2005/01/13 by M. Rudnev, Rudnev, M., V. Ten +1
Mathematics · Physics and Astronomy · #70H08 #70H20 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.DS #math.MP #msc:70H08 #msc:70H20
paper · pdf · doi:10.48550/arxiv.math/0501208
24 pages
arxiv created 2005/01/13 · arxiv updated 2009/12/01
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 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 kth Fourier coefficient satisfies the estimate O(e- ρ|k⋅ω| - |k|σ), k∈\Zn∖\0\, where ω∈\Rn is a Diophantine rotation vector of the system of rotators; ρ∈(0,π\over2) 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(\eps), in which case ωj∼\omeps, j=1,...,n.