Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolic
This is an introductory textbook on the analysis of nonlinear partial differential equations using techniques from dynamical systems theory. The main goal is to present techniques and results for nonlinear PDEs on unbounded domains and for modulation equations in a self-contained format. This is achieved by developing the theory in four parts: Part I is an introduction to finite-dimensional dynamics defined by ordinary differential equations; Part II is focused on dynamics in countably many dimensions, typically defined by PDEs on bounded domains; Part III is concerned with PDEs on the real line; Part IV deals with modulation theory and applications. The end of each chapter contains useful exercises, and there are many references to related or more advanced results. I think this is an excellent book, and I would be excited to use it for a topics course for our graduate students interested in PDEs and dynamics.
Chapters 2-4, which cover ODEs and form Part I of the book, are relatively comprehensive and include (albeit briefly) most topics typically covered in a graduate introduction to the subject, including existence and uniqueness, linear systems, Floquet theory, the Hartman-Grobman theorem, \(\omega\)-limit sets and attractors, chaos, stable/unstable/center manifold theory, bifurcations, and Hamiltonian dynamics. The authors have used this as a nice way to introduce a certain perspective on nonlinear dynamics and begin to build a toolkit that will be utilized and further developed in later chapters on PDEs. This is quite useful because often a proof technique appears for the first time in this finite-dimensional setting, and this base case can then be referred to when studying similar ideas in more complicated settings in the later chapters. A distinction is made between dissipative and Hamiltonian dynamics and the types of methods that are typically useful in each setting. Personally, I would probably not use these chapters as a replacement for a standard graduate course on ODEs, simply because the treatment of each topic is (necessarily) rather brief. However, one could easily do so, especially with the addition of a bit of supplementary material.
Part II consists of Chapters 5-6. Chapter 5 discusses PDEs on an interval, including a detailed treatment of the Chafee-Infante problem as an example. This setting is used to introduce a variety of key ideas in PDE theory, including semigroup theory, Sobolev spaces, and Fourier analysis, as well as to discuss important issues like smoothing and compactness. I found it very helpful to see these ideas presented first in this countable context, because one can really see where the finite-dimensional theory starts to breaks down. Chapter 6 is focused on the Navier-Stokes equation on a bounded domain, typically a torus, and includes basic results on local existence and uniqueness and a discussion of the important open millennium problem and of why standard methods fail to address it. Although this chapter seemed somewhat distinct from the overall theme of modulation equations, I still appreciated its inclusion, since the Navier-Stokes equation is such a well known system and a natural place to apply many of the techniques developed in the book up to this point.
Partial differential equations on the infinite line are the focus on Chapters 7-9, which comprise Part III of the book. Many of the key ideas are presented through examples, such as the KPP equation, Allen-Cahn equation, Burgers equation, reaction-diffusion systems, and the canonical modulation equations: the Nonlinear Schrödinger (NLS), Ginzburg-Landau (GL), and Korteweg-de Vries (KdV) equations. Topics covered include existence and uniqueness, the maximum principle, attractors, traveling waves, the Fourier transform, shocks, solitary waves, and Turing patterns. At this point the book starts to feel less like a book about an overall theory, which is illustrated by examples, and more like a book about specific equations and techniques that can be used in those particular contexts. This is not necessarily a bad thing, and perhaps in some situations necessary.
Finally, Chapters 10-14 from Part IV are on modulation equations and their applications. The GL equation is covered in Chapter 10, the NLS equation in Chapter 11, and the KdV and other equations concerning long wavelength in Chapter 12. The derivation of each of the modulation equations includes a significant number of rigorous results on their validity, as well as formal and physical intuition, which is a nice complementary perspective. There is also a discussion of why these equations can be viewed as universal reduced models, which describe a wide variety of physical phenomenon. Chapter 13 is focused on center manifold reduction and spatial dynamics, particularly in the context of modulation equations. Although quite brief, it is a nice introduction to the subject. Finally, Chapter 14 is concerned with diffusive stability, including the notions of relevant and irrelevant nonlinearities; it is a very nice summary of the main ideas involved, which keeps the treatment as straightforward as possible.
When I think of other books that approach PDEs from a dynamical systems point of view, the first two that come to mind are “Spectral and Dynamical Stability of Nonlinear Waves”, by Kapitula and Promislow, and “Geometric Theory of Semilinear Parabolic Equations”, by Henry. The former is largely focused on linear theory, and this book by Schneider and Uecker is a nice complement to it. The topics in the latter have a nontrivial overlap with this book, but this book is a more modern treatment, and it goes further and includes many additional topics.
Overall, this is a very nice book that collects a wide variety of interesting results on nonlinear PDEs from the dynamical systems point of view that I have not seen elsewhere in a textbook. It would be an excellent starting point for students to learn about this perspective and recent results from the last few decades, and also a very useful reference book for researchers in this or related fields.