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Global property of a Duffing-van der Pol oscillator with two external periodic excitations is investigated by generalized cell mapping digraph method. As the bifurcation parameter varies, a chaotic transient appears in a regular boundary crisis. Two kinds of transient boundary crises are discovered to reveal some reasons for the discontinuous changes for domains of attraction and boundaries. A chaotic saddle collides with the stable manifold of a periodic saddle at the fractal boundary of domains when the crisis occurs, if the chaotic saddle lies in the basin of attraction, the basin of attraction decreases suddenly while the boundary increases after the crisis; if the chaotic saddle is at a boundary, the two boundaries merge into one because of the crisis. In addition, two chaotic saddles can be merged into a new one, when they touch each other in a transient merging crisis. Finally the chaotic transient disappears in an interior crisis. The characteristics of these generalized crises are quite important for the study of chaotic transients.
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Keywords:
- generalized cell mapping digraph method /
- Duffing-van der Pol /
- chaotic transients /
- generalized crises
[1] Venkatesan A, Lakshmanan M 1997 Phys. Rev. E 56 6321
[2] Xu W, He Q, Rong H W, Fang T 2003 Acta Phys. Sin. 52 1365 (in Chinese) [徐伟, 贺群, 戎海武, 方同 2003 52 1365]
[3] Kakmeni F M, Bowong S, Tchawoua C, Kaptouom E 2004 J. Sound Vib. 227 783
[4] Yagasaki K 1994 Proceedings of the Royal Society of London A 445 597
[5] Hsu C S 1980 J. Appl. Mech. 47 931
[6] Hsu C S 1981 J. Appl. Mech. 48 634
[7] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[8] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 743 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱2008 57 743]
[9] Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[10] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 4021 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱 2008 57 4021]
[11] Yue X L, Xu W, Zhang Y 2011 Nonlinear Dyn. 69 437
[12] Hsu C S 1995 Int. J. Bifurcat. Chaos 5 1085
[13] Zou H L, Xu J X 2009 Sci. China E 52 787
[14] Wang W X, Lu Y Q, Chen H S, Ma M Q, Zhu Y Z, He D R 2002 Chin. Phys. Lett. 19 901
[15] Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181
[16] Hong L, Xu J X 2000 Acta Phys. Sin. 49 1228 (in Chinese) [洪灵, 徐健学 2000 49 1228]
[17] Hong L, Xu J X 2001 Int. J. Bifurcat. Chaos 11 723
[18] Xu W, He Q, Fang T, Rong H W 2005 Chaos Soliton. Fract. 23 141
[19] Stewart H B, Ueda Y, Grebogi C, Yorke J A 1995 Phys. Rev. Lett. 75 2478
[20] Kozlov A K, Sushchik M M, Molkov Y I, Kuznetsov A S 1999 Int. J. Bifurcat. Chaos 9 2271
[21] Wolf A, Swift B, Swinney H L, Vastano A 1985 Physica D 16 285
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[1] Venkatesan A, Lakshmanan M 1997 Phys. Rev. E 56 6321
[2] Xu W, He Q, Rong H W, Fang T 2003 Acta Phys. Sin. 52 1365 (in Chinese) [徐伟, 贺群, 戎海武, 方同 2003 52 1365]
[3] Kakmeni F M, Bowong S, Tchawoua C, Kaptouom E 2004 J. Sound Vib. 227 783
[4] Yagasaki K 1994 Proceedings of the Royal Society of London A 445 597
[5] Hsu C S 1980 J. Appl. Mech. 47 931
[6] Hsu C S 1981 J. Appl. Mech. 48 634
[7] Hong L, Xu J X 1999 Phys. Lett. A 262 361
[8] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 743 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱2008 57 743]
[9] Jiang J, Xu J X 1994 Phys. Lett. A 188 137
[10] He Q, Xu W, Li S, Xiao Y Z 2008 Acta Phys. Sin. 57 4021 (in Chinese) [贺群, 徐伟, 李爽, 肖玉柱 2008 57 4021]
[11] Yue X L, Xu W, Zhang Y 2011 Nonlinear Dyn. 69 437
[12] Hsu C S 1995 Int. J. Bifurcat. Chaos 5 1085
[13] Zou H L, Xu J X 2009 Sci. China E 52 787
[14] Wang W X, Lu Y Q, Chen H S, Ma M Q, Zhu Y Z, He D R 2002 Chin. Phys. Lett. 19 901
[15] Grebogi C, Ott E, Yorke J A 1983 Physica D 7 181
[16] Hong L, Xu J X 2000 Acta Phys. Sin. 49 1228 (in Chinese) [洪灵, 徐健学 2000 49 1228]
[17] Hong L, Xu J X 2001 Int. J. Bifurcat. Chaos 11 723
[18] Xu W, He Q, Fang T, Rong H W 2005 Chaos Soliton. Fract. 23 141
[19] Stewart H B, Ueda Y, Grebogi C, Yorke J A 1995 Phys. Rev. Lett. 75 2478
[20] Kozlov A K, Sushchik M M, Molkov Y I, Kuznetsov A S 1999 Int. J. Bifurcat. Chaos 9 2271
[21] Wolf A, Swift B, Swinney H L, Vastano A 1985 Physica D 16 285
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