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

TL;DR

This work extends the dynamics of contact Hamiltonian systems beyond monotone regimes by developing a global variational framework that links action-minimizing trajectories to weak KAM solutions. The authors introduce Mañé slices $\widetilde{\mathcal{N}}_{u}$ via backward/forward weak KAM solutions and establish the existence of semi-infinite orbits asymptotic to these slices, as well as heteroclinic connections between slices. Crucially, they show that, in non-monotone settings, connecting orbits can carry nonzero energy and need not be semi-static, highlighting a richer asymptotic structure than in classical Tonelli/Hamiltonian dynamics. The results rely on a global characteristics method and a robust variational representation of the solution semigroups, with a detailed examination of model systems to illustrate the full range of possible asymptotic behaviors and the decomposition of the Mañé set into time-free action-minimizing components.

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 $v$.

Figures (1)

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Theorems & Definitions (107)

• Theorem 1.1
• Theorem 1.2
• Example 1.3
• Proposition 1.4
• Definition 1.5
• Proposition 1.6
• Definition 1.7
• Definition 1.8
• Remark 1.9
• Theorem A