##
**Introduction to applied nonlinear dynamical systems and chaos.**
*(English)*
Zbl 0701.58001

Texts in Applied Mathematics, 2. New York etc.: Springer-Verlag. xiv, 672 p. DM 98.00/hbk (1990).

This significant textbook is a thorough introduction to applied nonlinear dynamics and chaos. As pointed out in the preface, the book is developed from the material presented in a year-long graduate-level course in nonlinear dynamics, that the author has taught at Caltech over the past five years.

In the study of a given dynamical system, the main goal is to give a complete characterization of the geometry of the orbit structure and, if the dynamical system depends on parameters, to characterize the change in the orbit structure as the parameters are varied. As to succeed in this goal is often a sort of an art, the author provides students with a strong theoretical base, by systematically introducing a large number of techniques. So, their chances of success when faced with a nonlinear problem are considerable increased.

Chapter 1 introduces the geometrical point of view of dynamical systems. An in depth knowledge of V. I. Arnold’s “Ordinary Differential Equations” (1973; Zbl 0296.34001), or M. W. Hirsch and S. Smale’s “Differential Equations, Dynamical Systems, and Linear Algebra” (1974; Zbl 0309.34001), is recommended by the author to be an ideal prerequisite for a nonlinear dynamics course. However, for those who do not satisfy this prerequisite, a detailed and very well structured review of the necessary background material is given in Section 1.1. Section 1.2 is devoted to the theory and construction of PoincarĂ© maps. The author develops the material presented in Chapter 1 around a specific dynamical system: the damped, forced Duffing oscillator. In this way, the main “strategy” in the study of dynamical systems, which consists in applying a variety of ideas and techniques in order to obtain as much information on the specific system as possible, is clearly illustrated from the beginning.

In Chapter 2, “Methods for Simplifying Dynamical Systems”, the center manifold theory and the method of normal forms are presented. These techniques form the foundation for the development of the local bifurcation theory in the next chapter.

In Chapter 3, the bifurcations occurring in the neighbourhood of fixed points of vector fields and maps are studied. The author does not consider the problems that appear in the case of dynamical systems having parameters that change in time and pass through bifurcation values, for which the behaviour is often very different from the situations where the parameters are constant. As it is pointed out, these problems fit better into the context of singular perturbation theory.

Chapter 4 is concerned with global aspects of dynamics. The mechanisms that give rise to chaotic dynamics are studied and analytical techniques for predicting when these mechanisms occur in specific dynamical systems, in terms of the system parameters, are introduced. In order to help the student to more easily develop his or her geometrical intuition, the author has limited most of the geometrical constructions to two dimensions for maps and to three dimensions for vector fields. For a discussion of the results for higher dimensions, the reader may consult a previous book written by the author, “Global Bifurcations and Chaos. Analytical Methods” (1988; Zbl 0661.58001). In Section 4.1 the author describes and analyzes the prototypical map possessing a chaotic invariant set: the Smale horseshoe. He proceeds to a geometrical construction of the invariant set of the map and uses the nature of this construction in such a way, as to motivate a description of the dynamics on its invariant set by symbolic dynamics. Section 4.2 discusses some aspects of symbolic dynamics viewed as an independent subject. In Section 4.3, “How to Prove That a Dynamical System is Chaotic”, the author introduces the Conley-Moser conditions, and in Section 4.4, entitled “Dynamics Near Homoclinic Points of Two-Dimensional Maps”, he shows that the existence of transverse homoclinic orbits to a hyperbolic fixed point of a two-dimensional map, implies that in a sufficiently small neighborhood of a point on the homoclinic orbit, the Conley-Moser conditions hold. Section 4.5 introduces the Melnikov’s method for proving the existence of transverse homoclinic orbits to hyperbolic periodic orbits in a class of two-dimensional, time-periodic vector fields. Section 4.6, “Geometry and Dynamics in the Tangle”, explores some aspects of the relationships between dynamics and geometry in the homoclinic (or heteroclinic) tangle. In Section 4.7 the author shows that under certain conditions, the creation of the complicated dynamics associated with a transverse homoclinic orbit to a hyperbolic periodic orbit is due to an infinite sequence of period-doubling and saddle-node bifurcations. Section 4.8 studies the orbit structure near orbits homoclinic to hyperbolic fixed points of three-dimensional autonomous vector fields; by the term “near”, the author refers to both phase space and parameter space. Section 4.9 considers the study of global bifurcations arising from local codimension-two bifurcations. In Section 4.10 the author introduces the concept of Lyapunov exponents, and in the final section - 4.11, he explains what is meant by the term “chaos” as applied to deterministic dynamical systems, as well as by the notion of “strange attractor”.

The book is written in an exceptionally clear style. Each chapter includes an extensive set of problems keyed to the subject matter discussed. A vast and up to day bibliography is proposed. There is a detailed table of contents, that will be very helpful in locating specific subjects. Besides, a wide listing of details is offered in the Index.

