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

Gen Ge,Wang Wei

doi:10.1155/2013/294162

Abstract

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.

Highlights

In recent years, there has been a great deal interest in understanding the dynamics and bifurcations in problems governed by 3D autonomous differential systems [1,2,3]
The dynamics of bifurcation near the structurally unstable homoclinic orbits in the Shilnikov sense requires the description on a two-parameter group (μ, ρ)
These foregoing 1D and 2D invariant manifolds form the Shilnikov type principal homoclinic orbit H in its analytical form. It benefits the construction of the second order Poincaremap shown in the Section 4

Summary

Introduction

There has been a great deal interest in understanding the dynamics and bifurcations in problems governed by 3D autonomous differential systems [1,2,3]. The common adapted tool for analyzing multipulse behaviors of the 3D nonlinear dynamical system is the well-known Poincaremap [4] It starts by a standard analysis of the local and global cross-sections to the principal homoclinic orbit of the equilibrium, where the stable and unstable manifolds can intersect along the tubular neighborhood of the principal homoclinic loop, giving rise to Poincarehomoclinic structure. With a further extension of the methodology, the Shilnikov sense principal homoclinic loop will be analytically obtained from the composition of series expression of the invariant manifolds surrounding the saddle-focus equilibrium These quantified manifolds intersect the cross-sections locally and globally produce more explicit coordinates up to a desired order and characterize the Poincarehomoclinic map of the system. It gives further facility of studying the complex dynamics, such as subsidiary or multipulse homoclinicity surrounding the principal homoclinic orbit

The Shilnikov Unfolding Based on the HopfZero Normal Form

Analytical Expression of the Principal Homoclinic Orbit

The Second Order Poincaré Map

Conclusions