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Noisy homoclinic pulse dynamics.

T S Eaves1, Neil J Balmforth2

  • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom.

Chaos (Woodbury, N.Y.)
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Summary

Stochastic perturbations significantly impact pulse trains by affecting trajectories outside fixed points. This research quantifies noise effects using singular perturbation theory, revealing new insights into bifurcations in complex systems.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Statistical Physics

Background:

  • Homoclinic pulse trains are crucial in many dynamical systems.
  • The influence of stochastic perturbations (noise) on these systems is not fully understood.
  • Common assumptions about noise effects near fixed points may be inaccurate.

Purpose of the Study:

  • To investigate the effect of stochastic perturbations on nearly homoclinic pulse trains.
  • To identify the primary mechanisms through which noise influences system dynamics.
  • To explore how noise affects bifurcations in complex models.

Main Methods:

  • Analysis of three model systems: Duffing oscillator, Shimizu-Morioka model, and a co-dimension-three normal form.
  • Application of singular perturbation theory to quantify noise effects.
  • Construction and analysis of stochastic pulse spacing maps.

Main Results:

  • Noise primarily perturbs trajectories outside the fixed point neighborhood, contrary to common assumptions.
  • Singular perturbation theory effectively quantifies noise-induced effects.
  • Stochastic maps reveal how noise influences bifurcation sequences.

Conclusions:

  • The study redefines the understanding of noise's role in homoclinic pulse trains.
  • Singular perturbation theory provides a robust framework for analyzing stochastic effects.
  • Noise significantly alters bifurcation dynamics in Lorenz-like and Shilnikov-type flows.