Bifurcation Theory
In this chapter we investigate “qualitative changes” in parametrized families of RDS and call these studies “stochastic bifurcation theory”.
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Summary In this chapter we investigate "qualitative changes" in parametrized families of RDS and call these studies "stochastic bifurcation theory". The first problem is to develop a mathematical formalization (called Dbifurcation) of "qualitative change" which is connected to the stability of an RDS under an invariant measure (i.e. to the Lyapunov exponents) and reduces to the deterministic definition of bifurcation in the absence of noise (Subsect. 9.2.1). We propose as a first task to study the branching of new invariant measures from a family of reference measures at a parameter value at which the reference measure has a vanishing Lyapunov exponent. We also discuss an older concept on the level of densities in state space (called Pbifurcation) and discuss its relation with D-bifurcation (Subsects. 9.2.2 and 9.5.1). As the theory of stochastic bifurcation is still in its infancy, we proceed mainly by way of instructive examples. Sect. 9.3 treats explicitly solvable one-dimensional examples of stochastic transcritical and pitchfork bifurcation. In Subsect. 9.3.4 we prove a general criterion for a pitchfork bifurcation in dimension one using the theory of random attractors, to which we give a brief introduction. Sect. 9.4 is devoted to the study of the prototypical noisy Duffing-van der Pol oscillator. We report on the state of the art as far as rigorous results are concerned and also present numerical findings supporting conjectures about the "correct" scenario of stochastic Hopf bifurcation. Sect. 9.5 is devoted to the only available general condition (due to Baxendale) for a D-bifurcation out of the fixed point x = 0 (and an associated P-bifurcation of the new branch of measures) in a family of SDE in JRd.
9.1 Introduction Despite of its rapid development in the last decade and the fact that it is probably the most relevant part of the whole theory of RDS, bifurcation theory of RDS is still in its infancy. There are few rigorous general theorems
L. Arnold, Random Dynamical Systems © Springer-Verlag Berlin Heidelberg 1998
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Chapter 9. Bifurcation Theory
and criteria, and many phenomena have been "proved" only by computer simulations, or for particular models. Nevertheless, this chapter was included although it is less self-contained and less mathematically rigorous than the rest of the book. The interest in understanding the changes encountered in a deterministic bifurcation scenario "in the presence of noise" is historically at the beginning of the theory of RDS, and has been one of its strong driving forces. This interest arose in the 1960's and prompted numerous investigations by engineers (S. T. Ariaratnam, F. Kozin, Y. K. Lin, N. Sri Namachchivaya, C. W. S. To, W. Wedig, and many others), physicists and chemists (W. Ebeling, P. Coullet, R. Graham, H. Haken, K. H. Hoffmann, W. Horsthemke, R. Lefever, M. Lucke, F. Moss, G. Nicolis, E. Tirapegui, K. Wiesenfeld, and many others), and mathematicians (L. Arnold, P. Baxendale, R. Khasminskii, W. Kliemann, P. Kloeden, G. Papanicolaou, M. Scheutzow, among ot
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