Nonlinear Differential Equations and Dynamical Systems (Universitext) by Ferdinand Verhulst

by Ferdinand Verhulst

On the subject of differential equations many elementary books have been written. This book bridges the gap between elementary courses and research literature. The basic concepts necessary to study differential equations - critical points and equilibrium, periodic solutions, invariant sets and invariant manifolds - are discussed first. Stability theory is then developed starting with linearisation methods going back to Lyapunov and Poincare. In the last four chapters more advanced topics like relaxation oscillations, bifurcation theory, chaos in mappings and differential equations, Hamiltonian systems are introduced, leading up to the frontiers of current thus the reader can start to work on open research problems, after studying this book. This new edition contains an extensive analysis of fractal sets with dynamical aspects like the correlation and information dimension. In Hamiltonian systems, topics like Birkhoff normal forms and the Poincare-Birkhoff theorem on periodic solutions have been added. There are now 6 appendices with new material on invariant manifolds, bifurcation of strongly nonlinear self-excited systems and normal forms of Hamiltonian systems. The subject material is presented from both the qualitative and the quantitative point of view, and is illustrated by many examples.

Genres: Mathematics

Paperback

First published December 21, 1989

About the author

Ferdinand Verhulst is retired professor of Mathematics at the University of Utrecht in the Netherlands.

He is also the founder of the publisher Epsilon Uitgaven, in 1985. This publisher aims to publish scientific books in Dutch, with a special eye to Mathematics, both because it is easier to learn in your own language, and also to promote mathematics among non-mathematicians. In Dutch (as published on the website) "Epsilon Uitgaven beoogt het tot stand brengen en verspreiden van wetenschappelijke boeken in de Nederlandse taal met de nadruk op wiskunde."

Reviews

[William Bies · Rating 3 out of 5]

The theory of classical dynamical systems has obvious relevance to the world around us: if one aims to undertake research as a physicist, some exposure to the subject beyond the undergraduate level will certainly be desirable. Thus, after an initial acquaintance with ordinary differential equations (from a textbook such as Earl A. Coddington’s classic An Introduction to Ordinary Differential Equations (1961), reviewed by us a while back here), a serious student of physics should want to proceed to the beginning graduate level and familiarize himself with a wide array of typical problems, going beyond Hamiltonian systems which, being a staple topic in the curriculum at both the undergraduate and graduate levels, he will surely learn about in other coursework, but not necessarily go all the way up to the level of a research monograph like Coddington and Norman Levinson’s standard Theory of Ordinary Differential Equations (McGraw-Hill, 1955).

Therefore, we wish to propose Ferdinand Verhulst’s Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag Universitext, second edition, 1996) as ideal for this purpose. This beginning graduate-level textbook is gently paced, not at all technically forbidding, furnished with good homework exercises and manageable in overall length. But beware: the blurb on the back cover claims, ‘Thus the reader can start to work on open research problems, after studying this book’ – which may well be overoptimistic, if we note that the level is just not high enough and many of the harder theorems are not proved, especially in the later chapters on bifurcation theory and Hamiltonian systems.

The first several chapters start out on a very elementary basis, with concepts such as phase space, critical points, linearization, periodicity, first integrals and integral manifolds, Liouville’s theorem, Bendixson’ criterion, the Poincaré-Bendixson theorem, linear stability resp. instability, asymptotical stability resp. instability, Lyapunov functions and naïve perturbation theory. Verhulst’s pedagogical method is to provide mathematically precise definitions and clear statements of the major theorems, then a derivation if at all possible without getting into too complicated an argument. So, the mathematical tools are basic and anything that would be challenging is deferred to the references.

Advanced topics occupy the remainder, from chapter 10 onwards. Chapters 10-11 cover the Poincaré-Lindstedt method of approximation of periodic solutions on arbitrary long time-scales and proof of their existence, respectively, the averaging method – Lagrange standard form, averaging in the general case or in the periodic case, adiabatic invariants, resonance manifolds, periodic solutions etc. In chapter 12, a rapid introduction to relaxation oscillations in mechanical systems with large friction, applied to the van der Pol and Volterra-Lotka equations.

Chapter 13 on bifurcation theory is impressionistic; it develops the subject from an elementary starting point by examining a few simple cases (up to the Hopf bifurcation), focusing on the formal transformation to a normal form and the definition of stable, unstable and center manifolds, but not getting as far as any general results (such as the classification of elementary catastrophes). Similarly, chapter 14 on chaos theory seems disjointed, too many topics are attempted with too little space for each: it would probably be better turn to Lichtenberg and Lieberman's Regular and chaotic dynamics (reviewed by us here) – but there aren’t any problems attached to this chapter anyway, so it is better skipped. The last chapter on Hamiltonian systems is also quite sketchy; at least it concludes with a few good problems that tie some typical examples back to the ideas investigated systematically, earlier in the book.

Maybe Verhulst’s sketchiness doesn’t matter so much, if one keep in mind the point that the proper reason to read him in the first place is mainly to hone one’s skill by attempting the exercises – which is indispensable, else one will get essentially nothing out of a perusal of this work. The homework problems themselves are never especially hard yet working through them will not, in any case, detain one for long since there are only 87 in the entire text. As long as one commit oneself to learning the material for real elsewhere, solving all of the exercises in Verhulst will help build one’s intuition. Thus, the present text may be awarded three and a half stars: one should plow through it quickly to pick up culture but not by any means to win expertise.