Computation of Heteroclinic Arcs with Application to the Volume Preserving Hénon Family

Let f : R3 → R3 be a diffeomorphism with p0, p1 ∈ R3 distinct hyperbolic fixed points. Assume that W (p0) and W (p1) are two dimensional manifolds which intersect transversally at a point q. Then the intersection is locally a one-dimensional smooth arc γ̃ through q, and points on γ̃ are orbits heteroclinic from p0 to p1. We describe and implement a numerical scheme for computing the jets of γ̃ to arbitrary order. We begin by computing high order polynomial approximations of some functions Pu, Ps : R2 → R3, and domain disks Du, Ds ⊂ R2, such that W u loc(p0) = Pu(Du) and W s loc(p1) = Ps(Ds) with W u loc(p0) ∩W s loc(p1) 6= ∅. Then the intersection arc γ̃ solves a functional equation involving Ps and Pu. We develop an iterative numerical scheme for solving the functional equation, resulting in a high order Taylor expansion of the arc γ̃. We present numerical example computations for the volume preserving Hénon family, and compute some global invariant branched manifolds.