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

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 $v$.