##
**Quasi-conservative systems: cycles, resonances and chaos.**
*(English)*
Zbl 0914.58012

World Scientific Series on Nonlinear Science. Series A. 30. Singapore: World Scientific. xii, 325 p. (1998).

This book studies the theory of non-conservative oscillatory systems and their attractors, for systems which are close to nonlinear integrable ones. The book begins with a nice introductory chapter and is divided into two parts. The first one is devoted to the global analysis of conservative nonlinear autonomous systems (integrable and non-integrable ones). The main results in this area are recalled and problems such as the existence of quasi-attrators or the stability of the stationary regimes are discussed through the detailed study of representative examples.

The second part studies non-conservative perturbations in integrable systems, that is, quasi-conservative systems. The problem of the estimate of the maximal number of limit cycles is analyzed for the Duffing and the pendulum equations. The topology of the resonance zones for non-autonomous systems periodic in time is also discussed in detail.

Then the extension and generalization of the previous results to higher dimensions is illustrated by the Lorentz system. The final chapter is concerned with quasi-conservative systems which are not quasi-integrable. The extent of applicability of the method of using a small parameter is discussed; numerical results and computer experiments are given. The book ends with a review of some recent developments and notions of chaotic dynamics (Hausdorff dimension, Lyapunov exponents, etc.).

This book requires some basic knowledge of the subject and is a very valuable addition to the literature on nonlinear dynamics. An index is missing.

The second part studies non-conservative perturbations in integrable systems, that is, quasi-conservative systems. The problem of the estimate of the maximal number of limit cycles is analyzed for the Duffing and the pendulum equations. The topology of the resonance zones for non-autonomous systems periodic in time is also discussed in detail.

Then the extension and generalization of the previous results to higher dimensions is illustrated by the Lorentz system. The final chapter is concerned with quasi-conservative systems which are not quasi-integrable. The extent of applicability of the method of using a small parameter is discussed; numerical results and computer experiments are given. The book ends with a review of some recent developments and notions of chaotic dynamics (Hausdorff dimension, Lyapunov exponents, etc.).

This book requires some basic knowledge of the subject and is a very valuable addition to the literature on nonlinear dynamics. An index is missing.

Reviewer: V.BerthĂ© (Marseille)

### MSC:

37Cxx | Smooth dynamical systems: general theory |

34Cxx | Qualitative theory for ordinary differential equations |

70K30 | Nonlinear resonances for nonlinear problems in mechanics |

70K05 | Phase plane analysis, limit cycles for nonlinear problems in mechanics |

37-02 | Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory |

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 |