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Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 8585290, 10 pages
Research Article

A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Padé Approximation Method

1Tianjin Key Laboratory for Control Theory & Applications in Complicated Systems, School of Mechanical Engineering, Tianjin University of Technology, Tianjin 300384, China
2Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin University, Tianjin 300072, China

Received 24 December 2015; Revised 20 June 2016; Accepted 4 July 2016

Academic Editor: Stefano Lenci

Copyright © 2016 Jingjing Feng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In dynamic systems, some nonlinearities generate special connection problems of non-Z2 symmetric homoclinic and heteroclinic orbits. Such orbits are important for analyzing problems of global bifurcation and chaos. In this paper, a general analytical method, based on the undetermined Padé approximation method, is proposed to construct non-Z2 symmetric homoclinic and heteroclinic orbits which are affected by nonlinearity factors. Geometric and symmetrical characteristics of non-Z2 heteroclinic orbits are analyzed in detail. An undetermined frequency coefficient and a corresponding new analytic expression are introduced to improve the accuracy of the orbit trajectory. The proposed method shows high precision results for the Nagumo system (one single orbit); general types of non-Z2 symmetric nonlinear quintic systems (orbit with one cusp); and Z2 symmetric system with high-order nonlinear terms (orbit with two cusps). Finally, numerical simulations are used to verify the techniques and demonstrate the enhanced efficiency and precision of the proposed method.