Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian Systemстатья
Статья опубликована в журнале из списка RSCI Web of Science
Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 30 декабря 2020 г.
Аннотация:We consider a Hamiltonian system depending on a parameter which slowly changes with rate ε≪1. If trajectories of the frozen autonomous system are periodic, then the system has adiabatic invariant which changes much slower than energy. For a system with 1 degree of freedom and a figure 8 separatrix, Anatoly Neishtadt [18] showed that for trajectories crossing the separatrix, the adiabatic invariant, and hence the energy, have quasirandom jumps of order ε. We prove a partial analog of Neishtadt's result for a system with n degrees of freedom such that the frozen system has a hyperbolic equilibrium possessing several homoclinic orbits. We construct trajectories staying near the homoclinic set with energy having jumps of order ε at time intervals of order |lnε|, so the energy may grow with rate ε/|lnε|. Away from the homoclinic set faster energy growth is possible: if the frozen system has chaotic behavior, Gelfreich and Turaev [16] constructed trajectories with energy growth rate of order ε.