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Embedding dynamics for round-off errors near a periodic orbit.

J. H. Lowenstein1, F. Vivaldi

  • 1Department of Physics, New York University, 2 Washington Place, New York, New York 10003.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
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This study analyzes round-off error propagation in periodic orbits of linear maps. Algebraic number theory reveals new insights into orbit classification and density, showing a disappearance of hierarchical structures and algebraic growth in orbit numbers.

Area of Science:

  • Dynamical Systems and Chaos Theory
  • Number Theory and Algebraic Geometry
  • Computational Mathematics

Background:

  • Investigates round-off error propagation near periodic orbits of linear maps.
  • Focuses on maps conjugate to planar rotations with rational rotation numbers.
  • Builds upon prior work on hierarchical arrangements of orbits in quadratic cases.

Purpose of the Study:

  • To develop efficient algorithms for classifying and computing orbits and their densities.
  • To analyze the behavior of round-off errors in a higher-dimensional embedding.
  • To examine the disappearance of hierarchical structures and the growth rate of orbits.

Main Methods:

  • Embedding the 2D discrete phase space (lattice) into a higher-dimensional torus.

Related Experiment Videos

  • Utilizing linear, discontinuous dynamics with algebraic integer coefficients.
  • Applying efficient algorithms for orbit classification and density computation.

    • Main Results:

    • Points with identical round-off errors are uniformly distributed in convex polyhedra.
    • Demonstrates algebraic growth in the number of orbits with increasing period.
    • Provides evidence for the disappearance of hierarchical orbit arrangements seen in quadratic cases.

    Conclusions:

    • The study offers a novel representation for analyzing round-off errors using algebraic number theory.
    • Efficient computational methods are established for orbit analysis.
    • The findings suggest a shift from hierarchical to algebraic structures in orbit growth for specific linear maps.