Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems

An extension of different lectures given by the authors, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite Dimensional Dynamical Systems provides the reader with a comprehensive overview of these topics.

Starting with the simplest bifurcation problems arising for ordinary differential equations in one- and two-dimensions, this book describes several tools from the theory of infinite dimensional dynamical systems, allowing the reader to treat more complicated bifurcation problems, such as bifurcations arising in partial differential equations. Attention is restricted to the study of local bifurcations with a focus upon the center manifold reduction and the normal form theory; two methods that have been widely used during the last decades.

Through use of step-by-step examples and exercises, a number of possible applications are illustrated, and allow the less familiar reader to use this reduction method by checking some clear assumptions. Written by recognised experts in the field of center manifold and normal form theory this book provides a much-needed graduate level text on bifurcation theory, center manifolds and normal form theory. It will appeal to graduate students and researchers working in dynamical system theory.

• bifurcations
• center manifold reduction
• infinite dimensional dynamical systems
• normal forms
• travelling waves
• ordinary differential equations
• partial differential equations

From the reviews:

'This book relies  on versions of the center manifold theorem that apply to infinite-dimensional dynamical systems...Chapter 4 of the book is distinctive in its presentation of normal forms for bifurcations in ‘reversible’ systems. These are systems in which there is a symmetry that reverses the orientation of time. When this symmetry is a reflection, it leads to systems that have large families of periodic orbits because forward and backward trajectories that start on the subspace must meet if they return to this subspace. This is an intricate subject, and this book makes it much more accessible than ever before. As for much of Iooss’ work throughout his career, this book gives many concrete examples of problems described by PDEs with an excellent balance between theory and applications of that theory' (SIAM Review, December 2011)

“This book relies on versions of the center manifold theorem that apply to infinite-dimensional dynamical systems. … This is an intricate subject, and this book makes it much more accessible than ever before. As for much of Iooss’ work throughout his career, this book gives many concrete examples of problems described by PDEs with an excellent balance between theory and applications of that theory.”­­­ (John Guckenheimer, SIAM Review, Vol. 53 (4), 2011)

• Laboratoire de Mathématiques, Université de Franche-Comté, Besançon cedex, France

  Mariana Haragus

• Laboratoire J.A.Dieudonné, IUF, Université de Nice, Nice Cedex 02, France

  Gérard Iooss