Meet the editor

Dr. Paul Bracken is currently a professor in the Mathematics Department at the University of Texas, Edinburg, TX. He obtained his BSc from the University of Toronto and PhD from the University of Waterloo in Canada. His research interests include mathematical problems in quantum mechanics and quantum field theory, differential geometry and partial differential equations as well as their overlap with the area of gravity. He has published

more than 150 papers in journals and books and has given about 20 talks at various meetings and conferences at different levels. This is the fifth volume he has contributed with IntechOpen.

Contents

Section 1

Section 2

by Paul Bracken

Loop-like Solitons

Dmitri B. Vengrovich

Game with Consumer Surplus by Wei Zhou and Tong Chu

Timoléon Crépin Kofané

Section 3

the Complex Ginzburg-Landau Equation

Ubiquitous Filtering for Nonlinear Problems by Valeri Kontorovich and Fernando Ramos-Alarcon

Chaos in Human Brain Phase Transition

Serge Yamigno Doka and Kofane Timoleon Crepin

by Mibaile Justin, Malwe Boudoue Hubert, Gambo Betchewe,

Preface III

Chaos in Quantum Mechanics 1

Chapter 1 3

Nonlinear Dynamics 11

Chapter 2 13

Chapter 3 41

Chapter 4 61

Chapter 5 89

Biology and Economics 105

Chapter 6 107

Introductory Chapter: Dynamical Symmetries and Quantum Chaos

Complex Dynamical Behavior of a Bounded Rational Duopoly

Nonlinear Dynamical Regimes and Control of Turbulence through

by Joël Bruno Gonpe Tafo, Laurent Nana, Conrad Bertrand Tabi and

by Vyacheslav O. Vakhnenko, E. John Parkes and

### Contents


#### Chapter 7 117

Global Indeterminacy and Invariant Manifolds Near Homoclinic Orbit to a Real Saddle by Beatrice Venturi

Preface

It is not exaggerating to state that the subject of chaos has reached nearly every branch of the natural sciences. It has become common to talk about chaotic weather patterns, chaotic evolution in population dynamics, and chaos in atomic physics and chemical reactions. The latter is perhaps a bit surprising at first, since at the atomic level, physics seems governed by the linear laws of quantum mechanics, while an

The current volume presents seven very good contributions to this area of research. One of the main contributions of the book is to illustrate the diversity of subjects that have been influenced by this area of research. Of these seven chapters, there are four that examine the issue of chaos in nonlinear and dynamical systems. These look at chaos as it manifests itself in various types of differential equations and systems of differential equations as well as its impact on mathematics in general. There is also a chapter that briefly discusses chaos in quantum mechanics. This will continue to be an important area of research in the future, as experimental techniques have advanced to a level that can be investigated at the atomic and molecular levels. The final two chapters track the progress of chaos into such diverse areas as

The book has been assembled out of the hard work of an international group of invited authors. It is a pleasure to thank them for their work and scientific contributions. I am grateful to acknowledge with much thanks the assistance provided by Ms. Sara Bacvarova, Author Service Manager, as well as the IntechOpen publishing

> Paul Bracken Professor,

> > USA

University of Texas,

Department of Mathematics,

group for the opportunity to work on this volume.

essential ingredient of chaos is nonlinearity in the dynamical equations.

biology and economics.
