Chaos in fractional-order autonomous nonlinear systems.

*(English)*Zbl 1033.37019Summary: 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.

##### MSC:

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 |

##### Keywords:

electronic chaotic oscillator; mechanical chaotic jerk model; numerical simulations; control; chaotic attractors; chaotic behavior; effect of fractional order; robustness##### Software:

Sprott's Software
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\textit{W. M. Ahmad} and \textit{J. C. Sprott}, Chaos Solitons Fractals 16, No. 2, 339--351 (2003; Zbl 1033.37019)

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