Periodic motions.

*(English)*Zbl 0805.34037
Applied Mathematical Sciences. 104. New York, NY: Springer-Verlag. xiii, 577 p. (1994).

This book collects the results obtained for periodic solutions of ordinary differential equations by means of mostly standard methods. Some of them belong to the author. It is intended for graduate and post- graduate students in mathematics as well as in applied sciences. One can also use it as a standard reference for the classical results in this field.

Although there are already several monographs dealing with periodic boundary value problems from the topological point of view, the overlapping with Farkas’s book is not big. It is not a typical book on the nonlinear oscillations theory either, because the parts related to the linear theory are also included. So, it is more a suitable complement to those mentioned above. Moreover, several topics, like the concept of derive-periodic solutions or a zip-bifurcation, are systematically considered for the first time here.

The text is self-consistent and does not require a special preparation of the reader; the advanced calculus and linear algebra are mostly sufficient with this respect. Nevertheless, for the reader’s convenience, the background for more advanced parts (involving the advanced matrix theory, topological degree, fixed point theorems and differentiable manifolds) is contained in three appendices.

Formally, the material is divided into seven chapters: 1) Introduction, 2) Periodic solutions of linear systems, 3) Autonomous systems in the plane, 4) Periodic solutions of periodic systems, 5) Autonomous systems of arbitrary dimension, 6) Perturbations, 7) Bifurcations.

Because of the author’s recent interest, the majority of sensitively chosen illustrating examples is related to population dynamics. Some misprints do not mar a good readability of this nice work.

Although there are already several monographs dealing with periodic boundary value problems from the topological point of view, the overlapping with Farkas’s book is not big. It is not a typical book on the nonlinear oscillations theory either, because the parts related to the linear theory are also included. So, it is more a suitable complement to those mentioned above. Moreover, several topics, like the concept of derive-periodic solutions or a zip-bifurcation, are systematically considered for the first time here.

The text is self-consistent and does not require a special preparation of the reader; the advanced calculus and linear algebra are mostly sufficient with this respect. Nevertheless, for the reader’s convenience, the background for more advanced parts (involving the advanced matrix theory, topological degree, fixed point theorems and differentiable manifolds) is contained in three appendices.

Formally, the material is divided into seven chapters: 1) Introduction, 2) Periodic solutions of linear systems, 3) Autonomous systems in the plane, 4) Periodic solutions of periodic systems, 5) Autonomous systems of arbitrary dimension, 6) Perturbations, 7) Bifurcations.

Because of the author’s recent interest, the majority of sensitively chosen illustrating examples is related to population dynamics. Some misprints do not mar a good readability of this nice work.

Reviewer: J.Andres (Olomouc)

##### MSC:

34C25 | Periodic solutions to ordinary differential equations |

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

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

92D25 | Population dynamics (general) |

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