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"Coarse" stability and bifurcation analysis using time-steppers: a reaction-diffusion example.

C Theodoropoulos1, Y H Qian, I G Kevrekidis

  • 1Department of Chemical Engineering, Princeton University, Princeton, NJ 08544, USA.

Proceedings of the National Academy of Sciences of the United States of America
|August 30, 2000
PubMed
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This study adapts the Recursive Projection Method for bifurcation analysis of microscopic models. It enables effective coarse bifurcation analysis of kinetic theory-based models, demonstrated using FitzHugh-Nagumo and Lattice-Boltzmann models.

Area of Science:

  • Computational physics
  • Mathematical modeling
  • Theoretical chemistry

Background:

  • Partial differential equations (PDEs) model molecular processes but their dynamics are complex.
  • Analyzing PDE behavior often involves direct simulation or stability/bifurcation calculations.
  • Time-stepper methods offer a framework for systematic analysis.

Purpose of the Study:

  • To adapt time-stepper methods for direct bifurcation analysis of microscopic kinetic models.
  • To enable effective "coarse" bifurcation analysis of complex molecular systems.
  • To compare the behavior of a PDE model with its microscopic counterpart.

Main Methods:

  • Adaptation of the Recursive Projection Method for bifurcation analysis.
  • Application to microscopic, kinetic theory-based models.

Related Experiment Videos

  • Comparative study using the FitzHugh-Nagumo PDE and a Lattice-Boltzmann model.

    • Main Results:

    • Demonstrated a direct and effective method for coarse bifurcation analysis of microscopic models.
    • Successfully applied the adapted Recursive Projection Method to kinetic-based systems.
    • Provided a comparative analysis of PDE and Lattice-Boltzmann models.

    Conclusions:

    • The adapted Recursive Projection Method facilitates direct bifurcation analysis of microscopic models.
    • This approach offers an effective way to study pattern formation in molecular systems.
    • The study validates the method by comparing a well-known PDE with its kinetic model.