The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

doi:10.1088/1674-1056/20/9/090202

中国物理B ›› 2011, Vol. 20 ›› Issue (9): 90202-090202.doi: 10.1088/1674-1056/20/9/090202

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

冯晶晶, 张琪昌, 王炜

Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China;State Key Laboratory of Engines, Tianjin University, Tianjin 300072, China

收稿日期:2011-01-05 修回日期:2011-05-16 出版日期:2011-09-15 发布日期:2011-09-15

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

Feng Jing-Jing(冯晶晶)^a)b)c), Zhang Qi-Chang(张琪昌)^a)b)c)†, and Wang Wei(王炜)^a)b)c)

^aDepartment of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China; ^bTianjin Key Labortory of Nonlinear Dynamics and Chaos Control, Tianjin University, Tianjin 300072, China; ^bState Key Laboratory of Engines, Tianjin University, Tianjin 300072, China

Received:2011-01-05 Revised:2011-05-16 Online:2011-09-15 Published:2011-09-15

摘要/Abstract

摘要： In this paper, the extended Pad'e approximant is used to construct the homoclinic and the heteroclinic trajectories in nonlinear dynamical systems that are asymmetric at origin. Meanwhile, the conservative system, the autonomous system, and the nonautonomous system equations with quadratic and cubic nonlinearities are considered. The disturbance parameter varepsilon is not limited to being small. The ranges of the values of the linear and the nonlinear term parameters, which are variables, can be determined when the boundary values are satisfied. New conditions for the potentiality and the convergence are posed to make it possible to solve the boundary-value problems formulated for the orbitals and to evaluate the initial amplitude values.

关键词: bifurcation, Pad'e approximant, strongly nonlinearity, homoclinic and heteroclinic orbitals

Abstract: In this paper, the extended Padé approximant is used to construct the homoclinic and the heteroclinic trajectories in nonlinear dynamical systems that are asymmetric at origin. Meanwhile, the conservative system, the autonomous system, and the nonautonomous system equations with quadratic and cubic nonlinearities are considered. The disturbance parameter ε is not limited to being small. The ranges of the values of the linear and the nonlinear term parameters, which are variables, can be determined when the boundary values are satisfied. New conditions for the potentiality and the convergence are posed to make it possible to solve the boundary-value problems formulated for the orbitals and to evaluate the initial amplitude values.

Key words: bifurcation, Padé approximant, strongly nonlinearity, homoclinic and heteroclinic orbitals

中图分类号: (Ordinary differential equations)

02.30.Hq

47.20.Ky (Nonlinearity, bifurcation, and symmetry breaking) 82.40.Bj (Oscillations, chaos, and bifurcations)

冯晶晶, 张琪昌, 王炜. The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant[J]. 中国物理B, 2011, 20(9): 90202-090202.

Feng Jing-Jing(冯晶晶), Zhang Qi-Chang(张琪昌), and Wang Wei(王炜) . The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant[J]. Chin. Phys. B, 2011, 20(9): 90202-090202.

参考文献 19

[1]	Guckenheimer J and Holmes P 1993 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (New York: Springer)
[2]	Wiggins S 1990 Introduction to Applied Nonlinear Dynamical Systems and Chaos (New York: Springer)
[3]	Liu N and Guan Z H 2009 Chin. Phys. B 18 1769
[4]	Yang Z Q and Lu Q S 2006 Chin. Phys. 15 518
[5]	Chen X W 2002 Chin. Phys. 11 441
[6]	Chen J J and Yu S M 2009 Acta Mech. Sin. 58 7525
[7]	Wang Z, Li Y X, Hui X J and Lü L 2011 Acta Mech. Sin. 60 010513
[8]	Jiang B, Han X J and Bi Q S 2010 Acta Mech. Sin. 59 8343
[9]	Melnikov V K 1963 Trans. Moscow Math. 12 1
[10]	Emaci E, Vakakis A F, Andrianov I V and Mikhlin Y V 1997 Nonlinear Dyn. 13 324
[11]	Mikhlin Y V 2000 J. Sound Vib. 230 971
[12]	Manucharyan G V and Mikhlin Y V 2005 Appl. Math. Mech. 69 39
[13]	Zhang Q C, Wang W and Li W Y 2008 Chin. Phys. Lett. 25 1905
[14]	Zhang Q C, Wang W and He X J 2008 Acta Mech. Sin. 57 5384
[15]	Chen S S, Chen Y Y and Sze K Y 2009 J. Sound Vib. 322 381
[16]	Chen Y Y, Chen S S and Sze K Y 2009 Acta Mech. Sin. 25 721
[17]	Baker Jr G A and Gammel J L (edn.) 1970 The Pad'e Approximant in Theoretical Physics (New York: Academic Press)
[18]	Baker Jr G A and Graves-Morris P R 1981 Pad'e Approximants, part 2: Extension and Applications (London: Addison—Wesley Publishing Company)
[19]	Mikhlin Y V 1995 J. Sound Vib. 182 577

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pad'e approximant

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