Ja n 20 05 A model for separatrix splitting near multiple resonances

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 ≥ 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 setup 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 kth Fourier coefficient satisfies the estimate O e − ρ|k·ω|−|k|σ , k ∈ Z n \ {0}, where ω ∈ R n is a Diophantine rotation vector of the system of rotators; ρ ∈ (0, π 2) and σ > 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(ε), in which case ω j ∼