Jumps of energy near a separatrix in slow-fast Hamiltonian systems
Sergey Bolotin (University of Wisconsin-Madison)
MSRI: Simons Auditorium
We consider a Hamiltonian system slowly depending on time: $$ H=H(x,\tau),\qquad \dot\tau=\epsilon\ll 1. $$ For small $\epsilon$ the energy $H$ changes slowly: $\dot H=O(\epsilon)$. If the frozen autonomous system with Hamiltonian $H(\cdot,\tau)$ has one degree of freedom and energy levels are closed curves, there is an adiabatic invariant $I$ which changes much slower. Then the energy $H=h(\tau,I)$ changes gradually. However, the adiabatic invariant is destroyed for trajectories passing near equilibria. Neishtadt showed that if the frozen system has a hyperbolic equilibrium with a figure eight separatrix, then generically the energy will have quasi-random jumps of order $\epsilon$ with frequency of order $1/|\ln\epsilon|$. We partly extend Neishtadt's result to multidimensional systems such that for each $\tau$ the frozen system has a hyperbolic equilibrium possessing several transverse homoclinics. The trajectories we construct have quasirandom jumps of energy of order $\epsilon$ with frequency $1/|\ln\epsilon|$ while staying distance of order $\epsilon$ away from the homoclinic set. Gelfreigh and Turayev showed that if the frozen system has compact uniformly hyperbolic chaotic invariant sets on energy levels, then generically there exist trajectories with energy having quasirandom jumps of order $\epsilon$ with frequency of order 1. Thus the energy may grow with rate of order $\epsilon$. However, this result does not work near a homoclinic set, where dynamics of the frozen system is slow, so there is no uniform hyperbolicity.