Nonlinear Dynamics

Many nonlinear systems around us can generate a very complex and counter-intuitive dynamics that contrasts with their simplicity, but their understanding requires concepts that are outside the basic training of most science students. This textbook, which is the fruit of graduate courses that the authors have taught at their respective universities, provides a richly illustrated introduction to nonlinear dynamical systems and chaos and a solid foundation for this fascinating subject. It will satisfy those who want discover this field, including at the undergraduate level, but also those who need a compact and consistent overview, gathering the concepts essential to nonlinear scientists. The first and second chapters describe the essential concepts needed to describe nonlinear dynamical systems as well as their stability. The third chapter introduces the concept of bifurcation, where the qualitative dynamical behavior of a system changes. The fourth chapter deals with oscillations, from their birth to their destabilization, and how they respond to external driving. The fifth and sixth chapters discuss complex behaviors that only occur in state spaces of dimension three and higher: quasi-periodicity and chaos, from their general properties to quantitative methods of characterization. All chapters are supplemented by exercises ranging from direct applications of the notions introduced in the corresponding chapter to elaborate problems involving concepts from different chapters, as well as numerical explorations.

Axelle Amon is Associate Professor at Université de Rennes and at the Institut de Physique de Rennes (UMR CNRS 6251) in the Soft Matter Department. Her current research activities are devoted to the modeling and understanding of mechanical systems in which disorder, elasticity and friction govern the dynamics, mainly using an experimental approach. She has a particular interest in granular media as model systems to study the plasticity of amorphous materials. Recently, with her co-workers, she reproduced earthquake dynamics in a granular experiment. She has also a modeling activity in nonlinear dynamics in collaboration with experimentalists from different fields (traffic in microfluidics, pattern formation, genetic regulation). She has taught a master course on Nonlinear Dynamics and Chaos for ten years at the University of Rennes. Marc Lefranc is a professor of Physics at the University of Lille, France and works at the Laboratoire de Physique des Lasers, Atomes, Molécules (UMR CNRS 8523), in the physics department of the Faculty of Science and Technology. His research interests are in nonlinear dynamics and its applications. He has long worked on topological methods for characterizing chaotic behavior in experimental systems, a field of research on which he co-authored a book with Robert Gilmore ("The Topology of Chaos"). His current research activities are devoted to the mathematical modeling of circadian clocks, the biological oscillators that keep the time inside us, and of their synchronization with daily rythms. He teaches master courses in nonlinear dynamics both at an introductory and at an advanced level. M.L. has presided over the French nonlinear dynamics conference (la "Rencontre du Non-Linéaire") from 2006 to 2013 and has been a member of the organizing committee of the Experimental Chaos and Complexity Conference from 2010 to 2018. Marion Erpelding has a PhD in physics from the University of Rennes, France. She works as a freelance illustrator.