It introduces and rigorously develops the central concepts and methods in dynamical systems and their applications to a wide variety of topics. What this book contains We begin with a detailed discussion of a series of elementary but fundamental examples. This is achieved by exploring hyperbolicity, transversality, global topological invariants, and variational methods. The methods include the study of stable and unstable manifolds, bifurcations, index and degree, and construction of orbits as minima and minimaxes of action functionals. In the third and fourth parts the general program outlined in the first part is carried out to considerable depth for low-dimensional and hyperbolic dynamical systems which are particularly amenable to such analysis.
Hyperbolic systems are the prime example of well-understood complexity.
This manifests itself in an orbit structure that is rich both from the topological and statistical point of view and stable under perturbation. At the same time the principal features can be described qualitatively and quantitatively with great precision. In low-dimensional dynamical systems on the other hand there are two situations. Although we develop most themes related to differentiable dynamics in some depth we have not tried to write an encyclopedia of differentiable dynamics. Even if this were possible, the resulting work would be strictly a reference source and not useful as an introduction or a text.