Numerical analysis of a simplest fractional-order hyperchaotic system
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摘要
In this paper, a simplest fractional-order hyperchaotic(SFOH) system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system, which possesses seven terms without any quadratic or higher-order polynomials. The numerical solution of the SFOH system is investigated based on the Adomian decomposition method(ADM). The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics. Dynamics of this system are demonstrated by means of phase portraits, bifurcation diagrams, Lyapunov exponent spectrum(LEs) and Poincaré section. The results show that the system has a wide chaotic range with order change, and large Lyapunov exponent when the order is very small, which indicates that the system has a good application prospect. Besides, the parameter a is a partial amplitude controller for the SFOH system. Finally, the system is successfully implemented by digital signal processor(DSP). It lays a foundation for the application of the SFOH system.
In this paper, a simplest fractional-order hyperchaotic(SFOH) system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system, which possesses seven terms without any quadratic or higher-order polynomials. The numerical solution of the SFOH system is investigated based on the Adomian decomposition method(ADM). The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics. Dynamics of this system are demonstrated by means of phase portraits, bifurcation diagrams, Lyapunov exponent spectrum(LEs) and Poincaré section. The results show that the system has a wide chaotic range with order change, and large Lyapunov exponent when the order is very small, which indicates that the system has a good application prospect. Besides, the parameter a is a partial amplitude controller for the SFOH system. Finally, the system is successfully implemented by digital signal processor(DSP). It lays a foundation for the application of the SFOH system.
In this paper, a simplest fractional-order hyperchaotic(SFOH) system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system, which possesses seven terms without any quadratic or higher-order polynomials. The numerical solution of the SFOH system is investigated based on the Adomian decomposition method(ADM). The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics. Dynamics of this system are demonstrated by means of phase portraits, bifurcation diagrams, Lyapunov exponent spectrum(LEs) and Poincaré section. The results show that the system has a wide chaotic range with order change, and large Lyapunov exponent when the order is very small, which indicates that the system has a good application prospect. Besides, the parameter a is a partial amplitude controller for the SFOH system. Finally, the system is successfully implemented by digital signal processor(DSP). It lays a foundation for the application of the SFOH system.
引文
[1]Y.Guan,Z.Zhao,X.Lin,On the existence of solutions for impulsive fractional differential equations,Adv.Math.Phys.(2017)1-12.
[2]A.S.Deshpande,V.Daftardargejji,On disappearance of chaos in fractional systems,Chaos.Solitons.Fract.102(2017)119-126.
[3]M.D.Ortigueira,J.A.T Machado,What is a fractional derivative?,J.Comput.Phys.293(2015)4-13.
[4]D.Peng,K.H.Sun,A.O.Alamodi,Dynamics analysis of fractional-order permanent magnet synchronous motor and its DSPimplementation,Int.J.Mode.Phys.B 33(2019)1950031.
[5]V.Daftardar-Gejji,S.Bhalekar,Chaos in fractional ordered Liu system,Comp.Math.Appl.59(2010)1117.
[6]X.Y.Wang,M.J.Wang,Dynamic analysis of the fractional order Liu system and its synchronization,Chaos 17(2007)033106.
[7]H.Sun,A.Abdelwahab,B.Onaral,Linear approximation of transfer function with a pole of fractional power,IEEE Trans.Auto.Cont.29(1984)441.
[8]K.Diethelm,N.J.Ford,A.D.Freed,A predictor-corrector approach for the numerical solution of fractional differential equations,Trans.Numer.Anal.5(1997)1.
[9]G.Adomian,A new approach to nonlinear partial differential equations,J.Math.Anal.Appl.102(1984)420.
[10]D.Cafagna,G.Grassi,Bifurcation and chaos in the fractionalorder Chen system via a time-domain approach,Int.J.Bifur.Chaos 18(2008)1845.
[11]D.Cafagna,G.Grassi,Hyperchaos in the fractional-order R?ssler system with lowest-order,Int.J.Bifur.Chaos 19(2009)339.
[12]Y.X.Xu,K.H.Sun,S.B.He,et al.,Solution and dynamics of a fractional-order 5-d hyperchaotic system with four wings,Eur.Phys.J.Plus.131(2016)186.
