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Effective dynamics in Hamiltonian systems with mixed phase space.

Adilson E Motter1, Alessandro P S de Moura, Celso Grebogi

  • 1Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany. motter@mpipks-dresden.mpg.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 21, 2005
PubMed
Summary

Finite-scale Hamiltonian dynamics are governed by novel effective invariants, differing from asymptotic descriptions. These scale-dependent invariants offer new insights into complex system behavior.

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Area of Science:

  • Physics
  • Dynamical Systems Theory

Background:

  • Characterizing Hamiltonian dynamics at physically relevant scales remains a significant challenge.
  • Existing models often focus on asymptotic behavior, neglecting finite-scale effects.

Purpose of the Study:

  • To investigate and characterize the dynamics of Hamiltonian systems at finite, physically relevant scales.
  • To identify and define the governing principles of finite-scale Hamiltonian dynamics.

Main Methods:

  • Analysis of Hamiltonian systems dynamics at varying resolution scales.
  • Development of a theoretical framework modeling nonhyperbolic dynamics as a chain of hyperbolic systems.

Main Results:

  • Demonstration that finite-scale Hamiltonian dynamics are governed by effective dynamical invariants.

Related Experiment Videos

  • Identification of effective invariants that are distinct from asymptotic dynamical invariants.
  • Observation that effective invariants are dependent on both the scale of resolution and the phase space region.

    • Conclusions:

    • Finite-scale Hamiltonian dynamics possess unique characteristics governed by scale-dependent effective invariants.
    • The proposed framework provides a new interpretation for understanding nonhyperbolic dynamics in Hamiltonian systems.
    • This research addresses a fundamental gap in the characterization of Hamiltonian systems at practical scales.