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Covering dynamical systems: twofold covers.

C Letellier1, R Gilmore

  • 1CORIA UMR 6614-Université de Rouen, Place Emile Blondel, F-76821 Mont Saint-Aignan Cedex, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 17, 2001
PubMed
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This study explores how discrete symmetry groups affect dynamical systems in R3. We examine how systems with symmetry relate to those without, introducing a new "peeling bifurcation."

Area of Science:

  • Dynamical Systems Theory
  • Group Theory
  • Topology

Background:

  • Dynamical systems can exhibit symmetries, meaning their behavior remains unchanged under certain transformations.
  • Understanding these symmetries is crucial for classifying system behavior and predicting long-term dynamics.
  • Locally identical systems may possess different global symmetries or lack them entirely.

Purpose of the Study:

  • To investigate the relationship between equivariant dynamical systems and systems lacking residual symmetry.
  • To explore the converse process of lifting a symmetric system to a multiple cover that is equivariant under a discrete symmetry group.
  • To analyze and compare the resulting equations of motion, strange attractors, and classifying branched manifolds.

Main Methods:

Related Experiment Videos

  • Analysis of dynamical systems in R3 under the action of discrete symmetry groups (rotation and inversion).
  • Comparison of mathematical structures including equations of motion and strange attractors.
  • Utilizing branched manifolds as topological invariants for classifying attractors.
  • Introduction and analysis of a novel global bifurcation termed the "peeling bifurcation."

    • Main Results:

    • Established connections between symmetric and non-symmetric dynamical systems.
    • Demonstrated the existence of multiple inequivalent equivariant multiple covers for a single dynamical system, distinguished by a topological index.
    • Presented concrete examples illustrating these concepts.
    • Described the characteristics and implications of the peeling bifurcation.

    Conclusions:

    • Symmetry plays a fundamental role in shaping the behavior and classification of dynamical systems.
    • The concept of multiple covers provides a richer framework for understanding symmetry in dynamical systems.
    • The peeling bifurcation represents a new mechanism for qualitative change in dynamical systems behavior.