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Bifurcation scenarios for bubbling transition.

Aleksey V Zimin1, Brian R Hunt, Edward Ott

  • 1Department of Physics, Box 240, Physics Building, University of Maryland, College Park, Maryland 20742, USA. alekseyz@physics.umd.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 15, 2003
PubMed
Summary

Dynamical systems exhibiting chaos can display bubbling, characterized by intermittent bursts due to perturbations. This study unifies the analysis of four bubbling bifurcation types and their transition behaviors.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Bifurcation Theory

Background:

  • Dynamical systems with chaos on invariant submanifolds can exhibit bubbling, a phenomenon of intermittent bursting triggered by small perturbations.
  • Bubbling occurs when a periodic orbit within a chaotic attractor becomes unstable to transverse perturbations.
  • This transverse instability can arise from generic bifurcations: pitchfork, transcritical, period-doubling, or Hopf.

Purpose of the Study:

  • To present a unified theoretical treatment of four types of bubbling bifurcations.
  • To determine conditions for soft versus hard transitions to bubbling.
  • To derive scaling laws for burst amplitude and interburst time.

Main Methods:

  • Unified theoretical analysis of bubbling bifurcations.
  • Derivation of conditions for soft/hard transitions.
  • Analysis of scaling laws for burst amplitude and interburst time.
  • Consideration of both random (noise) and fixed (mismatch) perturbations.
  • Numerical experiments to validate theoretical predictions.

Main Results:

  • A unified framework for analyzing four types of bubbling bifurcations is presented.
  • Conditions distinguishing between soft (continuous) and hard (discontinuous) transitions are established.
  • Scaling laws for maximum burst amplitude in soft transitions are derived.
  • Scaling laws for average interburst time for both soft and hard transitions are deduced.
  • Numerical results confirm the theoretical predictions for various perturbation types.

Conclusions:

  • The study provides a comprehensive understanding of bubbling bifurcations in chaotic dynamical systems.
  • The findings clarify the nature of transitions to bubbling and quantify key scaling behaviors.
  • The unified approach and validated predictions offer valuable insights for analyzing complex dynamical phenomena.