[13]L.M.Zhang,K.H.Sun,S.B.He,et al.,Solution and dynamics of a fractional-order 5-D hyperchaotic system with four wings,Eur.Phy.J.Plus.132(2017)31.
[14]X.Y.Wang,Y.L.Zhang,Modified projective synchronization of a fractional-order hyperchaotic system with a single driving variable,Chinese.Phys.B 20(2011)100506.
[15]C.X.Zhu,A novel image encryption scheme based on improved hyperchaotic sequences,Opt.Commun.285(2012)29.
[16]Z.Wang,X.Huang,Y.X.Li,et al.,A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system,Chinese.Phys.B 22(2013)124-130.
[17]X.Y.Wang,M.J.Wang,A hyperchaos generated from Lorenz system,Phys.A 387(2008)3751-3758.
[18]X.Y.Wang,Y.F.Gao,Y.X.Zhang,Hyperchaos Qi system,Int.J.Mod.Phys.B 24(2010)4771-4778.
[19]C.B.Li,J.C.Sprott,W.Thio,et al.,A new piecewise-linear hyperchaotic circuit,IEEE.Trans.Circuits.Syst.II 61(2014)977.
[20]M.F.Danca,Lyapunov exponents of a class of piecewise continuous systems of fractional order,Nonlinear.Dyn.81(2014)227.
[21]Y.J.Niu,X.Y.Wang,M.J.Wang,et al.,A new hyperchaotic system and its circuit implementation,Commun.Nonlinear.Sci.15(2010)3518-3524.
[22]Q.Han,C.X.Liu,L.Sun,et al.,A fractional order hyperchaotic system derived from a Liu system and its circuit realization,Chinese.Phys.B 22(2013)133-138.
[23]A.M.A.El-Sayed,H.M.Nour,A.Elsaid,et al.,Dynamical behaviors,circuit realization,chaos control,and synchronization of a new fractional order hyperchaotic system,Appl.Math.Model.40(2016)3516-3534.
[24]H.H.Wang,K.H.Sun,S.B.He,Dynamic analysis and implementation of a digital signal processor of a fractionalorder Lorenz-Stenflo system based on the Adomian decomposition method,Phys.Scr.90(2015)015206.
[25]S.B.He,K.H.Sun,H.H.Wang,Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system,Entropy 17(2015)8299.
[26]R.Gorenflo,F.Mainardi,Fractional Calculus:Integral and Differential Equations of Fractional Order.In:A.Carpinteri and F.Mainardi,Eds.,Fractals and Fractional Calculus in Continuum Mechanics,Springer Verlag,Wien and New York,1997
[27]R.Caponetto,S.Fazzino,An application of adomian decomposition for analysis of fractional-order chaotic systems,Int.J.Bifurca.Chaos.23(2013)1350050.
[28]S.Momani,K.Al-Khaled,Numerical solutions for systems of fractional differential equations by the decomposition method,Appl.Math.Comput.162(2005)1351.
[29]V.Daftardar-Gejji,H.Jafari,Adomian decomposition:a tool for solving a system of fractional differential equations,J.Math.Anal.Appl.301(2005)508.
[30]C.Li,Z.Gong,D.Qian,On the bound of the Lyapunov exponents for the fractional differential systems,Chaos 20(2010)261.
[31]H.F.Von Bremen,F.E.Udwadia.W.Proskurowski,An efficient QR based method for the computation of Lyapunov exponents,Phys.D.Nonlinear.Pheno.101(1997)1.
[32]K.H.Sun,J.C.Sprott,Periodically forced chaotic system with signum nonlinearity,Int.J.Bifurcat.Chaos 20(2010)1499-1507.
[33]J.Biazar,S.M.Shafiof,A simple algorithm for calculating adomian polynomials,Int.J.Contemp.Math.Sci.2(2007)975-982.
[34]D.Peng,K.H.Sun,S.B.He,et al.,what is the lowest order of the fractional-order nonlinear systems to behave chaotically?,Chaos.Solitons.Fract.119(2019)163-170.
[35]K.H.Sun,X.Liu,C.X.Zhu,The 0-1 test algorithm for chaos and its applications,Chinese.Phys.B 19(2010)200-206.
