## Bifurcation Theory and ApplicationsThis book covers comprehensive bifurcation theory and its applications to dynamical systems and partial differential equations (PDEs) from science and engineering, including in particular PDEs from physics, chemistry, biology, and hydrodynamics.The book first introduces bifurcation theories recently developed by the authors, on steady state bifurcation for a class of nonlinear problems with even order nondegenerate nonlinearities, regardless of the multiplicity of the eigenvalues, and on attractor bifurcations for nonlinear evolution equations, a new notion of bifurcation.With this new notion of bifurcation, many longstanding bifurcation problems in science and engineering are becoming accessible, and are treated in the second part of the book. In particular, applications are covered for a variety of PDEs from science and engineering, including the Kuramoto-Sivashinsky equation, the Cahn-Hillard equation, the Ginzburg-Landau 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, 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 expected that the book will greatly advance the study of nonlinear dynamics for many problems in science and engineering. |

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### Contents

Introduction to Steady State Bifurcation Theory | 1 |

Introduction to Dynamic Bifurcation | 21 |

Reduction Procedures and Stability | 41 |

Steady State Bifurcations | 75 |

Finite Dimensional Case | 105 |

Infinite Dimensional Case | 151 |

Bifurcations for Nonlinear Elliptic Equations | 197 |

ReactionDiffusion Equations | 241 |

Pattern Formation and Wave Equations | 267 |

Fluid Dynamics | 327 |

Bibliography | 367 |

373 | |

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assertions hold true assume attractor bifurcation bifurcated attractor bifurcated branch bifurcated solutions bifurcation equation bifurcation point bifurcation theorem Boussinesq equations Brouwer degree center manifold function center manifold theorem compact operator completely continuous field consider decomposed defined deg(id dimensional Dirichlet boundary condition domain dynamic bifurcation eigenfunctions eigenspace eigenvectors equilibrium point exists finite following assertions hold following equation following theorem Ginzburg-Landau equations given Hence homeomorphic implies invariant set Jn Jn Lemma linear operator locally asymptotically stable mapping matrix Morse index neighborhood nondegenerate nonlinear open set parameter periodic orbit problem proof is complete proof of Theorem prove real number Remark resp saddle-node bifurcation satisfies sectorial operator semi-group shown in Figure simple eigenvalue Sobolev spaces space stable manifold steady state solutions Step sufficiently small superconducting symmetric Theorem 5.2 theory unique vector field Wang