Uniform hyperbolicity of invariant cylinder

Abstract

For a nearly integrable Hamiltonian systems H = h(p) + ǫP (p, q) with (p, q) ∈ R 3 × T 3 , large normally hyperbolic invariant cylinders exist along the whole resonant path, except for the √ ǫ 1+d -neighborhood of finitely many double resonant points. It allows one to construct diffusion orbits to cross double resonance.[Z2] Zhou M., Infinity of minimal homoclinic orbits, Nonlinearity 24 (2011) 931-939.

Lasry-Lions, Lax-Oleinik and generalized characteristics

“…From the theorem above, the location of singularities affords possible information to construct "local" minimal orbits for Tonelli Lagrangian systems, which is totally unknown before. In the previous works of variational approach of Hamiltonian dynamical instability problem like Arnold diffusion (see, for instance, [7], [19], [20], [15], [16], [17] and [18]), the diffusion orbits shadow the variational minimizers which are not local ones.…”