Chaotic dynamics and synchronization of fractional-order Arneodo’s systems

In this paper, we numerically investigate the chaotic behaviors of the fractional-order Arneodo’s system. We find that chaos exists in the fractional-order Arneodo’s system with order less than 3. The lowest order we find to have chaos is 2.1 in this fractional-order Arneodo’s system. Our results are validated by the existence of a positive Lyapunov exponent. The linear and nonlinear drive–response synchronization methods are also presented for synchronizing the fractional-order chaotic Arneodo’s systems only using a scalar drive signal. The two approaches, based on stability theory of fractional-order systems, are simple and theoretically rigorous. They do not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization methods.

[1]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

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

[3]  Changpin Li,et al.  Chaos in Chen's system with a fractional order , 2004 .

[4]  Chi-Tsong Chen,et al.  Linear System Theory and Design , 1995 .

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

[6]  Ahmad Harb,et al.  On nonlinear control design for autonomous chaotic systems of integer and fractional orders , 2003 .

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

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

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

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

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

[12]  P. Arena,et al.  Chaotic behavior in noninteger-order cellular neural networks , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[13]  K. Moore,et al.  Discretization schemes for fractional-order differentiators and integrators , 2002 .

[14]  T. Hartley,et al.  Dynamics and Control of Initialized Fractional-Order Systems , 2002 .

[15]  P. Arena,et al.  Bifurcation and Chaos in Noninteger Order Cellular Neural Networks , 1998 .

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

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

[18]  I. Podlubny,et al.  Analogue Realizations of Fractional-Order Controllers , 2002 .

[19]  Julien Clinton Sprott,et al.  Chaos in fractional-order autonomous nonlinear systems , 2003 .

[20]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[21]  Juebang Yu,et al.  Synchronization of fractional-order chaotic systems , 2005, Proceedings. 2005 International Conference on Communications, Circuits and Systems, 2005..

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

[23]  Chyi Hwang,et al.  A note on time-domain simulation of feedback fractional-order systems , 2002, IEEE Trans. Autom. Control..

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

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

[26]  Alain Oustaloup,et al.  Frequency-band complex noninteger differentiator: characterization and synthesis , 2000 .

[27]  Ahmed S. Elwakil,et al.  Fractional-order Wien-bridge oscillator , 2001 .