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Production of computational portraits of bounded invariant manifolds

Authors
Journal
Journal of Sound and Vibration
0022-460X
Publisher
Elsevier
Publication Date
Volume
135
Issue
1
Identifiers
DOI: 10.1016/0022-460x(89)90754-2
Disciplines
  • Computer Science
  • Mathematics

Abstract

Abstract To develop an effective procedure for describing manifold tangencies, an automative process for describing the invariant manifolds involved is required. This paper presents such an automative process. The accurate location of the saddle is achieved by a modified Powell Hybrid procedure which finds the local solution to a non-linear system of algebraic equations. The resulting eigenvalue problem, based on the local Jacobian matrix of the Poincaré map, is solved and the eigenvectors evaluated. These are then used to extrapolate the initial points on the invariant manifold. By repeated iteration of the Poincaré map other points on this manifold can be located. By storing the resultant data in a singly linked list and making use of a stack reference structure and negatively signed pointers, ordering and inserting/deletion of points can be achieved while keeping a bound on computational time and space. The technique provides a basis for investigative procedures for describing global bifurcation events.

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