Breakdown mechanisms of normally hyperbolic invariant manifolds in terms of unstable periodic orbits and homoclinic/heteroclinic orbits in Hamiltonian systems

We analyze the mechanism of breakdown of normally hyperbolic invariant manifolds (NHIMs) based on unstable periodic orbits, homoclinic and heteroclinic orbits in NHIMs for Hamiltonian systems. First, we classify the breakdown mechanism in terms of the characteristic multipliers (the eigenvalues of the phase space Jacobian matrix) of unstable periodic orbits in the NHIM and elucidate the classification for systems of three degrees of freedom. Second, we also present an index that provides a sufficient condition under which the NHIM ceases to exist in more global regions along the homoclinic and heteroclinic orbits in the NHIM. We demonstrate local and global features of the breakdown of the NHIM for a hydrogen atom in crossed electric and magnetic fields. Our analysis of unstable periodic orbits of shorter periods indicates that local features of the breakdown are highly inhomogeneous, depending on each periodic orbit. This manifests an inherent dynamical feature for systems of more than two degrees of freedom. Contrastingly, our analysis of homoclinic orbits indicates that the energy at which the NHIM breaks down does not depend on each homoclinic orbit. This result suggests the existence of a global breakdown of the NHIM behind these homoclinic orbits since these homoclinic orbits run through broader phase space regions.