The various phenomena discussed in the preceding chapters are among the large number of problems described by systems of nonlinear differential equations which depend on one or more control parameters (e.g., the temperature gradient across the boundaries of a Rayleigh-Bénard cell, the angular velocity of the cylinders of a Taylor-Couette apparatus, the concentration of some reactive species in chemically active media, etc.). When studying these problems, one often wishes to know what are the fixed points of the dynamics, their stability, and their dependence on the control parameters. Furthermore, since multistability easily occurs in these systems, it is important to know, for practical purposes, how and why a particular state may be selected. The basic concepts that are needed to answer these questions and that are at the origin of the methods used throughout this book will be summarized here.
KeywordsHopf Bifurcation Nematic Liquid Crystal Linear Stability Analysis Unstable Mode Amplitude Equation
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- [14.3]H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, vol.I, Paris (1892).Google Scholar
- [14.4]A. A. Andronov and A. Witt, C. R. Acad. Sci. Paris,190, 256 (1930).Google Scholar
- [14.5]E. Hopf, Ber. Math. Phys., Sachsische Akademie der Wissenschaften Leipzig 94, 1 (1942).Google Scholar
- [14.7]A. Kelley in Transversal Mappings and Flows, R. Abraham and J. Robbin, Benjamin, New York (1967); see also O.E.Lanford in Nonlinear problems in the Physical Sciences and Biology,Springer Lecture Notes 322 (1973).Google Scholar
- [14.9]M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Nonlinear Equations, Noordhoff, Leyden, Netherland (1974).Google Scholar
- [14.16]J. Lauzeral and D. Walgraef, Pattern Formation in the Anisotropic ProctorSivashinsky Model, submitted to Phys. Rev. E (1996).Google Scholar
- [14.23]A. C. Newell and J. V. Moloney, Nonlinear Optics, Addison-Wesley, Redwood City, California (1992).Google Scholar
- [14.26]G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, New York (1977).Google Scholar
- [14.30]H. Haken, Advanced Synergetics, Springer-Verlag, Berlin, 1987.Google Scholar
- [ 14.33 ]R. J. Field and M. Burger eds., Oscillations and Travelling Waves in Chemical Systems, Wiley, New York (1985).Google Scholar