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Constructing constrained invariant sets in multiscale continuum systems.

David Morgan1, Erik M Bollt, Ira B Schwartz

  • 1Naval Research Laboratory, Code 6792, Plasma Physics Division, Washington, D.C. 20375, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 20, 2003
PubMed
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We developed a constrained invariant manifold method to visualize stable and unstable sets on slow manifolds. This method aids in analyzing complex systems, like structural mechanics, by computing Lyapunov exponents for chaotic saddles.

Area of Science:

  • Dynamical Systems and Control Theory
  • Computational Mechanics
  • Nonlinear Dynamics

Background:

  • Analyzing complex dynamical systems often involves understanding invariant sets (stable and unstable manifolds).
  • Singularly perturbed systems, common in structural mechanics, exhibit dynamics on slow invariant manifolds.
  • Characterizing chaotic saddles is crucial for predicting system behavior.

Purpose of the Study:

  • To introduce a novel visualization tool, the constrained invariant manifold method.
  • To construct stable and unstable invariant sets constrained to slow invariant manifolds.
  • To extend existing methods for computing properties of chaotic saddles within these constraints.

Main Methods:

  • The constrained invariant manifold method is proposed for visualizing invariant sets.

Related Experiment Videos

  • A singularly perturbed model of a pendulum coupled to a viscoelastic rod is used as an example.
  • The step and stagger method is extended to calculate delta pseudoorbits on chaotic saddles.

    • Main Results:

    • The method successfully constructs stable and unstable sets constrained to slow invariant manifolds.
    • The application to a structural-mechanical model demonstrates the method's utility.
    • Lyapunov exponents of chaotic saddles constrained to slow manifolds can now be computed.

    Conclusions:

    • The constrained invariant manifold method provides a powerful tool for analyzing complex dynamical systems.
    • This approach enhances the understanding of systems with slow manifolds, particularly in structural mechanics.
    • The extended step and stagger method facilitates the computation of key dynamical properties for chaotic saddles.