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Two-parameter bifurcation study of the regularized long-wave equation.

O Podvigina1, V Zheligovsky1, E L Rempel2,3

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Summary

Researchers explored the driven-damped regularized long-wave equation, revealing four pathways to spatiotemporal chaos (STC) through various bifurcations. This work clarifies complex dynamical phenomena in chaotic systems.

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Area of Science:

  • Nonlinear Dynamics
  • Mathematical Physics
  • Complex Systems

Background:

  • The driven-damped regularized long-wave equation models complex physical phenomena.
  • Understanding the transition to spatiotemporal chaos (STC) is crucial for characterizing complex system dynamics.

Purpose of the Study:

  • To conduct a two-parameter bifurcation analysis of the driven-damped regularized long-wave equation.
  • To identify and characterize the routes leading to spatiotemporal chaos (STC) by varying driver amplitude and phase.

Main Methods:

  • Performed a two-parameter bifurcation study.
  • Investigated the effects of varying driver amplitude and phase.
  • Analyzed global bifurcations, including codimension-two and homoclinic bifurcations.

Main Results:

  • Increasing driver amplitude leads to spatiotemporal chaos (STC).
  • Identified four distinct routes to STC, involving various bifurcation types (crises, intermittency, Ruelle-Takens, Feigenbaum cascade, saddle-node, homoclinic).
  • Observed the formation of chaotic saddles in phase space through bifurcations involving three-tori and steady wave manifolds.

Conclusions:

  • The study elucidates complex dynamical phenomena in the driven-damped regularized long-wave equation.
  • Provides a detailed map of transitions to spatiotemporal chaos, dependent on driver parameters.
  • Offers insights into the mechanisms generating chaotic behavior in nonlinear systems.