Differentiable Dynamical Systems: An Introduction to Structural Stability and Hyperbolicity

This text is an introduction to differentiable dynamical systems at the graduate level. Its emphasis on structural stability and hyperbolicity gives it a clear focus and allows a more manageable treatment of a subject whose breadth has grown substantially over the last few decades. For simplicity the author limits his treatment to the discrete setting and works with iterates of diffeomorphisms.

For a vector field associated with a dynamical system to be structurally stable means that the qualitative behavior of trajectories is unaffected by small \(C^1\) perturbations. Structural stability is a central idea whose history goes back to pioneering work by Andronov and Pontryagin. In the 1960s, Peixoto revived their work and proved important results about structural stability on orientable closed surfaces. Smale thought Peixoto’s results could be extended to higher dimensions, but realized (with help from Levinson) that they could not. A great deal of fascinating work followed from that realization. Hyperbolicity plays a big role.

The author begins with basic concepts in dynamical systems: limit set, nonwandering set, transitive and minimal sets, topological conjugacy and structural stability. To illustrate the concepts he provides a condensed account of the classical theory of circle homeomorphisms. The first chapter concludes with Conley’s fundamental theorem of dynamical systems (every dynamical system is “gradient-like” except for chain recurrent points) and its proof.

The chapter that follows introduces hyperbolicity in the simplest case of a single fixed point. The author defines hyperbolic linear isomorphisms of finite-dimensional normed vector spaces with their characteristic splitting into invariant subspaces. He then considers the stability of a hyperbolic fixed point and persistence of hyperbolicity under perturbation. Next he states and proves the Hartman-Grobman and stable manifold theorems for a hyperbolic fixed point. Three examples critical in the historical development of the subject follow: the Smale horseshoe, Anosov’s toral automorphism and the solenoid attractor.

The most technically challenging material is in Chapter 4, which generalizes the idea of hyperbolicity from a fixed point to a general invariant set. Here the analytical foundation of the theory of structural stability is laid out. The author has arranged the text so that the difficult aspects here have parallels in Chapter 2 that are more easily understood.

In the final two chapters the author introduces Axiom A diffeomorphisms and proves Smale’s \(\Omega\)-stability theorem. He concludes by introducing the notions of quasi-hyperbolicity and linear transversality, and then looks briefly at outstanding conjectures in stability theory.

This is a clearly written and attractive introduction that has obviously been informed by extensive classroom experience. The English syntax is occasionally a little fractured, but the mathematics comes through well enough. Clark Robinson’s An Introduction to Dynamical Systems: Continuous and Discrete offers a broader range of topics at a less sophisticated level and might serve as a useful companion volume for the novice. 

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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.