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.