Nonlinear ordinary differential equations.

*(English)*Zbl 0417.34002
Oxford Applied Mathematics and Computing Science Series. Oxford: Clarendon Press; Oxford University Press. VIII, 360 p. £12.00 (1977).

This is a textbook on the qualitative theory of nonlinear ordinary differential equations. It is designed for advanced undergraduate students and is directed towards practical applications. There are over 400 examples and exercises.

Chapter 1 introduces the phase plane and autonomous equations of the form \(\ddot x= f(x,\dot x)\), while in Chapter 2 the more general autonomous system \(\dot x= X(x,y)\), \(\dot y= Y(x,y)\) is considered. In particular, the linear approximation \(\dot x= ax+by\), \(\dot y= cx+dy\) in the neighborhood of the critical point \((0,0)\) leads to the usual classification of equilibrium points. The authors continue the treatment of autonomous systems in Chapter 3 with a discussion of the index, and geometrical and computational aspects of the phase diagram. Perturbation methods (when the differential equation contains a small parameter) and second Jordan and Smith. Nonlinear ordinary differential equations order of the form \(\ddot x+f(x,\dot x)=\) periodic forcing function, receive substantial attention. Perturbation theory of nonautonomous equations, viz.: \(\ddot x=f(x,\dot x,t,\varepsilon)\) and equations such as \(\varepsilon\ddot x+\cdots\) are covered in Chapter 6. This is truly a book on nonlinear differential equations. Chapter 8 on linear systems covers linearly independent solutions, Wronskian, Green’s functions, Floquet theory, Hill’s and Mathieu’s equation in 30 pages – including exercises, I do not find this of liability. It is adequate for the author’s objectives.

The last three chapters treat stability, Lyapunov functions, the Poincaré-Bendixson theorem, and the existence of periodic solutions.

The book is commended to the reader who has some background in the elementary theory of differential equations (and some knowledge of linear differential equations) and wishes to pursue the subject of nonlinear differential equations. General existence and uniqueness theory is neither covered in this book, nor is it prerequisite.

Chapter 1 introduces the phase plane and autonomous equations of the form \(\ddot x= f(x,\dot x)\), while in Chapter 2 the more general autonomous system \(\dot x= X(x,y)\), \(\dot y= Y(x,y)\) is considered. In particular, the linear approximation \(\dot x= ax+by\), \(\dot y= cx+dy\) in the neighborhood of the critical point \((0,0)\) leads to the usual classification of equilibrium points. The authors continue the treatment of autonomous systems in Chapter 3 with a discussion of the index, and geometrical and computational aspects of the phase diagram. Perturbation methods (when the differential equation contains a small parameter) and second Jordan and Smith. Nonlinear ordinary differential equations order of the form \(\ddot x+f(x,\dot x)=\) periodic forcing function, receive substantial attention. Perturbation theory of nonautonomous equations, viz.: \(\ddot x=f(x,\dot x,t,\varepsilon)\) and equations such as \(\varepsilon\ddot x+\cdots\) are covered in Chapter 6. This is truly a book on nonlinear differential equations. Chapter 8 on linear systems covers linearly independent solutions, Wronskian, Green’s functions, Floquet theory, Hill’s and Mathieu’s equation in 30 pages – including exercises, I do not find this of liability. It is adequate for the author’s objectives.

The last three chapters treat stability, Lyapunov functions, the Poincaré-Bendixson theorem, and the existence of periodic solutions.

The book is commended to the reader who has some background in the elementary theory of differential equations (and some knowledge of linear differential equations) and wishes to pursue the subject of nonlinear differential equations. General existence and uniqueness theory is neither covered in this book, nor is it prerequisite.

Reviewer: K. S. Miller

##### MSC:

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

34A34 | Nonlinear ordinary differential equations and systems, general theory |

34Cxx | Qualitative theory for ordinary differential equations |

34Dxx | Stability theory for ordinary differential equations |

34Exx | Asymptotic theory for ordinary differential equations |