On Hopf bifurcation in fractional dynamical systems

Abstract Fractional order dynamical systems admit chaotic solutions and the chaos disappears when the fractional order is reduced below a threshold value [1]. Thus the order of the dynamical system acts as a chaos controlling parameter. Hence it is important to study the fractional order dynamical systems and chaos. Study of fractional order dynamical systems is still in its infancy and many aspects are yet to be explored. In pursuance to this in the present paper we prove the existence of fractional Hopf bifurcation in case of fractional version of a chaotic system introduced by Bhalekar and Daftardar-Gejji [2]. We numerically explore the (A, B, α) parameter space and identify the regions in which the system is chaotic. Further we find (global) threshold value of fractional order α below which the chaos in the system disappears regardless of values of system parameters A and B.

[1]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

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

[3]  S. Bhalekar,et al.  Chaos in fractional order financial delay system , 2016 .

[4]  Binoy Krishna Roy,et al.  Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control , 2014 .

[5]  Salman Ahmad,et al.  Analytical and numerical study of Hopf bifurcation scenario for a three-dimensional chaotic system , 2016 .

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

[7]  Chunguang Li,et al.  Chaos and hyperchaos in the fractional-order Rössler equations , 2004 .

[8]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[9]  Sachin Bhalekar,et al.  A new predictor-corrector method for fractional differential equations , 2014, Appl. Math. Comput..

[10]  Varsha Daftardar-Gejji,et al.  An iterative method for solving nonlinear functional equations , 2006 .

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

[12]  Sachin Bhalekar,et al.  Forming Mechanizm of Bhalekar-Gejji Chaotic Dynamical System , 2013 .

[13]  Varsha Daftardar-Gejji.  Fractional Calculus: Theory and Applications , 2014 .

[14]  Local stable manifold theorem for fractional systems , 2016 .

[15]  Mohammad Saleh Tavazoei,et al.  Limitations of frequency domain approximation for detecting chaos in fractional order systems , 2008 .

[16]  Changpin Li,et al.  A survey on the stability of fractional differential equations , 2011 .

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

[18]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[19]  James D. Meiss,et al.  Differential dynamical systems , 2007, Mathematical modeling and computation.

[20]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .