# Jumps of Energy Near a Homoclinic Set of a Slowly Time Dependent Hamiltonian System

*2019, Volume 24, Number 6, pp. 682-703*

Author(s):

**Bolotin S. V.**

We consider a Hamiltonian system depending on a parameter which slowly changes with rate $\varepsilon \ll 1$. 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 $\varepsilon$.
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 $\varepsilon$ at time intervals of order $|\ln\varepsilon|$, so the energy may grow with rate $\varepsilon/|\ln\varepsilon|$. 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 $\varepsilon$.

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