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Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
ht. 1. Engng Sci. Vol. 26, No. 2, pp. 221-222, 1988 Pergamon Press plc.Printed in Great Britain
BOOK REVIEW
and P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields: Applied Mathematical Sciences, Vol. 42, Springer-Verlag, New
J. GUCKENHEIMER
York, Berlin, Heidelberg,
Tokyo (1983). XVI + 453 pp., 206 Figs, DM 104.
This book is an excellent monograph on the modern treatment of nonlinear oscillations by geometrical methods of dynamical systems formulated in the style of the “Smale school”. It contains both qualitative and quantitative investigations of many nonlinear “real” systems which have ample applications in engineering and covers a detailed and rather rigorous analysis of some phenomena peculiar to those systems such as chaotic behaviour and several types of bifurcations. The mathematical models considered are confined to sets of nonlinear ordinary differential equations and nonlinear maps arisen from their discretizations on finite dimensional Euclidean manifolds. The book comprises seven chapters. The introductory first chapter reviews some basic topics in the geometrical approach to the theory of ordinary differential equations (flows) and nonlinear maps. The concepts of stable and unstable manifolds associated with fixed points, homoclinic orbits, Poincare maps and structural stability are introduced. The second chapter deals with an unsystematic treatment of four examples of nonlinear oscillations: van der Pol and Duffing oscillators, the Lorenz equations and the problem of a bouncing ball from a massive, sinusoidally vibrating table. It is shown that all these problems manifest chaotic behaviour and they possess strange attractors associated with the existence of homoclinic orbits. The third chapter discusses the local bifurcation theory for flows and maps with special emphasis on center manifold and normal form theorems. The analysis is restricted to one-parameter (codimension one) bifurcations of equilibria. The fourth chapter treats some classical analytical methods of averaging and perturbation to study periodically forced oscillations. It is shown that some global information are obtainable by these methods under suitable conditions such as approximations to Poincar6 maps. Melnikov’s method is described for two-dimensional forced system on the manifold R* x S’(R: real line, S’: circle) to study perturbations of homoclinic orbits and Hamiltonian systems. This chapter ends with a brief discussion of chaos and nonintegrability in Hamiltonian systems and KAM (Kolmogorov-Arnold-Moser) theory. Chapter five is concerned with quite a rigorous treatment of chaotic behaviour. Both the irregular character of individual solutions and the complicated geometric structures (invariant sets with hyperbolic structures, attracting sets, attractors and strange attractors) are described. The horseshoe map of Smale is discussed at length and the technique of symbolic dynamics is only briefly explained although it has been extensively employed as the principal method of attack. Invariant measures, Liapunov exponents and topological entropies of maps, capacity and Hausdorff dimension of metric spaces are defined in final sections. Chapter six involves dynamical properties which can not be deduced from local information. The global homoclinic and heteroclinic bifurcations, bifurcations of one-dimensional maps are discussed. Wild hiperbolic sets are introduced and the application of renormalization group methods (which were developed in condensed matter physics) in one-dimensional maps is outlined. The final chapter discusses bifurcations from equilibria which have multiple degeneracies. Bifurcations involve two parameters. The cases corresponding to eigenvalues which consist of double zero, a pure imaginary pair and a simple zero, two pure imaginary pairs are considered. A final section is devoted to two problems involving partial differential equations. The book is very well written and quality of its printing is excellent although there are a few but rather annoying misprints. We believe that the coverage would be very interesting to engineers, physicists and mathematicians. Mathematicians would be delighted to see many practical applications of the advanced mathematical approaches via microcomputers. However, engineers who are not well versed in the 221
222
Book review
underlying mathematical concepts and who are not much familiar with a long list of unproven theorems given in the text may need to make a large amount of collateral reading along the guidelines set in the book if they really want to gain an insight into this subject of ever-increasing importance and to acquire a mastery of the techniques involved. E. S. SUHUBI
BOOK REVIEW
and P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields: Applied Mathematical Sciences, Vol. 42, Springer-Verlag, New
J. GUCKENHEIMER
York, Berlin, Heidelberg,
Tokyo (1983). XVI + 453 pp., 206 Figs, DM 104.
This book is an excellent monograph on the modern treatment of nonlinear oscillations by geometrical methods of dynamical systems formulated in the style of the “Smale school”. It contains both qualitative and quantitative investigations of many nonlinear “real” systems which have ample applications in engineering and covers a detailed and rather rigorous analysis of some phenomena peculiar to those systems such as chaotic behaviour and several types of bifurcations. The mathematical models considered are confined to sets of nonlinear ordinary differential equations and nonlinear maps arisen from their discretizations on finite dimensional Euclidean manifolds. The book comprises seven chapters. The introductory first chapter reviews some basic topics in the geometrical approach to the theory of ordinary differential equations (flows) and nonlinear maps. The concepts of stable and unstable manifolds associated with fixed points, homoclinic orbits, Poincare maps and structural stability are introduced. The second chapter deals with an unsystematic treatment of four examples of nonlinear oscillations: van der Pol and Duffing oscillators, the Lorenz equations and the problem of a bouncing ball from a massive, sinusoidally vibrating table. It is shown that all these problems manifest chaotic behaviour and they possess strange attractors associated with the existence of homoclinic orbits. The third chapter discusses the local bifurcation theory for flows and maps with special emphasis on center manifold and normal form theorems. The analysis is restricted to one-parameter (codimension one) bifurcations of equilibria. The fourth chapter treats some classical analytical methods of averaging and perturbation to study periodically forced oscillations. It is shown that some global information are obtainable by these methods under suitable conditions such as approximations to Poincar6 maps. Melnikov’s method is described for two-dimensional forced system on the manifold R* x S’(R: real line, S’: circle) to study perturbations of homoclinic orbits and Hamiltonian systems. This chapter ends with a brief discussion of chaos and nonintegrability in Hamiltonian systems and KAM (Kolmogorov-Arnold-Moser) theory. Chapter five is concerned with quite a rigorous treatment of chaotic behaviour. Both the irregular character of individual solutions and the complicated geometric structures (invariant sets with hyperbolic structures, attracting sets, attractors and strange attractors) are described. The horseshoe map of Smale is discussed at length and the technique of symbolic dynamics is only briefly explained although it has been extensively employed as the principal method of attack. Invariant measures, Liapunov exponents and topological entropies of maps, capacity and Hausdorff dimension of metric spaces are defined in final sections. Chapter six involves dynamical properties which can not be deduced from local information. The global homoclinic and heteroclinic bifurcations, bifurcations of one-dimensional maps are discussed. Wild hiperbolic sets are introduced and the application of renormalization group methods (which were developed in condensed matter physics) in one-dimensional maps is outlined. The final chapter discusses bifurcations from equilibria which have multiple degeneracies. Bifurcations involve two parameters. The cases corresponding to eigenvalues which consist of double zero, a pure imaginary pair and a simple zero, two pure imaginary pairs are considered. A final section is devoted to two problems involving partial differential equations. The book is very well written and quality of its printing is excellent although there are a few but rather annoying misprints. We believe that the coverage would be very interesting to engineers, physicists and mathematicians. Mathematicians would be delighted to see many practical applications of the advanced mathematical approaches via microcomputers. However, engineers who are not well versed in the 221
222
Book review
underlying mathematical concepts and who are not much familiar with a long list of unproven theorems given in the text may need to make a large amount of collateral reading along the guidelines set in the book if they really want to gain an insight into this subject of ever-increasing importance and to acquire a mastery of the techniques involved. E. S. SUHUBI