Bifurcation Theory and Applications

Abstract

PrefaceThis book provides an introduction to a newly developed bifurcation theory and its applications to dynamical systems and partial differential equations (PDEs) from science and engineering. The first two chapters of the book contain a brief introduction to the standard bifurcation theory for nonlinear PDEs. The treatment of the classical theorems is unified by the Lyapunov-Schmidt reduction and the center manifold reduction procedures.The next four chapters introduce a new bifurcation theory developed recently by the authors. This theory is centered at a new notion of bifurcation, called attractor bifurcation for nonlinear evolution equations. The main ingredients of the theory include a) the attractor bifurcation theory, b) steady state bifurcation for a class of nonlinear problems with even order nondegenerate nonlinearities, regardless of the multiplicity of the eigenvalues, and c) new strategies for the Lyapunov-Schmidt reduction and the center manifold reduction procedures.With the bifurcation theory, many long standing bifurcation problems in science and engineering are becoming accessible, and are treated in the last four chapters of the book. In particular, applications are made for variety of PDEs from science and engineering, including, in particular, the Kuramoto-Sivashinshy equation, the Cahn-Hillard equation, the GinzburgLandau equation, Reaction-Diffusion equations in Biology and Chemistry, the Benard convection problem, and the Taylor problem. The applications provide, on the one hand, general recipes for other applications of the theory addressed in this book, and on the other hand, full classifications of the bifurcated attractor and the global attractor as the control parameters cross certain critical values, dictated usually by the eigenvalues of the linearized problems. It is hoped that the book will be helpful in advancing the study

Dynamical Bifurcation of the One Dimensional Modified Swift-Hohenberg Equation

“…On the other hand, This implies by Theorem 5.10 of [13] that (3.19) bifurcates from the trivial solution to an attractor A N (α) as α passes through α N . Moreover, A N (α) is homeomorphic to S 1 .…”

“…Then the dynamics of the system after the threshold of bifurcation is completely determined by its behavior on the center manifold. In particular, Ma and Wang showed in [13] that the system bifurcates to a nontrivial attractor on the center manifold which determines the final patterns of the system. The Swift-Hohenberg equation is a widely accepted model in the study of the formation of patterns [1,12].…”

“…In general, it is very difficult to calculate the center manifold function. Recently, Ma and Wang derived a rather simple formula to compute it [13]. We will use this formula to derive the reduced equations on the center manifold.…”

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

“…Since the critical eigenspace is 2l c +1 dimensional which is odd, using Krasnosel'skii Theorem (see Theorem 1.10 in [8]) the existence of a bifurcated nontrivial steady state solution of the main equations at σ = σ c can be shown exactly as in [17].…”

“…By the attractor bifurcation theorem in [8], the bifurcated attractor Σ σ is homeomorphic to the 2l c dimensional sphere S 2lc . Moreover since the main equations possesses S 2lc -symmetry, this steady state solution will generate a S 2lc set of steady states.…”