J. D. Mireles James, Hector Lomel\'{i}
Computation of Heteroclinic Arcs with Application to the Volume Preserving H\'{e}non Family
(2190K, pdf)

ABSTRACT.  Let $f:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a diffeomorphism 
with $p_0, p_1 \in \mathbb{R}^3$ distinct hyperbolic fixed points. 
Assume that $W^u(p_0)$ and $W^s(p_1)$ are two dimensional manifolds 
which intersect transversally at a point $q$. Then the intersection 
is locally a one-dimensional smooth arc $\tilde \gamma$ through $q$, 
and points on $\tilde \gamma$ are orbits heteroclinic from $p_0$ to 
$p_1$. 
We describe and implement a numerical scheme for computing the jets 
of $\tilde \gamma$ to arbitrary order. We begin by computing high 
order polynomial approximations of some functions $P_u, P_s: 
\mathbb{R}^2 \rightarrow \mathbb{R}^3$, and domain disks $D_u, D_s 
\subset \mathbb{R}^2$, such that $W_{loc}^u(p_0) = P_u(D_u)$ and 
$W_{loc}^s(p_1) = P_s(D_s)$ with $W_{loc}^u(p_0) \cap W_{loc}^s(p_1) 
\neq \emptyset$. Then the intersection arc $\tilde \gamma$ solves a 
functional equation involving $P_s$ and $P_u$. We develop an 
iterative numerical scheme for solving the functional equation, 
resulting in a high order Taylor expansion of the arc $\tilde 
\gamma$. We present numerical example computations for the volume 
preserving H\'{e}non family, and compute some global invariant 
branched manifolds.