Heteroclinic intersections between invariant circles of volume-preserving maps

Abstract

We develop a Melnikov method for volume-preserving maps with codimension one invariant manifolds. The Melnikov function is shown to be related to the flux of the perturbation through the unperturbed invariant surface. As an example, we compute the Melnikov function for a perturbation of a three-dimensional map that has a heteroclinic connection between a pair of invariant circles. The intersection curves of the manifolds are shown to undergo bifurcations in homology.

Canonical Melnikov theory for diffeomorphisms

“…We will show next that L is indeed defined by (27) and that its critical points correspond to zeros of the displacement.…”

“…These graphs contain saddle connections if we choose a pair of neighbouring It is known that a generalized standard map with a potential of the form (17) is typically nonintegrable [27]. An integrable example, f 0 , for each µ ∈ (0, 1) is obtained when the diffeomorphism c is given by…”

“…The pullback of the one-form i (D) ω to the saddle connection is welldefined and exact. In particular, there exists a function L : → R, determined uniquely up to additive constants that obeys (27).…”

“…In discrete volume geometry there are many open Melnikov problems, since the application of Melnikov methods to volume-preserving maps began just a few years ago [25,27]. We plan to continue this program in several ways.…”

Nilpotent normal form for divergence-free vector fields and volume-preserving maps

“…Volume-preserving maps are a natural generalization of areapreserving maps to higher dimensions. They also arise as the normal form for certain homoclinic bifurcations for three-dimensional systems [15,14], and as integrators for incompressible flows [27,26,18,32] and thus have intrinsic mathematical interest [6,3,34,28,20,13,21,22].…”