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The existence of heteoclinic loop which connects the saddle focus equilibrium points is analyzed for a three-dimensional differential system based on heteoclinic orbit Shilnikov method, which proves the system possesses "horseshoe" chaos. Then the system bifurcation, Lyapunov exponent, Poincare mapping are studied by numerical analysis. In addition, adaptive backstepping design is used to control this system with three unknown key parameters, and an algorithm of this controller is presented. Finally, we make some numerical simulations of the system in order to verify the analytic results.
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Keywords:
- heteoclinic loop /
- adaptive backstepping /
- Shilnikov criterion /
- Poincare mapping
[1] Sparrow C 1982 The Lorenz Equation: bifurcation, chaos, and strange attractor (NewYork: Springer).
[2] Chen G R, Dong X 1998 From Chaos to Order: Methodologies, Perspectives and Applications(Singapore, world Scientific).
[3] Lü J H, Lu J A, Chen S H 2002 Chaotic Time Series Analysis and Application (Wuhan, Wuhan University Press).
[4] Li Z, Chen G R, Halang W A 2004 Information Sciences 165 235
[5] Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. on Circuits and Systems 33 1073
[6] Ueta T, Chen G R 2000 International Journal of Bifurcation and Chaos 10 1917
[7] Wang Z 2007 Analysis in Theory and Applications 23 343
[8] Wang Z, Wu Y T, Li Y X, Zou Y J 2009 Proceedings of the 4thICCSE 441
[9] Lü J H, Chen G R, Zhang S C 2002 Chaos, Solitons and Fractals 14 669
[10] Chen J J, Yu S M 2009 Acta Phys. Sin. 58 7525 (in Chinese)[陈建军、禹思敏 2009 物理学报 58 7525]
[11] Wang Z, Mao P W 2008 Journal of Dynamics and Control 6 16 (in Chinese)[王 震、毛鹏伟 2008 动力学与控制学报6 16]
[12] Lü J H, Chen G R, Zhang S C 2002 International Journal of Bifurcation and Chaos 12 659
[13] Chen X Y, Li G L 2008 Journal of Electronics & Information Technology 30 1932 (in Chinese)[陈希有、李冠林 2008 电子与信息学报 30 1932]
[14] Zhou T S, Tang Y, Chen G R 2004 International Journal of Bifurcation and Chaos 14 3167
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[1] Sparrow C 1982 The Lorenz Equation: bifurcation, chaos, and strange attractor (NewYork: Springer).
[2] Chen G R, Dong X 1998 From Chaos to Order: Methodologies, Perspectives and Applications(Singapore, world Scientific).
[3] Lü J H, Lu J A, Chen S H 2002 Chaotic Time Series Analysis and Application (Wuhan, Wuhan University Press).
[4] Li Z, Chen G R, Halang W A 2004 Information Sciences 165 235
[5] Chua L O, Komuro M, Matsumoto T 1986 IEEE Trans. on Circuits and Systems 33 1073
[6] Ueta T, Chen G R 2000 International Journal of Bifurcation and Chaos 10 1917
[7] Wang Z 2007 Analysis in Theory and Applications 23 343
[8] Wang Z, Wu Y T, Li Y X, Zou Y J 2009 Proceedings of the 4thICCSE 441
[9] Lü J H, Chen G R, Zhang S C 2002 Chaos, Solitons and Fractals 14 669
[10] Chen J J, Yu S M 2009 Acta Phys. Sin. 58 7525 (in Chinese)[陈建军、禹思敏 2009 物理学报 58 7525]
[11] Wang Z, Mao P W 2008 Journal of Dynamics and Control 6 16 (in Chinese)[王 震、毛鹏伟 2008 动力学与控制学报6 16]
[12] Lü J H, Chen G R, Zhang S C 2002 International Journal of Bifurcation and Chaos 12 659
[13] Chen X Y, Li G L 2008 Journal of Electronics & Information Technology 30 1932 (in Chinese)[陈希有、李冠林 2008 电子与信息学报 30 1932]
[14] Zhou T S, Tang Y, Chen G R 2004 International Journal of Bifurcation and Chaos 14 3167
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