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**Nonlinear oscillations, dynamical systems, and bifurcations of vector fields.**
*(English)*
Zbl 0515.34001

Applied Mathematical Sciences, 42. New York etc.: Springer-Verlag. xvi, 453 p., 206 figs. (1983).

The interest in chaotic behavior has been growing for the last twenty years. The book is concerned with the study of nonlinear oscillations. It is known that the solutions to many problems in this area demonstrate a chaotic behavior. There exist some attracting motions which are neither periodic nor quasiperiodic (strange attractors). These phenomena are studied in the book. It consists of seven chapters, a bibliography, an index, a glossary of terms, and an appendix ”Suggestions for further reading”.

Chapter 1, Introduction: Differential equations and dynamical systems, is a review of basic results in the theory of dynamical systems described by differential equations or by discrete mappings. Chapter 2, An introduction to chaos: four examples, deals with the Van der Pol oscillators, Duffing’s equation, the Lorenz equations, and with a bouncing ball problem, (a ball bouncing by a sinusoidally vibrating table). Chapter 3, Local bifurcation, presents the center manifold theorem, the normal forms, the Hopf bifurcations, and some examples of bifurcations.

In Chapter 4, Averaging and perturbation from a geometric viewpoint, a study of nonlinear periodically forced oscillators by the analytical methods is given, the Kolmogorov-Arnold-Moser theory and nonintegrability in Hamiltonian systems are discussed. Much attention is paid to Poincaré maps.

Chapter 5, Hyperbolic sets, symbolic dynamics, and strange attractors, the chaotic behavior of solutions to (ordinary) differential equations is studied. The horseshoe map is discussed and the method of symbolic dynamics is explained. The chapter ends with a brief discussion of the smooth ergodic theory.

Chapter 6, Global bifurcations, treats the global aspects of flows. Bifurcations of planar homoclinic and heteroclinic orbits, rotation numbers, bifurcations of one-dimensional maps, the Lorenz attractor. Shilnikov’s study of a three-dimensional flow in which there is a homoclinic trajectory to a saddle point with complex eigenvalues, homoclinic bifurcations of periodic orbits, and wild hyperbolic sets are discussed. The chapter ends with the section in which the technique of renormalization is introduced in order to study some universal features of the transition to chaotic behavior.

In Chapter 7, Local codimension two bifurcations of flows, the authors discuss bifurcations from equilibria which have a multiple degeneracies. The double zero eigenvalue, a pure imaginary pairs of eigenvalues without resonance are treated. The chapter ends with a section in which two physical problems are discussed. The first one deals with connection in a fluid layer. The second one deals with motions of an elastic panel subject to an axial load and a fluid flow along its surface.

The book can be used by graduate students and researchers interested in nonlinear oscillations. It is a valuable contribution to the literature.

Chapter 1, Introduction: Differential equations and dynamical systems, is a review of basic results in the theory of dynamical systems described by differential equations or by discrete mappings. Chapter 2, An introduction to chaos: four examples, deals with the Van der Pol oscillators, Duffing’s equation, the Lorenz equations, and with a bouncing ball problem, (a ball bouncing by a sinusoidally vibrating table). Chapter 3, Local bifurcation, presents the center manifold theorem, the normal forms, the Hopf bifurcations, and some examples of bifurcations.

In Chapter 4, Averaging and perturbation from a geometric viewpoint, a study of nonlinear periodically forced oscillators by the analytical methods is given, the Kolmogorov-Arnold-Moser theory and nonintegrability in Hamiltonian systems are discussed. Much attention is paid to Poincaré maps.

Chapter 5, Hyperbolic sets, symbolic dynamics, and strange attractors, the chaotic behavior of solutions to (ordinary) differential equations is studied. The horseshoe map is discussed and the method of symbolic dynamics is explained. The chapter ends with a brief discussion of the smooth ergodic theory.

Chapter 6, Global bifurcations, treats the global aspects of flows. Bifurcations of planar homoclinic and heteroclinic orbits, rotation numbers, bifurcations of one-dimensional maps, the Lorenz attractor. Shilnikov’s study of a three-dimensional flow in which there is a homoclinic trajectory to a saddle point with complex eigenvalues, homoclinic bifurcations of periodic orbits, and wild hyperbolic sets are discussed. The chapter ends with the section in which the technique of renormalization is introduced in order to study some universal features of the transition to chaotic behavior.

In Chapter 7, Local codimension two bifurcations of flows, the authors discuss bifurcations from equilibria which have a multiple degeneracies. The double zero eigenvalue, a pure imaginary pairs of eigenvalues without resonance are treated. The chapter ends with a section in which two physical problems are discussed. The first one deals with connection in a fluid layer. The second one deals with motions of an elastic panel subject to an axial load and a fluid flow along its surface.

The book can be used by graduate students and researchers interested in nonlinear oscillations. It is a valuable contribution to the literature.

Reviewer: A. G. Ramm

### MSC:

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

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

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

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

37E45 | Rotation numbers and vectors |

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

70H05 | Hamilton’s equations |

34C25 | Periodic solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

34C29 | Averaging method for ordinary differential equations |

37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |

70H08 | Nearly integrable Hamiltonian systems, KAM theory |