10-3 J. D. Mireles James, Hector Lomel\'{i}

Computation of Heteroclinic Arcs with Application to the Volume Preserving H\'{e}non Family (2190K, pdf) Jan 7, 10

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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.

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