Computing unstable 2-manifolds in 3-dimensional phase space by computing a volume curvature
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Computing unstable 2-manifolds in 3-dimensional phase space by computing a volume curvature

  • Author(s): Stepanian, Haik
  • Advisor(s): Mitchell, Kevin
  • et al.
Abstract

We develop a numerical method for computing a two dimensional unstable manifold defined in a 3-dimensional "phase space". The method relies on the calculation of a volume formed by a local linear approximation and a local quadratic approximation. We consider mappings on 3-dimensional phase space that preserve volumes under iterations. Approximating these volumes is a natural way of measuring the curvature of a 2-manifold. This calculated volume provides an efficient means of imposing a threshold criteria for refining a manifold; the measure of this volume makes the calculation of curvatures more feasible than approximating curvatures that depend on the direct use of derivatives, especially near cusps and corners. We start by first deriving a method for a 1-manifold embedded in 2-dimensional phase space with an area-preserving map, then apply our findings to the Hènon map. Then we discuss a way to extend this method to 2-manifolds in 3-dimensional phase space.

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