Numerical Bifurcation Analysis for Reaction-Diffusion Equations

1. Front Matter

   Pages i-xiv

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2. Reaction-Diffusion Equations

   • Zhen Mei

   Pages 1-6

3. Continuation of Nonsingular Solutions

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   Pages 7-29

4. Detecting and Computing Bifurcation Points

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   Pages 31-68

5. Branch Switching at Simple Bifurcation Points

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   Pages 69-84

6. Bifurcation Problems with Symmetry

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   Pages 85-100

7. Liapunov-Schmidt Method

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   Pages 101-127

8. Center Manifold Theory

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   Pages 129-150

9. A Numerical Bifurcation Function for Homoclinic Orbits

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   Pages 151-172

10. One-Dimensional Reaction-Diffusion Equations

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    Pages 173-198

11. Reaction-Diffusion Equations on a Square

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    Pages 199-229

12. Normal Forms for Hopf Bifurcations

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    Pages 231-254

13. Steady/Steady State Mode Interactions

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    Pages 255-281

14. Hopf/Steady State Mode Interactions

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    Pages 283-303

15. Homotopy of Boundary Conditions

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    Pages 305-329

16. Bifurcations along a Homotopy of Boundary Conditions

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    Pages 331-359

17. A Mode Interaction on a Homotopy of Boundary Conditions

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    Pages 361-388

18. Back Matter

    Pages 389-414

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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.

• Numerics
• Numerik
• Reaktion-Diffusionsgleichungen
• Verzweigung
• bifurcation
• calculus
• differential equation
• numerical analysis
• reaction-diffusion equations

“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)

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

  Zhen Mei

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