Stability and chaos in the fractional Chen system

Abstract The paper provides a theoretical analysis of some local bifurcations in the fractional Chen system. Contrary to the integer-order case, basic bifurcation properties of the fractional Chen system are shown to be qualitatively different from those described previously for the fractional Lorenz system. Further, the fractional Hopf bifurcation in the Chen system is expressed rigorously with respect to general entry parameters. Based on these observations, some particularities of the fractional dynamics of the Chen system are documented and its chaotic behavior for low derivative orders is discussed.

[1]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[2]  C. Lubich.  Discretized fractional calculus , 1986 .

[3]  Elsayed Ahmed,et al.  On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems , 2006 .

[4]  Mohammad Saleh Tavazoei,et al.  A proof for non existence of periodic solutions in time invariant fractional order systems , 2009, Autom..

[5]  Guanrong Chen,et al.  A note on the fractional-order Chen system , 2006 .

[6]  Mohd. Salmi Md. Noorani,et al.  Homotopy analysis method for solving fractional Lorenz system , 2010 .

[7]  I. Podlubny.  Fractional differential equations , 1998 .

[8]  N. D. Cong,et al.  On fractional lyapunov exponent for solutions of linear fractional differential equations , 2014 .

[9]  Roberto Garrappa,et al.  Trapezoidal methods for fractional differential equations: Theoretical and computational aspects , 2015, Math. Comput. Simul..

[10]  Nikolay V. Kuznetsov,et al.  Matlab Code for Lyapunov Exponents of Fractional-Order Systems , 2018, Int. J. Bifurc. Chaos.

[11]  Shao-Fang Wen,et al.  Dynamical response of Mathieu–Duffing oscillator with fractional-order delayed feedback , 2017 .

[12]  Zhen Wang,et al.  Dynamic Analysis of a Delayed Fractional-Order SIR Model with Saturated Incidence and Treatment Functions , 2018, Int. J. Bifurc. Chaos.

[13]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[14]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[15]  Junwei Lu,et al.  Stability and bifurcation of a delayed generalized fractional-order prey-predator model with interspecific competition , 2019, Appl. Math. Comput..

[16]  A. Alomari.  A novel solution for fractional chaotic Chen system , 2015 .

[17]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[18]  Changpin Li,et al.  Fractional dynamical system and its linearization theorem , 2013 .

[19]  Guanrong Chen,et al.  Complex dynamics in Chen’s system☆ , 2006 .

[20]  Guanrong Chen,et al.  Local bifurcations of the Chen System , 2002, Int. J. Bifurc. Chaos.

[21]  Giuseppe Grassi,et al.  Bifurcation and Chaos in the fractional-Order Chen System via a Time-Domain Approach , 2008, Int. J. Bifurc. Chaos.

[22]  A. K. Alomari,et al.  A new analytic solution for fractional chaotic dynamical systems using the differential transform method , 2011, Comput. Math. Appl..

[23]  G. Adomian.  A review of the decomposition method and some recent results for nonlinear equations , 1990 .

[24]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[25]  Chunguang Li,et al.  Chaos in the fractional order Chen system and its control , 2004 .

[26]  Guanrong Chen,et al.  On a Generalized Lorenz Canonical Form of Chaotic Systems , 2002, Int. J. Bifurc. Chaos.

[27]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[28]  Varsha Daftardar-Gejji,et al.  On Hopf bifurcation in fractional dynamical systems , 2017 .

[29]  B. Onaral,et al.  Fractal system as represented by singularity function , 1992 .

[30]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[31]  Jan Cermák,et al.  Local Bifurcations and Chaos in the Fractional Rössler System , 2018, Int. J. Bifurc. Chaos.

[32]  Eva Kaslik,et al.  Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions☆ , 2012 .

[33]  Luděk Nechvátal,et al.  The Routh–Hurwitz conditions of fractional type in stability analysis of the Lorenz dynamical system , 2017 .

[34]  Nikolay V. Kuznetsov,et al.  On differences and similarities in the analysis of Lorenz, Chen, and Lu systems , 2014, Appl. Math. Comput..

[35]  Nathalie Corson,et al.  Synchronization of Chaotic fractional-Order Systems via Linear Control , 2010, Int. J. Bifurc. Chaos.