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Juliano A de Oliveira1, R A Bizão, Edson D Leonel

  • 1Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Av 24A, 1515 Bela Vista, CEP 13506-700 Rio Claro, SP, Brazil.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 21, 2010
PubMed
Summary
This summary is machine-generated.

This study investigates the transition from order to chaos in two-dimensional Hamiltonian mappings. Researchers identified universal scaling behaviors and critical exponents characterizing this complex dynamical system.

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Area of Science:

  • Dynamical systems theory
  • Statistical mechanics
  • Nonlinear dynamics

Background:

  • Two-dimensional Hamiltonian mappings often exhibit mixed phase space, featuring both regular (Kolmogorov-Arnold-Moser islands) and chaotic regions.
  • Understanding the transition from integrable to nonintegrable dynamics is crucial for characterizing complex systems.

Purpose of the Study:

  • To analyze the transition from integrability to nonintegrability in 2D Hamiltonian mappings with mixed phase space.
  • To characterize the scaling behavior of average quantities within the chaotic sea.
  • To define universality classes based on critical exponents.

Main Methods:

  • Utilized scaling functions to describe average quantities within the chaotic sea.
  • Performed extensive numerical simulations to obtain critical exponents.
  • Investigated mappings parameterized by an exponent gamma in dynamical variables.

Main Results:

  • Identified scaling functions that describe the transition.
  • Determined critical exponents characterizing the scaling behavior.
  • Found that critical exponents depend on the parameter gamma, leading to defined universality classes.

Conclusions:

  • The transition from integrability to nonintegrability in these systems can be effectively described by universal scaling functions.
  • Universality classes are established based on the obtained critical exponents, dependent on system parameters.
  • This work provides insights into the statistical properties of chaotic dynamics in Hamiltonian systems.