This book discusses the generalized Duffing equation and its periodic perturbations, with special emphasis on the existence and multiplicity of periodic solutions, subharmonic solutions and different approaches to prove rigorously the presence of chaotic dynamics. Topics in the book are presented at an expository level without entering too much into technical detail. It targets to researchers in the field of chaotic dynamics as well as graduate students with a basic knowledge of topology, analysis, ordinary differential equations and dynamical systems. The book starts with a study of the autonomous equation which represents a simple model of dynamics of a mechanical system with one degree of freedom. This special case has been discussed in the book by using an associated energy function. In the case of a centre, a precise formula is given for the period of the orbit by studying the associated period map.

The book also deals with the problem of existence of periodic solutions for the periodically perturbed equation. An important operator, the Poincaré map, is introduced and studied with respect to the existence and multiplicity of its fixed points and periodic points. As a map of the plane into itself, complicated structure and patterns can arise giving numeric evidence of the presence of the so-called chaotic dynamics. Therefore, some novel topological tools are introduced to detect and rigorously prove the existence of periodic solutions as well as analytically prove the existence of chaotic dynamics according to some classical definitions introduced in the last decades. Finally, the rest of the book is devoted to some recent applications in different mathematical models. It carefully describes the technique of stretching along the paths, which is a very efficient tool to prove rigorously the presence of chaotic dynamics.

provides a comprehensive review of recent advances in the study of chaos for the Duffing equation illustrates the theory of topological horseshoes with new applications focuses on fixed points and periodic points of Poincaré map, applying topological degree and Poincaré-Birkhoff theorem

Autorentext

Lakshmi Burra is Visiting Professor of mathematics at the International Institute of Information Technology, Hyderabad. Earlier, she has been Associate Professor at Jawaharlal Nehru Technological University, Hyderabad. She earned her PhD from the University of Udine, Italy, as well as from Osmania University, Hyderabad. She also has done postgraduate in philosophy. With more than 20 years of teaching and research career, her interests of interest include mathematical modeling of real-life problems, dynamical systems, and chaotic dynamics. In addition to her teaching responsibilities, she has very actively been involved in research. She has authored several research papers as well as a book, Chaotic Dynamics in Nonlinear Theory (Springer). She has delivered several invited talks in several universities in India, and conferences. Fabio Zanolin is Professor Emeritus of Mathematics at the University of Udine, Italy, where he joined in 1987 as Full Professor of Mathematical Analysis. He also served as Chairman of the department and Coordinator of the PhD program on Mathematics and Physics at the University of Udine. He graduated from the University of Trieste, Italy, where he served from 1976-1987 as Assistant Professor and Associate Professor. He also delivered PhD courses at the International School of Advanced Studies (SISSA) in Trieste. He has been Visiting Professor, as well as an invited speaker at several universities across the world: Louvain-la-Neuve (Belgium), Krakow (Poland), Beijing (China), Santiago (Chile), Lisbon (Portugal), Granada and Madrid (Spain), Edmonton (Canada), San Antonio (USA). Invited speaker at many international conferences, he has been supervisor of several PhD theses. He was responsible for research contracts of the European Union and principal coordinator of Italian national research projects. With more than 40 years of teaching and research experience, his main research areas are differential equations, nonlinear analysis and topology. He is author/coauthor of more than 200 research papers in these fields.

Inhalt

Preface.- The Autonomous Duffing Equation.- The Periodically Forced Duffing Equation.- Chaos in the Duffing Equation: With Some Simulations.- Topological Methods for the Detection of Chaos.- Applications to the Superlinear Duffing Equation.- Laser.- The Forced Pendulum.- Chaos in the Duffing-type Equation related to Tides.- Index.