Dynamical systems. Stability, symbolic dynamics, and chaos. 2nd corr. and enlarged ed.

*(English)*Zbl 0914.58021
Studies in Advanced Mathematics. Boca Raton, FL: CRC Press. 506 p. (1999).

The second edition of this book improves slightly the first one (see our review in (1995; Zbl 0853.58001)) which fully characterizes its present state, too).

The minor changes are mentioned by the author in Preface to the second edition:

“The second edition of this book has provided the opportunity for correcting many minor typographical errors or mistakes. Needless to say, the basic approach and content of the book have stayed the same. The discussion of the saddle node bifurcation has been rewritten using notation to make it easier to understand. In an attempt to expand the comparison of results for diffeomorphisms and flows, I have added a section on the horseshoe for a flow with a transverse homoclinic point. This section makes explicit the meaning and interpretation of a horseshoe in the case of a flow instead of a diffeomorphism. Another subsection on horseshoes for nontransverse homoclinic points indicates some recent extensions to the understanding of how horseshoes arise. Also added is a section proving the ergodicity of a hyperbolic toral automorphism. This proof is fairly simple, but introduces an important technique which is used to prove ergodicity in other situations. Finally, a new chapter on Hamiltonian systems has been added. This chapter treats mainly local properties near fixed points, but should prove of interest to some of the readers”.

The minor changes are mentioned by the author in Preface to the second edition:

“The second edition of this book has provided the opportunity for correcting many minor typographical errors or mistakes. Needless to say, the basic approach and content of the book have stayed the same. The discussion of the saddle node bifurcation has been rewritten using notation to make it easier to understand. In an attempt to expand the comparison of results for diffeomorphisms and flows, I have added a section on the horseshoe for a flow with a transverse homoclinic point. This section makes explicit the meaning and interpretation of a horseshoe in the case of a flow instead of a diffeomorphism. Another subsection on horseshoes for nontransverse homoclinic points indicates some recent extensions to the understanding of how horseshoes arise. Also added is a section proving the ergodicity of a hyperbolic toral automorphism. This proof is fairly simple, but introduces an important technique which is used to prove ergodicity in other situations. Finally, a new chapter on Hamiltonian systems has been added. This chapter treats mainly local properties near fixed points, but should prove of interest to some of the readers”.

Reviewer: J.Andres (Olomouc)

##### MSC:

37C75 | Stability theory for smooth dynamical systems |

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37E99 | Low-dimensional dynamical systems |