Global bifurcations and chaos. Analytical methods.

*(English)*Zbl 0661.58001
Applied Mathematical Sciences, 73. New York etc.: Springer-Verlag. xiv, 494 p. DM 98.00 (1988).

The book is devoted to chaotic phenomena in deterministic nonlinear dynamical systems. It is self-contained, and may be used as the first introduction on the subject. There are many excellent figures and examples, that allows to the author to set forth new and modern results easily and simply.

Chapter One contains some introductory material. In Chapter Two the techniques of Conley-Moser and Afrajmovich-Bykov-Silnikov for proving that an invertible map has a hyperbolic, chaotic invariant Cantor set are generalized to arbitrary (finite) dimension and subshifts of finite type. Similar techniques are developed for the nonhyperbolic case. In Chapter Three the nonhyperbolic techniques are applied to the study of the orbit structure near orbits homoclinic to normally hyperbolic invariant tori. In Chapter Four the author develops a class of global perturbation techniques that enable one to detect orbits homoclinic or heteroclinic to hyperbolic fixed points, hyperbolic periodic orbits, and normally hyperbolic tori in a large class of systems.

Chapter One contains some introductory material. In Chapter Two the techniques of Conley-Moser and Afrajmovich-Bykov-Silnikov for proving that an invertible map has a hyperbolic, chaotic invariant Cantor set are generalized to arbitrary (finite) dimension and subshifts of finite type. Similar techniques are developed for the nonhyperbolic case. In Chapter Three the nonhyperbolic techniques are applied to the study of the orbit structure near orbits homoclinic to normally hyperbolic invariant tori. In Chapter Four the author develops a class of global perturbation techniques that enable one to detect orbits homoclinic or heteroclinic to hyperbolic fixed points, hyperbolic periodic orbits, and normally hyperbolic tori in a large class of systems.

Reviewer: Yu.Latushkin

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

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