[36]X.Y.Wang,T.Lin,Q.Xue,A novel colour image encryption algorithm based on chaos,Signal.Process.92(2012)1101-1108.
[37]C.B.Li,W.J.Thio,J.C.Sprott,et al,Linear synchronization and circuit implementation of chaotic system with complete amplitude control,Chinese.Phys.B 26(2017)120501.
[38]C.B.Li,J.C.Sprott,H.Y.Xing,Crisis in amplitude control hides in metastability,Int.J.Bifurcation.Chaos 26(2016)1650233.
[39]Y.J.Zhang,X.Y.Wang,A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice,Inform.Sciences 273(2014)329-351.
[40]K.H.Sun,S.B.He,Y.He,et al.,Complexity analysis of chaotic pseudo-random sequences based on spectral entropy algorithm,Acta.Phys.Sin.62(2013)010501.
[41]S.B.He,K.H.Sun,R.X.Wang,Fractional fuzzy entropy algorithm and the complexity analysis for nonlinear time series,Eur.Phys.J.Spec.Top.227(2018)943-957.
[2]A.S.Deshpande,V.Daftardargejji,On disappearance of chaos in fractional systems,Chaos.Solitons.Fract.102(2017)119-126.
[3]M.D.Ortigueira,J.A.T Machado,What is a fractional derivative?,J.Comput.Phys.293(2015)4-13.
[4]D.Peng,K.H.Sun,A.O.Alamodi,Dynamics analysis of fractional-order permanent magnet synchronous motor and its DSPimplementation,Int.J.Mode.Phys.B 33(2019)1950031.
[5]V.Daftardar-Gejji,S.Bhalekar,Chaos in fractional ordered Liu system,Comp.Math.Appl.59(2010)1117.
[6]X.Y.Wang,M.J.Wang,Dynamic analysis of the fractional order Liu system and its synchronization,Chaos 17(2007)033106.
[7]H.Sun,A.Abdelwahab,B.Onaral,Linear approximation of transfer function with a pole of fractional power,IEEE Trans.Auto.Cont.29(1984)441.
[8]K.Diethelm,N.J.Ford,A.D.Freed,A predictor-corrector approach for the numerical solution of fractional differential equations,Trans.Numer.Anal.5(1997)1.
[9]G.Adomian,A new approach to nonlinear partial differential equations,J.Math.Anal.Appl.102(1984)420.
[10]D.Cafagna,G.Grassi,Bifurcation and chaos in the fractionalorder Chen system via a time-domain approach,Int.J.Bifur.Chaos 18(2008)1845.
[11]D.Cafagna,G.Grassi,Hyperchaos in the fractional-order R?ssler system with lowest-order,Int.J.Bifur.Chaos 19(2009)339.
[12]Y.X.Xu,K.H.Sun,S.B.He,et al.,Solution and dynamics of a fractional-order 5-d hyperchaotic system with four wings,Eur.Phys.J.Plus.131(2016)186.
[13]L.M.Zhang,K.H.Sun,S.B.He,et al.,Solution and dynamics of a fractional-order 5-D hyperchaotic system with four wings,Eur.Phy.J.Plus.132(2017)31.
[14]X.Y.Wang,Y.L.Zhang,Modified projective synchronization of a fractional-order hyperchaotic system with a single driving variable,Chinese.Phys.B 20(2011)100506.
[15]C.X.Zhu,A novel image encryption scheme based on improved hyperchaotic sequences,Opt.Commun.285(2012)29.
[16]Z.Wang,X.Huang,Y.X.Li,et al.,A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system,Chinese.Phys.B 22(2013)124-130.
[17]X.Y.Wang,M.J.Wang,A hyperchaos generated from Lorenz system,Phys.A 387(2008)3751-3758.
[18]X.Y.Wang,Y.F.Gao,Y.X.Zhang,Hyperchaos Qi system,Int.J.Mod.Phys.B 24(2010)4771-4778.
[19]C.B.Li,J.C.Sprott,W.Thio,et al.,A new piecewise-linear hyperchaotic circuit,IEEE.Trans.Circuits.Syst.II 61(2014)977.
