From equilibrium to chaos. Practical bifurcation and stability analysis. (English) Zbl 0652.34059

New York etc.: Elsevier. xv, 367 p. £35.00 (1988).
This is a very interesting book in which the emphasis is on the part of the title indicated by “practical bifurcation”; this has to be understood in the sense of a computational and quantitative approach, especially by numerical methods. Let’s describe its contents. Chapter 1 introduces the basic theory of equilibria and stationary solutions, their stability and parameter dependence. Chapter 2, “basic nonlinear phenomena”, introduces the elementary bifurcations, transcritical, pitchfork, Hopf etc. Examples are buckling and oscillation of a beam, Lorenz’s equations, FitzHugh nerve impulse theory, the continuous stirred tank reactor. The idea of generic branching is illustrated by imbedding one-parameter bifurcations in a two-parameter setting. Chapter 3 briefly discusses practical problems arising when using numerical software to tackle multi-parameter problems. To the uninitiated this chapter may seem confusing but it is all worked out in the subsequent chapters. Chapter 4 for instance deals with the basic method of branch tracing by principles of continuation. The basic elements, predictor, parameterisation, corrector, step control, leads to various continuation methods, none of which can be recommended exclusively. A long chapter (5) is devoted to branching behaviour, branch switching, symmetry breaking and the various numerical methods needed. Interesting examples are taken from nonlinear parabolic pde’s with as a case discussed in detail a coupled cell reaction. Classical problem formulations are met in chapter 6, branching behaviour of ODE boundary-value problems and chapter 7, stability of periodic solutions. Two smaller chapters are devoted finally to qualitative instruments (singularity theory, catastrophes) and chaos (Cantor sets, strange attraction, Lyapunov exponents). The book concludes with 6 technical appendices and 372 references. This is a workbook for problem solvers and as such a unique text. Although numerical methods prevail in this and much could and should be added on analytical methods, the author has produced a very stimulating practical bifurcation book.
Reviewer: F.Verhulst


34C99 Qualitative theory for ordinary differential equations
37-XX Dynamical systems and ergodic theory
34C25 Periodic solutions to ordinary differential equations