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Numerical Bifurcation Analysis for Reaction-Diffusion Equations

  • Book
  • © 2000

Overview

Authors:
  1. Zhen Mei

    1. Department of Mathematics, University of Marburg, Marburg, Germany

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  • First monograph on numerical aspects of bifurcation theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 28)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xiv
    Download chapter PDF

  2. Reaction-Diffusion Equations

    • Zhen Mei
    Pages 1-6

  3. Continuation of Nonsingular Solutions

    • Zhen Mei
    Pages 7-29

  4. Detecting and Computing Bifurcation Points

    • Zhen Mei
    Pages 31-68

  5. Branch Switching at Simple Bifurcation Points

    • Zhen Mei
    Pages 69-84

  6. Bifurcation Problems with Symmetry

    • Zhen Mei
    Pages 85-100

  7. Liapunov-Schmidt Method

    • Zhen Mei
    Pages 101-127

  8. Center Manifold Theory

    • Zhen Mei
    Pages 129-150

  9. A Numerical Bifurcation Function for Homoclinic Orbits

    • Zhen Mei
    Pages 151-172

  10. One-Dimensional Reaction-Diffusion Equations

    • Zhen Mei
    Pages 173-198

  11. Reaction-Diffusion Equations on a Square

    • Zhen Mei
    Pages 199-229

  12. Normal Forms for Hopf Bifurcations

    • Zhen Mei
    Pages 231-254

  13. Steady/Steady State Mode Interactions

    • Zhen Mei
    Pages 255-281

  14. Hopf/Steady State Mode Interactions

    • Zhen Mei
    Pages 283-303

  15. Homotopy of Boundary Conditions

    • Zhen Mei
    Pages 305-329

  16. Bifurcations along a Homotopy of Boundary Conditions

    • Zhen Mei
    Pages 331-359

  17. A Mode Interaction on a Homotopy of Boundary Conditions

    • Zhen Mei
    Pages 361-388

  18. Back Matter

    Pages 389-414
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Keywords

About this book

Reaction-diffusion equations are typical mathematical models in biology, chemistry and physics. These equations often depend on various parame­ ters, e. g. temperature, catalyst and diffusion rate, etc. Moreover, they form normally a nonlinear dissipative system, coupled by reaction among differ­ ent substances. The number and stability of solutions of a reaction-diffusion system may change abruptly with variation of the control parameters. Cor­ respondingly we see formation of patterns in the system, for example, an onset of convection and waves in the chemical reactions. This kind of phe­ nomena is called bifurcation. Nonlinearity in the system makes bifurcation take place constantly in reaction-diffusion processes. Bifurcation in turn in­ duces uncertainty in outcome of reactions. Thus analyzing bifurcations is essential for understanding mechanism of pattern formation and nonlinear dynamics of a reaction-diffusion process. However, an analytical bifurcation analysis is possible only for exceptional cases. This book is devoted to nu­ merical analysis of bifurcation problems in reaction-diffusion equations. The aim is to pursue a systematic investigation of generic bifurcations and mode interactions of a dass of reaction-diffusion equations. This is realized with a combination of three mathematical approaches: numerical methods for con­ tinuation of solution curves and for detection and computation of bifurcation points; effective low dimensional modeling of bifurcation scenario and long time dynamics of reaction-diffusion equations; analysis of bifurcation sce­ nario, mode-interactions and impact of boundary conditions.

Reviews

“Literature on bifurcation theory is supplemented by one more excellent book highlighting its numerical aspect. The reviewed book will be very helpful for all specialists applying bifurcation theory mathods in their investigations.” (Boris V.Loginov, zbMATH 0952.65105, 2022)

Authors and Affiliations

  • Department of Mathematics, University of Marburg, Marburg, Germany

    Zhen Mei

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