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

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 $\{{\tilde{\mathcal{N}}}_u\}$ 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 ${\tilde{\mathcal{N}}}_u$ and heteroclinic orbits between ${\tilde{\mathcal{N}}}_u$ and ${\tilde{\mathcal{N}}}_v$ for two different solutions $u$ and $v$.