Minimal invariant tori in the resonant regions for nearly integrable Hamiltonian systems

Abstract

Abstract. Consider a real analytical Hamiltonian system of KAM type H(p, q) = N (p) + P (p, q) that has n degrees of freedom (n > 2) and is positive definite in p.In this paper we show that for most rotation vectors in Ω, in the sense of (n − 1)-dimensional Lebesgue measure, there is at least one (n − 1)-dimensional invariant torus. These tori are the support of corresponding minimal measures. The Lebesgue measure estimate on this set is uniformly valid for any perturbation.

Asymptotic trajectories of KAM torus

“…where (X, Y ) ∈ B(0, r) and H i,k (•) is a polynomial of order k, i = 1, 2. Notice that 13. In fact, U5 can be loosened.…”

“…Then the following holds: Theorem 1.6. [13] There exists a smooth exact symplectic transformation T ∞ f : D 0 → D 0 , where D 0 ⊂ T * T 2 × S 1 is a small neighborhood of {0} × T 2 × S 1 . For ǫ ≤ ǫ 0 and under this transformation, we can convert Hamiltonian (1.2) to (1.3) H(p, q, t) = h(p) + p t , f (q, t)p + O(p 3 ), (q, p, t) ∈ D 0 , here p t , f (q, t)p is a quadratic polynomial with p t = (p 1 , p 2 ), ∇h(0) = ω, and D 2 h(0) is strictly positively definite.…”

Homoclinic orbits and critical points of barrier functions

“…Unfortunately, this problem is still open in general. The only known generic results on this topic can be found in [12] for the codimension 1 case. When, for some c, the projected Aubry set A c is composed of a hyperbolic periodic orbit, the two methods are similar.…”

Hamiltonian Systems: Stable or Unstable?

“…Under generic conditions the existence of 2 k (n − k)-dimensional tori was proved in [21] when the n-torus is k-resonant. It appears that the whiskered tori are the support of some minimal measure ( [14,18]) when the system is convex in the action variables. There are many works in this direction, one can refer [43] for more references.…”