Anal. Theory Appl., 37 (2021), pp. 1-23.
Published online: 2021-04
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We prove the existence of trajectories shadowing chains of heteroclinic orbits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system. The results are quite different for real and complex eigenvalues. General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate $\varepsilon$. If the frozen autonomous system has a hyperbolic equilibrium possessing transverse homoclinic orbits, we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order $\varepsilon$ and changing with average rate of order $\varepsilon|\ln\varepsilon|$. This provides a partial multidimensional extension of the results of A. Neishtadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.
}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2021.pr80.11}, url = {http://global-sci.org/intro/article_detail/ata/18762.html} }We prove the existence of trajectories shadowing chains of heteroclinic orbits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system. The results are quite different for real and complex eigenvalues. General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate $\varepsilon$. If the frozen autonomous system has a hyperbolic equilibrium possessing transverse homoclinic orbits, we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order $\varepsilon$ and changing with average rate of order $\varepsilon|\ln\varepsilon|$. This provides a partial multidimensional extension of the results of A. Neishtadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.