Global Attractors of Non-Autonomous Dissipative Dynamical Systems
is an extended study of dissipative systems that include ordinary differential, partial differential, and functional differential equations for continuous and discrete time. The emphasis is on global properties, especially the existence and nature of global attractors. A prototypical example of these systems is the ODE u' = f (u,t) where all the solutions are eventually confined to a fixed bounded domain (as t goes to infinity) because of the natural effects of dissipation. (A simple example is harmonic motion with a time-dependent damping term.) The model for this work is the existing qualitative theory of ordinary differential equations.
The initial chapter introduces autonomous equations, describes several different versions of what "dissipative" can mean, and establishes when they are and are not equivalent. The second chapter begins the discussion of non-autonomous dissipative systems. Successive chapters explore special aspects and settings for non-autonomous dissipative systems: the analytic context, Lyapunov function methods, structure of global attractors, and triangle maps. One chapter focuses on global attractors for non-autonomous Navier-Stokes equations.
This is an extremely dense text that would be very challenging for anyone but experts in the field. The thread connecting individual chapters is pretty thin so it is difficult to understand how individual topics relate to each other. The index is sparse and it is very difficult to trace back to find definitions, for example. The writing is usually clear enough locally, in individual sections, but there are occasional non-idiomatic or ungrammatical phrases that are confusing. Overall, this book needs some serious editing and a far better index.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.