TY - JOUR
T1 - Shadowing Homoclinic Chains to a Symplectic Critical Manifold
AU - Bolotin , Sergey
JO - Analysis in Theory and Applications
VL - 1
SP - 1
EP - 23
PY - 2021
DA - 2021/04
SN - 37
DO - http://doi.org/10.4208/ata.2021.pr80.11
UR - https://global-sci.org/intro/article_detail/ata/18762.html
KW - Hamiltonian system, homoclinic orbit, shadowing.
AB - <p style="text-align: justify;">We prove the existence of trajectories&nbsp; 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&nbsp; degree of freedom and a figure 8 separatrix.</p>