Hopf bifurcation analysis of a new commensurate fractional-order hyperchaotic system

In this paper, a new fractional-order hyperchaotic system based on the Lorenz system is presented. The chaotic behaviors are validated by the positive Lyapunov exponents. Furthermore, the fractional Hopf bifurcation is investigated. It is found that the system admits Hopf bifurcations with varying fractional order and parameters, respectively. Under different bifurcation parameters, some conditions ensuring the Hopf bifurcations are proposed. Numerical simulations are given to illustrate and verify the results.

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

[2]  R. Wu,et al.  Dynamics of a New Hyperchaotic System with Only One Equilibrium Point , 2013 .

[3]  Nikolay V. Kuznetsov,et al.  Analytical-Numerical Localization of Hidden Attractor in Electrical Chua’s Circuit , 2013 .

[4]  Chongxin Liu,et al.  A new chaotic attractor , 2004 .

[5]  Elena Grigorenko,et al.  Erratum: Chaotic Dynamics of the Fractional Lorenz System [Phys. Rev. Lett.91, 034101 (2003)] , 2006 .

[6]  G. Leonov,et al.  On stability by the first approximation for discrete systems , 2005, Proceedings. 2005 International Conference Physics and Control, 2005..

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

[8]  Milad Siami,et al.  More Details on Analysis of Fractional-order Van der Pol Oscillator , 2009 .

[9]  Nikolay V. Kuznetsov,et al.  Time-Varying Linearization and the Perron Effects , 2007, Int. J. Bifurc. Chaos.

[10]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[11]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[12]  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 .

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

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

[15]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[16]  Sachin Bhalekar,et al.  Chaos in fractional ordered Liu system , 2010, Comput. Math. Appl..

[17]  Mohammad Saleh Tavazoei,et al.  A note on fractional-order derivatives of periodic functions , 2010, Autom..

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

[19]  O. Rössler.  CONTINUOUS CHAOS—FOUR PROTOTYPE EQUATIONS , 1979 .

[20]  C. F. Lorenzo,et al.  Chaos in a fractional order Chua's system , 1995 .

[21]  M. Ichise,et al.  An analog simulation of non-integer order transfer functions for analysis of electrode processes , 1971 .

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

[23]  Guanrong Chen,et al.  Dynamical Analysis of a New Chaotic Attractor , 2002, Int. J. Bifurc. Chaos.

[24]  Nikolay V. Kuznetsov,et al.  Hidden attractor in smooth Chua systems , 2012 .

[25]  Nikolay V. Kuznetsov,et al.  Hidden attractors in Dynamical Systems. From Hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman Problems to Hidden Chaotic Attractor in Chua Circuits , 2013, Int. J. Bifurc. Chaos.

[26]  B. Onaral,et al.  Linear approximation of transfer function with a pole of fractional power , 1984 .

[27]  Nikolay V. Kuznetsov,et al.  Analytical-numerical method for attractor localization of generalized Chua's system , 2010, PSYCO.

[28]  Mohammed-Salah Abdelouahab,et al.  Hopf bifurcation and chaos in fractional-order modified hybrid optical system , 2012 .

[29]  O. Rössler.  An equation for hyperchaos , 1979 .