中文摘要：

对一个非自治分数阶Duffing系统的激变现象进行了研究.首先介绍了一种研究分数阶非线性系统全局动力学的数值方法,即拓展的广义胞映射方法 （EGCM）.该方法是基于分数阶导数的短记忆原理,并结合了广义胞映射方法和改进的预估校正算法,根据胞空间的特点,将胞尺寸作为截断误差的参考值,以此得到了一步映射时间的估算公式.用EGCM方法分别研究了分数阶Duffing系统随分数阶导数的阶数和外激励强度变化发生的边界激变和内部激变.并基于此,将激变拓展定义为混沌基本集与周期基本集之间的碰撞,其中混沌基本集包括混沌吸引子,边界上的混沌集合以及吸引域内部的非混沌吸引子的混沌集合.所得结果进一步说明了EGCM方法对于分析分数阶系统全局动力学的有效性.

英文摘要：

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 furthe