Global bifurcations in fractional-order chaotic systems with an extended generalized cell mapping method.

Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.

[1]  Grebogi,et al.  Double crises in two-parameter dynamical systems. , 1995, Physical review letters.

[2]  Mohammad Saleh Tavazoei,et al.  Non-fragile control and synchronization of a new fractional order chaotic system , 2013, Appl. Math. Comput..

[3]  Jian-Qiao Sun.  A Nonuniform Cell Partition for the Analysis of Nonlinear Stochastic Systems , 1998 .

[4]  Juhn-Horng Chen,et al.  Chaotic dynamics of the fractionally damped van der Pol equation , 2008 .

[5]  Ueda,et al.  Safe, explosive, and dangerous bifurcations in dissipative dynamical systems. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Dumitru Baleanu,et al.  The Proposed Modified Liu System with Fractional Order , 2013 .

[7]  D. Baleanu,et al.  Discrete fractional logistic map and its chaos , 2014 .

[8]  J. Yorke,et al.  Crises, sudden changes in chaotic attractors, and transient chaos , 1983 .

[9]  D. Baleanu,et al.  Lattice fractional diffusion equation in terms of a Riesz–Caputo difference , 2015 .

[10]  Jun Jiang,et al.  A method of point mapping under cell reference for global analysis of nonlinear dynamical systems , 1994 .

[11]  Shangbo Zhou,et al.  Chaos in a fractional-order Rössler system , 2009 .

[12]  Kai Diethelm,et al.  Multi-order fractional differential equations and their numerical solution , 2004, Appl. Math. Comput..

[13]  Hongyong Zhao,et al.  The effect of vaccines on backward bifurcation in a fractional order HIV model , 2015 .

[14]  C. Hsu.  GLOBAL ANALYSIS OF DYNAMICAL SYSTEMS USING POSETS AND DIGRAPHS , 1995 .

[15]  Ling Hong,et al.  Crises and chaotic transients studied by the generalized cell mapping digraph method , 1999 .

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

[17]  Jianying Yang,et al.  Dynamical models of happiness with fractional order , 2010 .

[18]  Qigui Yang,et al.  Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci , 2011 .

[19]  Wei Xu,et al.  Stochastic bifurcation of an Asymmetric Single-Well Potential Duffing oscillator under Bounded Noise Excitation , 2010, Int. J. Bifurc. Chaos.

[20]  Yury F. Luchko,et al.  Algorithms for the fractional calculus: A selection of numerical methods , 2005 .

[21]  C. S. Hsu,et al.  Cell-to-Cell Mapping , 1987 .

[22]  Jun Jiang,et al.  AN ITERATIVE METHOD OF POINT MAPPING UNDER CELL REFERENCE FOR THE GLOBAL ANALYSIS OF NON-LINEAR DYNAMICAL SYSTEMS , 1996 .

[23]  Reyad El-Khazali,et al.  Fractional-order dynamical models of love , 2007 .

[24]  L. Hong,et al.  Global dynamics of fractional-order systems with an extended generalized cell mapping method , 2016 .

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

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

[27]  Weihua Deng,et al.  Short memory principle and a predictor-corrector approach for fractional differential equations , 2007 .

[28]  C. S. Hsu,et al.  GLOBAL ANALYSIS BY CELL MAPPING , 1992 .

[29]  Dumitru Baleanu,et al.  Synchronization in a nonidentical fractional order of a proposed modified system , 2015 .

[30]  Tong Fang,et al.  Global analysis of crisis in twin-well Duffing system under harmonic excitation in presence of noise , 2005 .

[31]  Sara Dadras,et al.  Four-wing hyperchaotic attractor generated from a new 4D system with one equilibrium and its fractional-order form , 2012 .

[32]  Junguo Lu.  Chaotic dynamics of the fractional-order Lü system and its synchronization , 2006 .

[33]  Ivo Petras,et al.  Fractional-Order Nonlinear Systems , 2011 .

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