Many exciting frontiers of science and engineering require understanding the spatiotemporal properties of sustained nonequilibrium systems such as fluids, plasmas, reacting and diffusing chemicals, crystals solidifying from a melt, heart muscle, and networks of excitable neurons in brains. This introductory textbook for graduate students in biology, chemistry, engineering, mathematics, and physics provides a systematic account of the basic science common to these diverse areas. This book provides a careful pedagogical motivation of key concepts, discusses why diverse nonequilibrium systems often show similar patterns and dynamics, and gives a balanced discussion of the role of experiments, simulation, and analytics. It contains numerous worked examples and over 150 exercises. This book will also interest scientists who want to learn about the experiments, simulations, and theory that explain how complex patterns form in sustained nonequilibrium systems.

• A systematic account of the basic science common to biology, chemistry, engineering, mathematics and physics • Discusses the role of experiments, simulation, and analytics, and gives a careful pedagogical motivation of key concepts • Contains over 150 exercises and numerous illustrative worked examples

### Contents

Preface; 1. Introduction; 2. Linear instability: basics; 3. Linear instability: application to reacting and diffusing chemicals; 4. Nonlinear states; 5. Models; 6. One-dimensional amplitude equation; 7. Amplitude equations for two-dimensional patterns; 8. Defects and fronts; 9. Patterns far from threshold; 10. Oscillatory patterns; 11. Excitable media; 12. Numerical methods; Appendixes; References; Index.

### Reviews

'This book gives an excellent didactic introduction to pattern formation in spatially extended systems. It can serve both as the basis for an advanced undergraduate or graduate course as well as a reference. It is one of those books that will never outlive its usefulness. It is a must for anyone interested in non-linear, non-equilibrium physics.' Eberhard Bodenschatz, MPI for Dynamics and Self-Organization, University of Goettingen, Cornell University

'This book fills a long-standing need, and is certain to be an instant classic. The physics of pattern forming systems is diverse but the theoretical core of the subject, along with many of the most important applications, can be learned from this splendid book. It is bound to be a key text for courses, as well as a much cited reference.' Stephen Morris, University of Toronto

'This book by Cross and Greenside presents a comprehensive introduction to an important area of natural science and assembles in one volume the essential conceptual, theoretical and experimental tools a serious student will need to obtain a modern understanding of pattern formation outside of equilibrium. The masterful 50-page introduction lays out the essential questions and provides motivation to the reader to explore the subsequent chapters, beginning with simple ideas and growing progressively in mathematical sophistication and physical depth. Careful attention is paid to the relationship between the theoretical methods and controlled laboratory experiments or numerical simulations. I can highly recommend this book to any student or researcher interested in a deepened understanding of nonequilibrium spatiotemporal patterns.' Pierre Hohenberg, New York University

'… clearly written and follows a logical structure, making and excellent introduction to the field at the graduate student level.' UK Nonlinear News