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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.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.
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Keywords:
- fractional-order systems /
- chaos /
- crises
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[15] Hsu C S 1992 Int. J. Bifurcation Chaos 2 727
[16] Xu J X, Guttalu R S, Hsu C S 1985 Int. J. Non-Linear Mech. 20 507
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[19] Xiong F R, Qin Z C, Xue Y, et al. 2014 Commun. Nonlinear Sci. Numer. Simul. 19 1465
[20] Tongue B H, Gu K 1988 J. Appl. Mech. Trans. ASME 55 461
[21] Zufiria P, Guttalu R 1993 Nonlinear Dyn. 4 207
[22] Levitas J, Weller T, Singer J 1994 J. Sound Vib. 176 641
[23] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[24] Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[25] Guder R, Kreuzer E 1999 Nonlinear Dyn. 20 21
[26] Liu X J, Hong L, Jiang J, Tang D F, Yang L X 2016 Nonlinear Dyn. 83 1419
[27] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) pp130-132
[28] Deng W H 2007 J. Comput. Appl. Math. 206 174
[29] Ford N J, Charles Simpson A 2001 Numer. Algorithms. 26 333
[30] Petr I 2011 Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing Higher: Education Press) pp238-242
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[1] Oldham K B, Spanier J 1974 The Fractional Calculus (New York: Academic Press) pp1-150
[2] Mandelbrot B B 1982 The fractal geometry of nature (San Francisco: W H Freeman) pp1-32
[3] Bagley R L, Torvik P J 1983 J. Rheol. 27 201
[4] Bagley R L, Torvik P J 1983 Aiaa J. 21 741
[5] Bagley R L, Torvik P J 1985 Aiaa J. 23 981
[6] Agrawal O P 2004 Nonlinear Dyn. 38 191
[7] Deng R, Davies P, Bajaj A K 2004 Nonlinear Dyn. 38 247
[8] Depollier C, Fellah Z E, Fellah M 2004 Nonlinear Dyn. 38 181
[9] Chen Y Q, Vinagre B M, Podlubny I 2004 Nonlinear Dyn. 38 355
[10] Zhang Y X, Kong G Q, Yu J N 2008 Acta Phys. Sin. 57 6182 (in Chinese) [张永祥，孔贵琴，俞建宁 2008 物理学报 57 6182]
[11] Zhang G J, Xu J X 2005 Acta Phys. Sin. 54 557 (in Chinese) [张广军，徐健学 2005 物理学报 54 557]
[12] Yu J J, Cao H F, Xu H B, Xu Q 2006 Acta Phys. Sin. 55 29 (in Chinese) [于津江，曹鹤飞，徐海波，徐权 2006 物理学报 55 29]
[13] Li C P, Chen G R 2004 Chaos, Solitons Fractals 22 549
[14] Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181
[15] Hsu C S 1992 Int. J. Bifurcation Chaos 2 727
[16] Xu J X, Guttalu R S, Hsu C S 1985 Int. J. Non-Linear Mech. 20 507
[17] Ushio T, Hsu C S 1986 Int. J. Non-Linear Mech. 21 183
[18] Guzzetta V, Franco B, Trask B J, et al. 1992 Genomics 13 551
[19] Xiong F R, Qin Z C, Xue Y, et al. 2014 Commun. Nonlinear Sci. Numer. Simul. 19 1465
[20] Tongue B H, Gu K 1988 J. Appl. Mech. Trans. ASME 55 461
[21] Zufiria P, Guttalu R 1993 Nonlinear Dyn. 4 207
[22] Levitas J, Weller T, Singer J 1994 J. Sound Vib. 176 641
[23] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[24] Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[25] Guder R, Kreuzer E 1999 Nonlinear Dyn. 20 21
[26] Liu X J, Hong L, Jiang J, Tang D F, Yang L X 2016 Nonlinear Dyn. 83 1419
[27] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) pp130-132
[28] Deng W H 2007 J. Comput. Appl. Math. 206 174
[29] Ford N J, Charles Simpson A 2001 Numer. Algorithms. 26 333
[30] Petr I 2011 Fractional-order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing Higher: Education Press) pp238-242
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