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Unstable and Stable Manifolds

  • Helena E. Nusse
  • James A. Yorke
  • Eric J. Kostelich
Part of the Applied Mathematical Sciences book series (AMS, volume 101)

Abstract

Poincaré called the periodic orbits of maps the soul of the dynamics. The study of the stable and unstable manifolds of these orbits does indeed reveal a great deal about the dynamics. The unstable manifold of a fixed point p of a map F may be defined as the set of points q(0) that have a backward orbit coming from p, that is, a sequence of points q(i) with i = −1, −2, ∦, so that F(q(i−1)) = q(i) for which q(i) → p as i → ∞. The stable manifold of a periodic orbit may be defined for invertible processes as the unstable manifold for the inverted system. If a point p is a saddle fixed point of a map in the plane, then the stable and unstable manifolds are both curves that pass through p. The routines described below are not restricted to planar systems, but the manifold being computed must be a curve.

Keywords

Periodic Orbit Periodic Point Unstable Manifold Chaotic Attractor Stable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References related to Dynamics

  1. Z. You, E. Kostelich, and J.A. Yorke, Calculating stable and unstable manifolds. International J. of Bifurcation and Chaos 1 (1991), 605–624MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1994

Authors and Affiliations

  • Helena E. Nusse
    • 1
    • 2
  • James A. Yorke
    • 2
  • Eric J. Kostelich
    • 3
  1. 1.Vakgroep EconometrieRijksuniversiteit GroningenGroningenThe Netherlands
  2. 2.Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA
  3. 3.Department of MathematicsArizona State UniversityTempeUSA

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