Chaos in fractional-order autonomous nonlinear systems. (English) Zbl 1033.37019

Summary: We numerically investigate the chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in \((0,1]\), based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study; an electronic chaotic oscillator, and a mechanical chaotic “jerk” model. In both models, numerical simulations are used to demonstrate that, for different types of model nonlinearities, and using the proper control parameters, chaotic attractors are obtained with system orders as low as 2.1. Consequently, we present a conjecture that third-order chaotic nonlinear systems can still produce a chaotic behavior with a total system order of \(2 + \varepsilon \), \(1 > \varepsilon > 0\), using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is studied. It is demonstrated that as the order is decreased, the chaotic range of the control parameter is affected by contraction and translation. Robustness against model order reduction is demonstrated.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C28 Complex behavior and chaotic systems of ordinary differential equations
37M05 Simulation of dynamical systems
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37N35 Dynamical systems in control
Full Text: DOI


[1] Cuomo, K.; Oppenheim, A.; Strogatz, S., Synchronization of Lorenz-based chaotic circuits with applications to communications, IEEE transactions on circuits and systems-II, 40, 10, (1993)
[2] Huang, X.; Xu, J., Realization of the chaotic secure communication based on interval synchronization, Journal of xi’an jiatong university, 33, 56-58, (1999)
[3] Bai, E.; Lonngren, K.; Sprott, J., On the synchronization of a class of electronic circuits that exhibit chaos, Chaos, solitons & fractals, 13, 1515-1521, (2002) · Zbl 1005.34041
[4] Hirsch, M.W.; Smale, S., Differential equations: dynamical systems and linear algebra, (1974), Academic Press New York, [chapter 11, pp. 238-54]
[5] Hartley, T.T.; Lorenzo, C.F.; Qammer, H.K., Chaos in a fractional order chua’s system, IEEE transactions on circuits and systems, 42, 8, (1995)
[6] Arena P, Caponetto R, Fortuna L, Porto D. Chaos in a fractional order Duffing system. In: Proceedings ECCTD, Budapest, September, 1997. p. 1259-62
[7] Oldham, K.; Spainer, J., Fractional calculus, (1974), Academic press New York
[8] Samavati, H.; Hajimiri, A.; Shahani, A.; Nasserbakht, G.; Lee, T., Fractal capacitors, IEEE journal of solid-state circuits, 33, 10, 2035-2041, (1998)
[9] Ahmad, W.; El-Khazali, R.; El-Wakil, A., Fractional-order wien-bridge oscillators, Electronics letters, 37, 18, 1110-1112, (2001)
[10] Elwakil, A.; Kennedy, M., Construction of classes of circuit-independent chaotic oscillators using passive-only nonlinear devices, IEEE transactions on circuits and systems-I, 48, 3, (2001) · Zbl 0998.94048
[11] Sprott, J.C., A new class of chaotic circuit, Physics letters A, 266, 19-23, (2000)
[12] Sprott, J.C., Simple chaotic systems and circuits, American journal of physics, 68, 758-763, (2000)
[13] Charef, A.; Sun, H.H.; Tsao, Y.Y.; Onaral, B., Fractal system as represented by singularity function, IEEE transactions on automatic control, 37, 9, (1992) · Zbl 0825.58027
[14] Sprott JC, Rowlands G. Chaos data analyzer: the professional version, Physics Academic Software, Raleigh, NC, 1995
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.