A model for separatrix splitting near multiple resonances

Abstract

We propose a model for local dynamics of a perturbed convex real-analytic Liouville-integrable Hamiltonian system near a resonance of multiplicity $1+m,m\ge 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouville-integrable system of $n$ non-linear rotators via a small analytic potential. The global bifurcation problem is set-up for the $n$-dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on an $n$-torus, whose $k$th Fourier coefficient satisfies the estimate $$O(e^{-\rho |k\cdot \omega |-|k|\sigma }),k\in {\mathbb{Z}}^n\setminus \{0\},$$ where $\omega \in {\mathbb{R}}^n$ is a Diophantine rotation vector of the system of rotators; $\rho \in (0,\frac{\pi }{2})$ and $\sigma >0$ are the analyticity parameters built into the model. The estimate, under suitable assumptions would generalize to a general multiple resonance normal form of a convex analytic Liouville integrable Hamiltonian system, perturbed by $O(\eps)$, in which case $ω_j\sim\omeps, j=1,...,n.$

24 pages