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Elegant chaos. Algebraically simple chaotic flows. Dedicated to the memory of Edward Norton Lorenz. (English) Zbl 1222.37005
Hackensack, NJ: World Scientific (ISBN 978-981-283-881-0/hbk). xv, 285 p. (2010).
In this book examples of simple differential equations whose solutions show chaotic behaviour are presented. The term “elegant” used by the author means that the differential equation is “simple”. With “chaotic behaviour” she means that the solutions have a “chaotic strange attractor”. However, although it is explained informally, exact definitions of the notions “chaos” and “strange attractor” are not given. In the examples computer experiments are used to find chaotic strange attractors (no rigorous proofs are given). To this end the differential equation is solved numerically using the Runge-Kutta method.
A lot of examples of such “elegant chaotic flows” form the main part of this book. Among them there are the inhomogeneous van der Pol equation and the Lorenz equation. For first order differential equations in two dimensions examples with singularities are given because otherwise such systems would not show chaotic behaviour according to the PoincarĂ©-Bendixson theorem. Moreover, among the examples there are the \(N\)-body gravitational system, the Lotka-Volterra system, simple partial differential equations and time-delay systems. In the final chapter chaotic electrical circuits are presented.
It is very nice to have this large collection of simple examples showing chaotic behaviour. This makes this book interesting for lecturers. The good motivation of the concept of chaotic behaviour makes it useful for students. On the other hand the lack of a more exact definition of chaotic behaviour is a disadvantage of this book, in particular for students. Nonetheless a lot of examples from applications makes it attractive for people mainly interested in applications of mathematics.
Reviewer: Peter Raith (Wien)

37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
34K23 Complex (chaotic) behavior of solutions to functional-differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C10 Dynamics induced by flows and semiflows
35B41 Attractors
Biographic References:
Lorenz, Edward Norton