Crises in a non-autonomous fractional-order Duffing system

Abstract

In this paper, the crises in a non-autonomous fractional-order Duffing system are investigated. Firstly, based on the short memory principle of fractional derivative, a global numerical method called an extended generalized cell mapping (EGCM), which combines the generalized cell mapping with the improved predictor-corrector algorithm, is proposed for fractional-order nonlinear systems. The one-step transition probability matrix of Markov chain of the EGCM is generated by the improved predictor-corrector approach for fractional-order systems. The one-step mapping time of the proposed method is evaluated with the help of the short memory principle for fractional derivative to deal with its non-local property and to properly define a bound of the truncation error by considering the features of cell mapping. On the basis of the characteristics of the cell state space, the bound of the truncation error is defined to ensure that the truncation error is less than half a cell size. For a fractional-order Duffing system, boundary and interior crises with varying the derivative order and the intensity of external excitation are determined by the EGCM method. A boundary crisis results from the collision of a chaotic (or regular) saddle in the fractal (or smooth) basin boundary with a periodic (or chaotic) attractor. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes a chaotic attractor to occur, and simultaneously the previous attractor and the unstable chaotic set are converted into a part of the chaotic attractor. 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 the chaotic set to have a sudden discontinuous change. Here the chaotic set involves three different kinds of chaotic basic sets: 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. The results further reveal that the EGCM is a powerful tool to determine the global dynamics of fractional-order systems.