[20]M.F.Danca,Lyapunov exponents of a class of piecewise continuous systems of fractional order,Nonlinear.Dyn.81(2014)227.
[21]Y.J.Niu,X.Y.Wang,M.J.Wang,et al.,A new hyperchaotic system and its circuit implementation,Commun.Nonlinear.Sci.15(2010)3518-3524.
[22]Q.Han,C.X.Liu,L.Sun,et al.,A fractional order hyperchaotic system derived from a Liu system and its circuit realization,Chinese.Phys.B 22(2013)133-138.
[23]A.M.A.El-Sayed,H.M.Nour,A.Elsaid,et al.,Dynamical behaviors,circuit realization,chaos control,and synchronization of a new fractional order hyperchaotic system,Appl.Math.Model.40(2016)3516-3534.
[24]H.H.Wang,K.H.Sun,S.B.He,Dynamic analysis and implementation of a digital signal processor of a fractionalorder Lorenz-Stenflo system based on the Adomian decomposition method,Phys.Scr.90(2015)015206.
[25]S.B.He,K.H.Sun,H.H.Wang,Complexity analysis and DSP implementation of the fractional-order Lorenz hyperchaotic system,Entropy 17(2015)8299.
[26]R.Gorenflo,F.Mainardi,Fractional Calculus:Integral and Differential Equations of Fractional Order.In:A.Carpinteri and F.Mainardi,Eds.,Fractals and Fractional Calculus in Continuum Mechanics,Springer Verlag,Wien and New York,1997
[27]R.Caponetto,S.Fazzino,An application of adomian decomposition for analysis of fractional-order chaotic systems,Int.J.Bifurca.Chaos.23(2013)1350050.
[28]S.Momani,K.Al-Khaled,Numerical solutions for systems of fractional differential equations by the decomposition method,Appl.Math.Comput.162(2005)1351.
[29]V.Daftardar-Gejji,H.Jafari,Adomian decomposition:a tool for solving a system of fractional differential equations,J.Math.Anal.Appl.301(2005)508.
[30]C.Li,Z.Gong,D.Qian,On the bound of the Lyapunov exponents for the fractional differential systems,Chaos 20(2010)261.
[31]H.F.Von Bremen,F.E.Udwadia.W.Proskurowski,An efficient QR based method for the computation of Lyapunov exponents,Phys.D.Nonlinear.Pheno.101(1997)1.
[32]K.H.Sun,J.C.Sprott,Periodically forced chaotic system with signum nonlinearity,Int.J.Bifurcat.Chaos 20(2010)1499-1507.
[33]J.Biazar,S.M.Shafiof,A simple algorithm for calculating adomian polynomials,Int.J.Contemp.Math.Sci.2(2007)975-982.
[34]D.Peng,K.H.Sun,S.B.He,et al.,what is the lowest order of the fractional-order nonlinear systems to behave chaotically?,Chaos.Solitons.Fract.119(2019)163-170.
[35]K.H.Sun,X.Liu,C.X.Zhu,The 0-1 test algorithm for chaos and its applications,Chinese.Phys.B 19(2010)200-206.
[36]X.Y.Wang,T.Lin,Q.Xue,A novel colour image encryption algorithm based on chaos,Signal.Process.92(2012)1101-1108.
[37]C.B.Li,W.J.Thio,J.C.Sprott,et al,Linear synchronization and circuit implementation of chaotic system with complete amplitude control,Chinese.Phys.B 26(2017)120501.
[38]C.B.Li,J.C.Sprott,H.Y.Xing,Crisis in amplitude control hides in metastability,Int.J.Bifurcation.Chaos 26(2016)1650233.
[39]Y.J.Zhang,X.Y.Wang,A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice,Inform.Sciences 273(2014)329-351.
[40]K.H.Sun,S.B.He,Y.He,et al.,Complexity analysis of chaotic pseudo-random sequences based on spectral entropy algorithm,Acta.Phys.Sin.62(2013)010501.
[41]S.B.He,K.H.Sun,R.X.Wang,Fractional fuzzy entropy algorithm and the complexity analysis for nonlinear time series,Eur.Phys.J.Spec.Top.227(2018)943-957.