Advanced students in engineering, physics, chemistry and biology, who do not necessarily have an extensive mathematical background, will certainly find this textbook as useful as students in mathematics.

In the study of a given dynamical system, the main goal is to give a complete characterization of the geometry of the orbit structure and, if the dynamical system depends on parameters, to characterize the change in the orbit structure as the parameters are varied. As to succeed in this goal is often a sort of an art, the author provides students with a strong theoretical base, by systematically introducing a large number of techniques. So, their chances of success when faced with a nonlinear problem are considerable increased.

Chapter 1 introduces the geometrical point of view of dynamical systems. An in depth knowledge of V. I. Arnold’s “Ordinary Differential Equations” (1973; Zbl 0296.34001), or M. W. Hirsch and S. Smale’s “Differential Equations, Dynamical Systems, and Linear Algebra” (1974; Zbl 0309.34001), is recommended by the author to be an ideal prerequisite for a nonlinear dynamics course. However, for those who do not satisfy this prerequisite, a detailed and very well structured review of the necessary background material is given in Section 1.1. Section 1.2 is devoted to the theory and construction of PoincarĂ© maps. The author develops the material presented in Chapter 1 around a specific dynamical system: the damped, forced Duffing oscillator. In this way, the main “strategy” in the study of dynamical systems, which consists in applying a variety of ideas and techniques in order to obtain as much information on the specific system as possible, is clearly illustrated from the beginning.

In Chapter 2, “Methods for Simplifying Dynamical Systems”, the center manifold theory and the method of normal forms are presented. These techniques form the foundation for the development of the local bifurcation theory in the next chapter.

In Chapter 3, the bifurcations occurring in the neighbourhood of fixed points of vector fields and maps are studied. The author does not consider the problems that appear in the case of dynamical systems having parameters that change in time and pass through bifurcation values, for which the behaviour is often very different from the situations where the parameters are constant. As it is pointed out, these problems fit better into the context of singular perturbation theory.

Chapter 4 is concerned with global aspects of dynamics. The mechanisms that give rise to chaotic dynamics are studied and analytical techniques for predicting when these mechanisms occur in specific dynamical systems, in terms of the system parameters, are introduced. In order to help the student to more easily develop his or her geometrical intuition, the author has limited most of the geometrical constructions to two dimensions for maps and to three dimensions for vector fields. For a discussion of the results for higher dimensions, the reader may consult a previous book written by the author, “Global Bifurcations and Chaos. Analytical Methods” (1988; Zbl 0661.58001). In Section 4.1 the author describes and analyzes the prototypical map possessing a chaotic invariant set: the Smale horseshoe. He proceeds to a geometrical construction of the invariant set of the map and uses the nature of this construction in such a way, as to motivate a description of the dynamics on its invariant set by symbolic dynamics. Section 4.2 discusses some aspects of symbolic dynamics viewed as an independent subject. In Section 4.3, “How to Prove That a Dynamical System is Chaotic”, the author introduces the Conley-Moser conditions, and in Section 4.4, entitled “Dynamics Near Homoclinic Points of Two-Dimensional Maps”, he shows that the existence of transverse homoclinic orbits to a hyperbolic fixed point of a two-dimensional map, implies that in a sufficiently small neighborhood of a point on the homoclinic orbit, the Conley-Moser conditions hold. Section 4.5 introduces the Melnikov’s method for proving the existence of transverse homoclinic orbits to hyperbolic periodic orbits in a class of two-dimensional, time-periodic vector fields. Section 4.6, “Geometry and Dynamics in the Tangle”, explores some aspects of the relationships between dynamics and geometry in the homoclinic (or heteroclinic) tangle. In Section 4.7 the author shows that under certain conditions, the creation of the complicated dynamics associated with a transverse homoclinic orbit to a hyperbolic periodic orbit is due to an infinite sequence of period-doubling and saddle-node bifurcations. Section 4.8 studies the orbit structure near orbits homoclinic to hyperbolic fixed points of three-dimensional autonomous vector fields; by the term “near”, the author refers to both phase space and parameter space. Section 4.9 considers the study of global bifurcations arising from local codimension-two bifurcations. In Section 4.10 the author introduces the concept of Lyapunov exponents, and in the final section - 4.11, he explains what is meant by the term “chaos” as applied to deterministic dynamical systems, as well as by the notion of “strange attractor”.

The book is written in an exceptionally clear style. Each chapter includes an extensive set of problems keyed to the subject matter discussed. A vast and up to day bibliography is proposed. There is a detailed table of contents, that will be very helpful in locating specific subjects. Besides, a wide listing of details is offered in the Index.

Advanced students in engineering, physics, chemistry and biology, who do not necessarily have an extensive mathematical background, will certainly find this textbook as useful as students in mathematics.

Reviewer: D.Savin

### MSC:

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

37Cxx | Smooth dynamical systems: general theory |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |

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