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Dressed symbolic dynamics.

Robert Gilmore1, Christophe Letellier

  • 1Physics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 12, 2003
PubMed
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Symmetry in strange attractors (SA) can be mapped to a simpler image SA. This image SA can be lifted to multiple equivariant SAs, each with distinct symbolic dynamics and structural stability.

Area of Science:

  • Dynamical Systems
  • Chaos Theory
  • Group Theory

Background:

  • Strange attractors (SA) exhibit complex dynamics.
  • Symmetry groups (G) can be associated with certain SAs.
  • Mapping SAs to images without symmetry is a known technique.

Purpose of the Study:

  • To explore the lifting of image strange attractors to equivariant strange attractors.
  • To analyze the symbolic dynamics and structural stability of these lifted attractors.
  • To develop methods for characterizing the relationship between image and equivariant attractors.

Main Methods:

  • Utilizing smooth mappings with singularities to map symmetric SAs to image SAs.
  • Investigating the concept of equivariant covers and their symbolic dynamics.

Related Experiment Videos

  • Defining an index based on group operators to distinguish covers.
  • Analyzing the role of subgroups (H) in determining the number of disconnected components.
  • Presenting an algorithm for orbit analysis in equivariant attractors.

    • Main Results:

    • An image SA can be lifted to multiple distinct, structurally stable, equivariant SAs.
    • The symbolic dynamics of equivariant covers require |G|s symbols for s symbols in the image SA.
    • There are |G|(s) distinct equivariant covers, distinguished by an index.
    • The subgroup H generated by the index determines the number of disconnected components (|G|/|H|).
    • An algorithm is provided for determining orbit numbers and periods in equivariant attractors.

    Conclusions:

    • The lifting process generates a rich structure of equivariant attractors from a single image attractor.
    • Symbolic dynamics and group theory provide a framework for understanding these structures.
    • The presented algorithm aids in the analysis of periodic orbits within these complex systems.