Ordinary differential equations with applications.

*(English)*Zbl 0937.34001
Texts in Applied Mathematics. 34. New York, NY: Springer. xv, 561 p. (1999).

This graduate-level textbook is a thorough introduction to the modern qualitative theory of ordinary differential equations (ODEs) written with geometrical point of view of dynamical systems and special attention to applications of ODEs to various problems in physical science and engineering.

In the study of a given dynamical system, the main goal is to give a complete characterization of the geometry of its orbit structure. To succeed in this goal the author provides the readers with a strong theoretical base by introducing a large number of classical and modern techniques. These techniques are illustrated in detail by studying various realistic and interesting problems of classical physics, mechanics, and engineering. The treatment of these problems is an integral part of the book and is of fundamental importance in presentation of the material. In particular, much attention is given to the origins of ODEs in mathematics and physical science.

Chapter 1 is a survey of the standard topics encountered in an introductory course of ODEs: existence theory, flows, invariant manifolds, Lyapunov stability, phase plane analysis. It also contains a review of calculus in Banach spaces.

Chapter 2 contains linear systems and linearized stability theories, including Floquet theory and Hill’s equation.

Chapter 3 is devoted to the origins and applications of ODEs. It is shown how ODEs arise from foundations of physical science (Euler-Lagrange equations, motion of charged particle, Kepler’s motion, diamagnetic Kepler problem and others). Dynamics of some mechanical systems (coupled pendula, inverted pendulum, and Fermi-Ulam-Pasta oscillator) is discussed. The author also discusses the origins of ODEs in the theory of partial differential equations (PDEs): parabolic PDEs, Galerkin approximations, and travelling wave solutions.

Chapter 4 is an introduction to the theory of hyperbolic structures in differential equations. Two fundamental theorems (the center and stable manifold theorem and the Hartman-Grobman theorem) are presented.

In Chapter 5, “Continuation of Periodic Solutions”, autonomous and in time periodic perturbations of periodic orbits of plane autonomous systems are considered. Methods of analysis given here are illustrated by studying forced oscillators, in particular, the forced van der Pol oscillator.

Chapter 6 contains a perturbation theory for homoclinic orbits of planar systems, Melnikov’s method, and transition to chaotic dynamics. As an application of these techniques, the chaotic dynamics of the ABC system (a system of ODEs derived from the Navier-Stokes equations) is discussed. Chapter 7 is an introduction to the method of averaging. Chapter 8 is concerned with two most important local bifurcations in ODEs associated with rest points: the saddle-node bifurcation and Hopf bifurcation.

Clearly and beautifully written, the book is a good introduction to the modern qualitative theory of differential equations and dynamical systems. It contains an extensive material explained in detail. The book is accessible to students who have studied differential equations during one undergraduate semester. Since the book contains some material that is not encountered in most treatments of the subject, it can be a good source of information for specialists. A great number of exercises enriches very much the presentation of the material.

In the study of a given dynamical system, the main goal is to give a complete characterization of the geometry of its orbit structure. To succeed in this goal the author provides the readers with a strong theoretical base by introducing a large number of classical and modern techniques. These techniques are illustrated in detail by studying various realistic and interesting problems of classical physics, mechanics, and engineering. The treatment of these problems is an integral part of the book and is of fundamental importance in presentation of the material. In particular, much attention is given to the origins of ODEs in mathematics and physical science.

Chapter 1 is a survey of the standard topics encountered in an introductory course of ODEs: existence theory, flows, invariant manifolds, Lyapunov stability, phase plane analysis. It also contains a review of calculus in Banach spaces.

Chapter 2 contains linear systems and linearized stability theories, including Floquet theory and Hill’s equation.

Chapter 3 is devoted to the origins and applications of ODEs. It is shown how ODEs arise from foundations of physical science (Euler-Lagrange equations, motion of charged particle, Kepler’s motion, diamagnetic Kepler problem and others). Dynamics of some mechanical systems (coupled pendula, inverted pendulum, and Fermi-Ulam-Pasta oscillator) is discussed. The author also discusses the origins of ODEs in the theory of partial differential equations (PDEs): parabolic PDEs, Galerkin approximations, and travelling wave solutions.

Chapter 4 is an introduction to the theory of hyperbolic structures in differential equations. Two fundamental theorems (the center and stable manifold theorem and the Hartman-Grobman theorem) are presented.

In Chapter 5, “Continuation of Periodic Solutions”, autonomous and in time periodic perturbations of periodic orbits of plane autonomous systems are considered. Methods of analysis given here are illustrated by studying forced oscillators, in particular, the forced van der Pol oscillator.

Chapter 6 contains a perturbation theory for homoclinic orbits of planar systems, Melnikov’s method, and transition to chaotic dynamics. As an application of these techniques, the chaotic dynamics of the ABC system (a system of ODEs derived from the Navier-Stokes equations) is discussed. Chapter 7 is an introduction to the method of averaging. Chapter 8 is concerned with two most important local bifurcations in ODEs associated with rest points: the saddle-node bifurcation and Hopf bifurcation.

Clearly and beautifully written, the book is a good introduction to the modern qualitative theory of differential equations and dynamical systems. It contains an extensive material explained in detail. The book is accessible to students who have studied differential equations during one undergraduate semester. Since the book contains some material that is not encountered in most treatments of the subject, it can be a good source of information for specialists. A great number of exercises enriches very much the presentation of the material.

Reviewer: E.Ershov (St.Peterburg)

##### MSC:

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

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

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

34C29 | Averaging method for ordinary differential equations |

34C28 | Complex behavior and chaotic systems of ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

34A26 | Geometric methods in ordinary differential equations |

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

34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |

34C23 | Bifurcation theory for ordinary differential equations |