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Modeling by nonlinear differential equations. Dissipative and conservative processes. (English) Zbl 1188.35002

This book describes the dynamics of some selected nonlinear ordinary and partial differential equations, including the Fisher equation, the van der Pol equation, the Lorenz equation, the Korteweg-de Vries equation, the FitzHugh-Nagumo equation and many others. A variety of topics on nonlinear dynamics of these equations are discussed, such as relaxation oscillations, solitons, pulse propagation, order and chaos. As stated in the introduction, the aim of this book is to provide students with the subjects in a transparent manner and stimulate further study in the area for researchers.
The book is divided into nine chapters. The first chapter introduces the reader to the contents to be covered. Chapter 2 is devoted to processes in closed and open systems. Particularly, autocatalytic chemical processes are studied. In a closed system, the reaction mixture asymptotically approaches thermodynamical equilibrium, whereas, in an open system, the reaction mixture is kept away from equilibrium. The dynamics of molecular evolution is investigated in Chapter 3. The ordinary differential equations, called replicator equations, to describe the replication process in biology are introduced. Two mechanisms of template induced autocatalysis, autocatalytic formation of oligonucleotides and replication of RNA molecules, are analyzed. Chapter 4 deals with relaxation oscillations of ordinary differential equations. The van der Pol equation and the Stoker-Haag equation are studied in detail in this respect. In Chapter 5, the authors discuss the ordered and chaotic features of nonlinear dynamics. The logistic map, the Lorenz equation, the Chua circuit equation and an autocatalytic reaction network are investigated. Chapter 6 is devoted to reaction diffusion dynamics, which includes pulse front solutions of the Fisher equation, diffusion driven spatial inhomogeneities, and the Turing mechanism of chemical pattern formation. The solitons of the Korteweg-de Vries equation, the Burgers equation, and the sine-Gordon equation are treated in Chapter 7. The neuron pulse propagation governed by the FitzHugh-Nagumo equation and the Hodgkin-Huxley equation are covered in Chapter 8. Finally, in Chapter 9, the authors briefly discuss the concepts of time reversal, dissipation and conservation in dynamical systems.

MSC:

35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
35Qxx Partial differential equations of mathematical physics and other areas of application
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