Stability and Periodic Solutions of Ordinary and Functional Differential Equations

This text, a Dover reprint of a book originally published in 1985, takes a somewhat unusual approach to the subject of ordinary differential equations. The author addresses a broad class of differential equations (ordinary differential equations, as well as Volterra equations and functional equations with bounded, unbounded and infinite delays) in a unified treatment. Most textbooks treat the latter subjects separately if at all, but the author makes a good case for treating them together.

The book is designed as a graduate-level text for students in mathematics, engineering and physics. It is divided into four parts. The first part presents the standard theory of the structure and stability of linear differential equations. The author then moves on to Volterra integrodifferential equations; he notes that their solution spaces are essentially indistinguishable from those of nonautonomous linear ordinary differential equations. Consequently, he believes that it is more natural to treat the topics together.

The second part consists primarily of examples designed to motivate and fill in some of the historical background. After starting with some standard fare on second-order equations that model mechanical and electrical damping, the author proceeds to two very nice examples of delay equations. One of them considers a model for controlling a ship that incorporates the delay between ordering a course change and the response of the ship. Another analyzes the motion of a sunflower plant in response to a growth hormone that acts to keep the plant as close to vertical as possible. A concluding set of examples looks at more generic applications of delay equations in biology, economics and epidemics.

The author addresses existence questions — of solutions and of periodic solutions — in the third part. His approach focuses almost exclusively on the use of fixed-point theory — first via the contraction-mapping theorem for linear differential equations, and then variations of Schauder’s theorem for nonlinear equations.

The book concludes with a long chapter on limit sets, periodicity and stability. Among other standard material it includes a discussion of the Poincaré-Bendixson theorem and Liapunov theory. Special topics that he treats here are equations with bounded delays, the Volterra equation with infinite delay, and stability theory for systems with unbounded delays.

This is an unusual text by the standards of many textbooks now used for graduate level courses. Many of the now-standard topics like local behavior of solutions near equilibria and qualitative behavior of solutions are mostly absent. Nonetheless, there are valuable features here that do not get much attention in standard first courses — differential equations with delay, for example. Using fixed-point theory to establish existence of solutions is also a nice feature and offers some desirable insights and connections to topology and geometry.

All the exercises here are incorporated inline within the text. They are very plentiful in Chapter 1 but considerably sparser in later chapters. The index is quite poor; it is very hard to find specific items in the book. The author also uses his own acronyms, and their meanings are also sometimes quite elusive. (VP for variation of parameters wasn’t difficult but I was stumped for quite a while by others, such as PMS for primary matrix solution.)

This would probably not be the text of choice for a first graduate course, but it does have some very good supplementary material on delay equations.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.