Multi-pulse Orbits and Homoclinic Trees in a Non-autonomous Resonant Hamiltonian System

Sha Zhou,Wei Zhang,Tian-Jun Yu,H Gong,B Yan,J.H Lee

doi:10.1051/matecconf/20164503004

Abstract

In this study, we develop the energy-phase method to deal with the high-dimensional non-autonomous nonlinear dynamical systems. Our generalized energy-phase method applies to integrable, two-degree-of freedom non-autonomous resonant Hamiltonian systems. As an example, we investigate the multi-pulse orbits and homoclinic trees for a parametrically excited, simply supported rectangular thin plate of two-mode approximation. In both the Hamiltonian and dissipative case we find homoclinic trees, which describe the repeated bifurcations of multi-pulse solutions, and we present visualizations of these complicated structures.

Highlights

IntroductionThe energy-phase method, which was first presented by Haller and Wiggins [1-10], was used to reveal fa milies of mu lti-pulse solutions and predict the parameter region for chaotic motion may occur
The energy-phase method, which was first presented by Haller and Wiggins [1-10], was used to reveal fa milies of mu lti-pulse solutions and predict the parameter region for chaotic motion may occur. the energy-phase method has been applied widely to engineering problems, it was used to solve autonomous perturbed Hamiltonian systems
We have generalized the energy-phase method to deal with the non-autonomous nonlinear system and derive the explicit formu las of the energydifferent function

Summary

Introduction

The energy-phase method, which was first presented by Haller and Wiggins [1-10], was used to reveal fa milies of mu lti-pulse solutions and predict the parameter region for chaotic motion may occur. The energy-phase method has been applied widely to engineering problems, it was used to solve autonomous perturbed Hamiltonian systems. It is worth to mention that the energy-phase method has never used in non-autonomous nonlinear dynamical systems. We develop the energy-phase method to deal with non-autonomous nonlinear dynamical systems for the first time. This paper develops the energy-phase method to deal with the high-dimensional non-autonomous nonlinear dynamical systems.

Generalized of the energy-phase method

Equations of motion of the buckled thin plate

The unperturbed dynamics

HPI 2HfI sin 2J cos I0

Conclusion