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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 294162, 6 pages
Research Article

Constructing the Second Order Poincaré Map Based on the Hopf-Zero Unfolding Method

1School of Mechanical Engineering, Tianjin Polytechnic University, Tianjin 300072, China
2Department of Mechanics, Tianjin University, Tianjin 300072, China

Received 12 August 2013; Revised 12 September 2013; Accepted 13 September 2013

Academic Editor: Massimiliano Ferrara

Copyright © 2013 Gen Ge and Wang Wei. 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.


We investigate the Shilnikov sense homoclinicity in a 3D system and consider the dynamical behaviors in vicinity of the principal homoclinic orbit emerging from a third order simplified system. It depends on the application of the simplest normal form theory and further evolution of the Hopf-zero singularity unfolding. For the Shilnikov sense homoclinic orbit, the complex form analytic expression is accomplished by using the power series of the manifolds surrounding the saddle-focus equilibrium. Then, the second order Poincaré map in a generally analytical style helps to portrait the double pulse dynamics existing in the tubular neighborhood of the principal homoclinic orbit.