##
**Dynamical systems and numerical analysis.**
*(English)*
Zbl 0869.65043

Cambridge Monographs on Applied and Computational Mathematics. 2. Cambridge: Cambridge University Press. xxi, 685 p. (1996).

The theory of dynamical systems is a very fascinating subject which appears in the modeling of naturally occurring phenomena and at the same time is closely related to many areas in pure and applied mathematics. In particular, numerical methods for differential equations are fundamental tools to understand the behaviour of many dynamical systems. Hence, the relationship between the numerical analysis of initial value problems and the theory of dynamical systems has increased considerably in the last years.

In many pictures that we can see of the phase space of dynamical systems what we really are seeing is a numerical simulation of the original problem, i.e. the dynamics of an approximate problem generated by the numerical method, and therefore it is important to know whether or not the numerical method preserves the main features of interest in the phase portraits and also if it may introduce undesirable spurious solutions to the original problem. This kind of problems has led in the last fifteen years to a growing interest on some aspects of the interaction between dynamical systems and numerical analysis. Thus, some well-known numerical analysis researchers in the field like Griffiths, Hairer, Iserles, Sanz-Serna as well as the authors of the present book have made important contributions on the conditions to be satisfied for a numerical method which does not introduce spurious solutions. Further, the existence and construction of numerical methods that preserve some structural properties of the underlying flow (such as symplecticity or orthogonality) have been the subject of many recent papers.

In view of the enormous literature on dynamical systems, too large for anyone but for a specialist to keep track of, together with the above mentioned research on numerical methods it is gratifying that the authors, which are well-known researchers in the field, give us their unified perspective on the subject.

The book contains eight chapters and three appendices with notations, linear algebra and fixed point theorems. In Chapters 1 and 2 the fundamental theory for maps and for ordinary differential equations is presented. The two chapters run parallel to one another in the sense that each concept on maps given in Chapter 1 has its corresponding for ordinary differential equations in Chapter 2. In Chapter 3 the classical theory of Runge-Kutta and multistep methods for ordinary differential equations is briefly revised. In Chapter 4, such standard methods are formulated as dynamical systems, studying some problems on the existence, uniqueness and continuous dependence of solutions to the method. Chapter 5 deals with the stability of numerical methods. Here besides the classical theories for linear problems (\(A\)-stability) and nonlinear contractive problems with respect to a scalar product (\(G\)- and \(B\)-stability), the authors study the behaviour of some numerical methods for dissipative and gradient systems (in the sense of dynamical systems). Chapter 6 is concerned with the convergence of invariant sets for numerical methods under some general assumptions. Further studies on stability and convergence are considered in Chapter 7, and the book concludes in Chapter 8 with a brief survey of numerical methods for Hamiltonian problems.

The book is self-contained and the excellent organization of the material makes it an ideal source for a graduate course or a seminar selecting suitably the corresponding material. The mathematical presentation is always exact and the writing style is very clear. Further, many examples that illustrate some aspects of the theory are included and each section is ended with a selection of exercises. However, in spite of the opinion stated by the authors that “computation and numerical analysis should be taught together”, the book under review does not address most computational aspects concerned with the practice of numerical methods. In any case the book provides an attractive presentation of a very active interdisciplinary field and can be recommended for all people related to dynamical systems or the numerical analysis of differential equations.

In many pictures that we can see of the phase space of dynamical systems what we really are seeing is a numerical simulation of the original problem, i.e. the dynamics of an approximate problem generated by the numerical method, and therefore it is important to know whether or not the numerical method preserves the main features of interest in the phase portraits and also if it may introduce undesirable spurious solutions to the original problem. This kind of problems has led in the last fifteen years to a growing interest on some aspects of the interaction between dynamical systems and numerical analysis. Thus, some well-known numerical analysis researchers in the field like Griffiths, Hairer, Iserles, Sanz-Serna as well as the authors of the present book have made important contributions on the conditions to be satisfied for a numerical method which does not introduce spurious solutions. Further, the existence and construction of numerical methods that preserve some structural properties of the underlying flow (such as symplecticity or orthogonality) have been the subject of many recent papers.

In view of the enormous literature on dynamical systems, too large for anyone but for a specialist to keep track of, together with the above mentioned research on numerical methods it is gratifying that the authors, which are well-known researchers in the field, give us their unified perspective on the subject.

The book contains eight chapters and three appendices with notations, linear algebra and fixed point theorems. In Chapters 1 and 2 the fundamental theory for maps and for ordinary differential equations is presented. The two chapters run parallel to one another in the sense that each concept on maps given in Chapter 1 has its corresponding for ordinary differential equations in Chapter 2. In Chapter 3 the classical theory of Runge-Kutta and multistep methods for ordinary differential equations is briefly revised. In Chapter 4, such standard methods are formulated as dynamical systems, studying some problems on the existence, uniqueness and continuous dependence of solutions to the method. Chapter 5 deals with the stability of numerical methods. Here besides the classical theories for linear problems (\(A\)-stability) and nonlinear contractive problems with respect to a scalar product (\(G\)- and \(B\)-stability), the authors study the behaviour of some numerical methods for dissipative and gradient systems (in the sense of dynamical systems). Chapter 6 is concerned with the convergence of invariant sets for numerical methods under some general assumptions. Further studies on stability and convergence are considered in Chapter 7, and the book concludes in Chapter 8 with a brief survey of numerical methods for Hamiltonian problems.

The book is self-contained and the excellent organization of the material makes it an ideal source for a graduate course or a seminar selecting suitably the corresponding material. The mathematical presentation is always exact and the writing style is very clear. Further, many examples that illustrate some aspects of the theory are included and each section is ended with a selection of exercises. However, in spite of the opinion stated by the authors that “computation and numerical analysis should be taught together”, the book under review does not address most computational aspects concerned with the practice of numerical methods. In any case the book provides an attractive presentation of a very active interdisciplinary field and can be recommended for all people related to dynamical systems or the numerical analysis of differential equations.

Reviewer: M.Calvo (Zaragoza)

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |