Extended Chen: a new class of chaotic fractional-order systems

A new class of chaotic fractional-order systems is introduced and its necessary conditions for chaotic behaviour of this class have been provided. These chaotic systems are constructed based on the extension of fractional-order chaotic Chen system by addition of a general function term, satisfying some necessary conditions. This property makes the chaotic behaviour of the extended system, to large extent, independent of the selected function and so a vast range of chaotic systems can be synthesized. The main application of the proposed chaotic systems may be in secure communication systems. To make the subject clearer and in order for validation of the proposed model, five examples are provided and their results are simulated. The results confirm the theoretic analyses very well.

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

[2]  Yousef Farid,et al.  A novel image encryption/decryption scheme based on chaotic neural networks , 2012, Eng. Appl. Artif. Intell..

[3]  L. O. Chua,et al.  The double scroll family. I: Rigorous of chaos. II: Rigorous analysis of bifurcation phenomena , 1986 .

[4]  R. Bagley,et al.  The fractional order state equations for the control of viscoelastically damped structures , 1989 .

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

[6]  Ivo Petráš,et al.  Chaos in the fractional-order Volta’s system: modeling and simulation , 2009 .

[7]  Alexander L. Fradkov.  Cybernetical Physics: From Control of Chaos to Quantum Control , 2007 .

[8]  Xiaojun Tong,et al.  Image encryption with compound chaotic sequence cipher shifting dynamically , 2008, Image Vis. Comput..

[9]  E. Ahmed,et al.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models , 2007 .

[10]  L. Chua,et al.  The double scroll family , 1986 .

[11]  Guanrong Chen,et al.  A chaos-based image encryption algorithm with variable control parameters , 2009 .

[12]  B. Ross,et al.  Fractional Calculus and Its Applications , 1975 .

[13]  N. Laskin.  Fractional market dynamics , 2000 .

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

[15]  I. S. Jesus,et al.  Fractional control of heat diffusion systems , 2008 .

[16]  Giuseppe Grassi,et al.  New 3D-scroll attractors in hyperchaotic Chua's Circuits Forming a Ring , 2003, Int. J. Bifurc. Chaos.

[17]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[18]  D. Kusnezov,et al.  Quantum Levy Processes and Fractional Kinetics , 1999, chao-dyn/9901002.

[19]  YangQuan Chen,et al.  Fractional-order Systems and Controls , 2010 .

[20]  J. Swinburne.  Electromagnetic Theory , 1894, Nature.

[21]  M. Haeri,et al.  Synchronization of chaotic fractional-order systems via active sliding mode controller , 2008 .

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

[23]  Rafael Martínez-Guerra,et al.  Chaotic Synchronization and Secure Communication via Sliding-Mode Observer , 2008, Int. J. Bifurc. Chaos.

[24]  Xinghuo Yu,et al.  Design and analysis of multiscroll chaotic attractors from saturated function series , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

[25]  Daoxing Xia,et al.  Analytic theory of subnormal operators , 2014 .

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

[27]  Anatoly N. Kochubei,et al.  General Fractional Calculus, Evolution Equations, and Renewal Processes , 2011, 1105.1239.

[28]  R. Bagley,et al.  Fractional order state equations for the control of viscoelasticallydamped structures , 1991 .

[29]  C. P. Silva,et al.  Shil'nikov's theorem-a tutorial , 1993 .

[30]  Ljupco Kocarev,et al.  On Chaotic Synchronization and Secure Communications , 1995 .

[31]  B. Ross,et al.  A BRIEF HISTORY AND EXPOSITION OF THE FUNDAMENTAL THEORY OF FRACTIONAL CALCULUS , 1975 .

[32]  Giuseppe Grassi,et al.  Fractional-Order Chua's Circuit: Time-Domain Analysis, bifurcation, Chaotic Behavior and Test for Chaos , 2008, Int. J. Bifurc. Chaos.

[33]  Mehdi Dalir,et al.  Applications of Fractional Calculus , 2010 .