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Space-time complexity in Hamiltonian dynamics.

V Afraimovich1, G M Zaslavsky

  • 1San Luis Potosi University-IICO, Av. Karakorum 1470, San Luis Potosi, SLP-78240, Mexico.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
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New complexity and entropy functions describe complex systems using trajectory separation. These functions reveal fractal dimensions and link to transport properties, offering new insights into system dynamics.

Area of Science:

  • Dynamical Systems and Chaos Theory
  • Statistical Mechanics
  • Nonlinear Dynamics

Background:

  • Existing complexity and entropy measures may not fully capture systems with intermittent behavior or zero Lyapunov exponents.
  • Describing systems with "flights," trappings, and weak mixing requires novel analytical tools.

Purpose of the Study:

  • Introduce new complexity (C(epsilon;t,s)) and entropy (S(epsilon;t,s)) functions for analyzing complex systems.
  • Develop a framework that accounts for epsilon-separation of initially close trajectories.
  • Explore the relationship between these new functions and system transport properties.

Main Methods:

  • Definition of new complexity and entropy functions incorporating epsilon-separation.
  • Transformation of variables (t to eta=ln t, s to xi=ln s) to analyze algebraic complexity.

Related Experiment Videos

  • Analysis of invariants (alpha,beta) of the entropy function in log-phase space.

    • Main Results:

    • The new functions effectively describe systems with nonzero/zero Lyapunov exponents and intermittent dynamics.
    • Invariants (alpha,beta) of the entropy function characterize fractal dimensions of trajectory space-time structures.
    • A link is established between invariants, transport properties, and Riemann invariants.

    Conclusions:

    • The introduced complexity and entropy functions provide a powerful new tool for understanding complex system dynamics.
    • The fractal dimensions and transport properties are quantifiable through the derived invariants.
    • The analogy to Riemann invariants offers a novel perspective on transport exponents in log-phase space.