For contact Hamiltonian systems, if solution semigroups converge to ordered weak KAM solutions, action-minimizing semi-infinite and heteroclinic orbits asymptotic to the associated Mane slices exist.
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On the dynamics of contact Hamiltonian systems II: Variational construction of asymptotic orbits
For contact Hamiltonian systems, if solution semigroups converge to ordered weak KAM solutions, action-minimizing semi-infinite and heteroclinic orbits asymptotic to the associated Mane slices exist.