On the dynamics of contact Hamiltonian systems II: Variational construction of asymptotic orbits

TL;DR

This paper investigates the behavior of action minimizing orbits in contact Hamiltonian systems without the monotonicity assumption, establishing the existence of asymptotic and heteroclinic orbits connecting invariant sets associated with solutions to the Hamilton-Jacobi equation.

Contribution

It extends the analysis of contact Hamiltonian systems by constructing semi-infinite and heteroclinic orbits without the monotonicity condition, using an extended characteristic method.

Findings

Existence of semi-infinite orbits asymptotic to invariant sets.

Construction of heteroclinic orbits between invariant sets.

Extension of characteristic method for contact Hamiltonian systems.

Abstract

This paper is a continuation of our study of the dynamics of contact Hamiltonian systems in \cite{JY}, but without monotonicity assumption. Due to the complexity of general cases, we focus on the behavior of action minimizing orbits. We pick out certain action minimizing invariant sets $\{\widetilde{\mathcal{N}}_u\}$ in the phase space naturally stratified by solutions $u$ to the corresponding Hamilton-Jacobi equation. Using an extension of characteristic method, we establish the existence of semi-infinite orbits that is asymptotic to some $\widetilde{\mathcal{N}}_u$ and heteroclinic orbits between $\widetilde{\mathcal{N}}_u$ and $\widetilde{\mathcal{N}}_v$ for two different solutions $u$ and $v$.