Exponentially small splitting of separatrices in the perturbed McMillan map

Abstract

The McMillan map is a one-parameter family of integrable symplectic\ud maps of the plane, for which the origin is a hyperbolic xed point\ud with a homoclinic loop, with small Lyapunov exponent when the parameter is\ud small. We consider a perturbation of the McMillan map for which we show\ud that the loop breaks in two invariant curves which are exponentially close one\ud to the other and which intersect transversely along two primary homoclinic\ud orbits. We compute the asymptotic expansion of several quantities related to\ud the splitting, namely the Lazutkin invariant and the area of the lobe between\ud two consecutive primary homoclinic points. Complex matching techniques are\ud in the core of this work. The coe cients involved in the expansion have a\ud resurgent origin, as shown inPeer ReviewedPostprint (published version

Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results

“…Both parameterizations (48) and (50) satisfy that, fixing τ = τ * , they give parameterizations of the invariant curves of the fixed point of the 2π -Poincaré map from the section τ = τ * to the section τ = τ * + 2π .…”

“…Actually the next theorem is a classical perturbative result. (50) and such that they are the analytic continuation of the parameterizations of the invariant manifolds obtained in Theorem 4.5.…”

“…(50). Moreover, (Q u 1 , P u 1 ) ∈ A σ × A σ are defined in I u ρ 3 ,ρ 4 × T σ and there exists a constant b 4 > 0 such that…”

“…which are solutions of the partial differential equation (50). Our strategy will be: • To obtain the parameterizations (Q u,s (v, τ ), P u,s (v, τ )) in a transition domain (Theorem 4.5).…”

Stability of the phase motion in race-track microtrons

“…The three separatrices (thick lines), some level curves (thin lines), and the four equilibrium points (small circles) of the the limit Hamiltonian(25) associated to the third order resonance. Therefore,f 3 = φ O( 2 ), and the integrable dynamics of the HamiltonianH 1 approximates the dynamics of the mapf 3 when µ 3.…”