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Probability phenomena due to separatrix crossing.

A. I. Neishtadt1

  • 1Space Research Institute Academy of Sciences of the USSR, Moscow 117810, USSR.

Chaos (Woodbury, N.Y.)
|July 1, 1991
PubMed
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Small perturbations can cause chaotic behavior in Hamiltonian systems, leading to unpredictable trajectory scattering and random jumps in adiabatic invariants. This phenomenon, known as adiabatic chaos, impacts classical mechanics and real-world physics like planetary science.

Area of Science:

  • * Classical mechanics
  • * Plasma physics
  • * Planetary science

Background:

  • * Hamiltonian systems with one degree of freedom are analyzed.
  • * The phase portrait is divided by separatrices into distinct regions.
  • * Perturbations allow phase points to cross these separatrices.

Purpose of the Study:

  • * To investigate probabilistic phenomena arising from separatrix crossings in perturbed Hamiltonian systems.
  • * To describe trajectory scattering, random adiabatic invariant jumps, and adiabatic chaos.
  • * To highlight the relevance of these phenomena in both theoretical and applied physics.

Main Methods:

  • * Analysis of small or slow perturbations on Hamiltonian systems.
  • * Examination of phase portraits and separatrix dynamics.

Related Experiment Videos

  • * Description of probabilistic outcomes including scattering and random jumps.

    • Main Results:

    • * Separatrix crossings induce probabilistic phenomena.
    • * Trajectories exhibit scattering.
    • * Adiabatic invariants experience random jumps, leading to adiabatic chaos.

    Conclusions:

    • * Separatrix crossings in perturbed Hamiltonian systems generate significant probabilistic effects.
    • * Adiabatic chaos is a key outcome, affecting system predictability.
    • * The findings are applicable to idealized mechanics and real physical systems.