Mathematics for dynamic modeling.

*(English)*Zbl 0625.58012
Boston etc.: Academic Press, Inc., Harcourt Brace Jovanovich, Publishers. XVI, 277 p.; $ 27.50 (1987).

The concepts of equilibrium and stability, limit cycles, bifurcation and chaos, attractors and repellors are now often used in mathematics, engineering and biophysical sciences. This comes from the fact that most of the accurate descriptions of physical phenomena are inherently nonlinear. The behaviour of the nonlinear world is essentially different from that of the linear one: existence, uniqueness of the solutions can be tested at most locally and qualitatively. This book in an attractive and quasi elementary way, offers the mathematical tools to understand the before mentioned concepts. Here is the contents of the book:

Preface, (1) Simple Dynamic Models, (2) Stable and Unstable Motion, I, (3) Stable and Unstable Motion, II, (4) Growth and Decay, (5) Motion in Time and Space, (6) Cycles and Bifurcations, (7) Bifurcation and Catastrophe, (8) Chaos, (9) There is Better Way, Appendix (Ordinary Differential Equations), References and a Guide to Further Readings, Notes on Individual chapters and an Index.

The exposition is very attractive and the examples there treated are taken from different areas: mechanics (pendulum), biology (struggle for life of populations, algae blooms, pollution of rivers), geophysics (earth magnetic field reversal). All the chapters are provided at the end with very nice problems, whose solutions give the reader a deeper insight into the matters of the chapter. Of great use is the guide for further reading which offers interesting books of different levels. The reading of the book offers an accessible way to understand the main theorems which governs the nonlinear world: Poincaré-Bendixson, Hopf and the famous one of René Thom. We believe that this book gives an exciting portrait of the actual nonlinear description of physical phenomena and the behaviour of the last ones and will soon become a classic introduction in this area. Therefore it should be read by every scientist, mathematician, engineer or physicist.

Preface, (1) Simple Dynamic Models, (2) Stable and Unstable Motion, I, (3) Stable and Unstable Motion, II, (4) Growth and Decay, (5) Motion in Time and Space, (6) Cycles and Bifurcations, (7) Bifurcation and Catastrophe, (8) Chaos, (9) There is Better Way, Appendix (Ordinary Differential Equations), References and a Guide to Further Readings, Notes on Individual chapters and an Index.

The exposition is very attractive and the examples there treated are taken from different areas: mechanics (pendulum), biology (struggle for life of populations, algae blooms, pollution of rivers), geophysics (earth magnetic field reversal). All the chapters are provided at the end with very nice problems, whose solutions give the reader a deeper insight into the matters of the chapter. Of great use is the guide for further reading which offers interesting books of different levels. The reading of the book offers an accessible way to understand the main theorems which governs the nonlinear world: Poincaré-Bendixson, Hopf and the famous one of René Thom. We believe that this book gives an exciting portrait of the actual nonlinear description of physical phenomena and the behaviour of the last ones and will soon become a classic introduction in this area. Therefore it should be read by every scientist, mathematician, engineer or physicist.

Reviewer: D.Stanomir

##### MSC:

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

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

37C75 | Stability theory for smooth dynamical systems |

37A99 | Ergodic theory |

70K20 | Stability for nonlinear problems in mechanics |

34C25 | Periodic solutions to ordinary differential equations |