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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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[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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[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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