Local Adiabatic Invariants Near a Homoclinic Set of a Slow--Fast Hamiltonian System

Sergey V. Bolotin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 19-32

Résumé

In slow–fast systems, fast variables change at a rate of the order of one, and slow variables, at a rate of the order of $\varepsilon \ll 1$. The system obtained for $\varepsilon =0$ is said to be frozen. If the frozen (fast) system has one degree of freedom, then in the region where the level curves of the frozen Hamiltonian are closed there exists an adiabatic invariant. A. Neishtadt showed that near a separatrix of the frozen system the adiabatic invariant exhibits quasirandom jumps of order $\varepsilon $. In this paper we partially extend Neishtadt's result to the multidimensional case. We show that if the frozen system has a hyperbolic critical point possessing several transverse homoclinics, then for small $\varepsilon $ there exist trajectories shadowing homoclinic chains. The slow variables evolve in a quasirandom way, shadowing trajectories of systems with Hamiltonians similar to adiabatic invariants. This paper extends the work of V. Gelfreich and D. Turaev, who considered similar phenomena away from critical points of the frozen Hamiltonian.