. Nonlinear Analsis: Real World Applicaions,, ): 9. [] C. Valls. Rikiake ssem: analic and darbouian inegrals. Proceedings of he Roal Socie of Edinburgh, Secion: A Mahemaics,, ): 9. WJMS for conribuion: submi@wjms.org.uk

10 World Journal of Modelling and Simulaion, Vol. 9 ) No., pp [] U. Vincen. Snchroniaion of rikiake chaoic aracor using acive conrol. Phsics Leers A,, -):. [] Z. Wei, Q. Yang. Dnamical analsis of a new auonomous -d chaoic ssem onl wih sable equilibria. Nonlinear Analsis: Real World Applicaions,, ):. []. Yajima, H. Nagahama. Geomerical unified heor of rikiake ssem and kcc-heor. Nonlinear Analsis: heor, Mehods & Applicaions, 9, 7): e e. WJMS for subscripion: info@wjms.org.uk