Generalized exchange lemmas and orbits heteroclinic to invariant manifolds

An elementary model of torus canards

“…The more general Fenichel theory of persistence of normally hyperbolic invariant manifolds 20,21 guarantees the persistence of these manifolds for sufficiently small e. In particular, we use the Fenichel theory presented in Ref. 35 to conclude that the full system possesses invariant manifolds, which we also label S e a;2 and S e r;2 , that are differentiably OðeÞ close to their unperturbed counterparts. Of course, in this regime, the dynamics in the h direction is fast and the dynamics in the z direction is slow.…”

“…In the remainder of this section, we briefly examine how these manifolds persist for 0 < e ( 1, using Fenichel theory, 21,34 following in particular the presentation of Ref. 35. We exclude small neighborhoods of the ring C SN and the point P 0 , where the manifolds are not normally hyperbolic.…”

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…Taking into account that f n vanishes at τ = 0, σ = 0 [see (46)], we obtain f n ≤ C τ, σ ≤ Cλ k J due to (55) and (58). Combining these inequalities, we find that…”

“…The straightening of the manifolds and foliations means that one can introduce C 1 -coordinates (u, v, z) in a neighbourhood of A such that the manifold W s loc will have equation z = 0, the manifold W u loc will be given by u = 0, and the leaves of the foliations E ss and E uu will all have the form {z = 0, v = const} and {u = 0, v = const} respectively (cf. [58]). The cylinder A thus lies in {u = 0, z = 0}.…”

Geometric singular perturbation theory with real noise

“…It was first proved by Jones and Kopell [15] and extended to more general settings by many authors [14,16,13,[19][20][21][22]4,5]. There are basically three approaches to the proof and we refer to [21] for detailed explanations.…”