Synopses & Reviews
"Resonances are widely studied in most areas of engineering and physics, but the approach remains mostly computational or experimental. The reason is that even reduced models of resonant interactions are typically higher dimensional and exhibit great complexity; therefore, they are inaccessible to textbook techniques from dynamical systems theory. This book offers the first systematic exposition of recent analytic results that can be used to understand and predict the global effect of resonances in phase space. The geometric methods discussed here enable one to identify complicated multi-time-scale solution sets and slow-fast chaos in physical problems. The topics include slow and partially slow manifolds, homoclinic and heteroclinic jumping, universal global bifurcations, generalized (\vS)ilnikov orbits and manifolds, disintegration of invariant manifolds near resonances, and high-codimension homoclinic jumping. The main emphasis is on near-integrable dissipative systems, but a separate chapter is devoted to resonance phenomena in Hamiltonian systems. A number of applications are described from the areas of fluid mechanics, rigid body dynamics, chemistry, atmospheric science, and nonlinear optics. In addition, the theory is extended to infinite dimensions to cover resonances in certain nonlinear partial differential equations, such as single and coupled nonlinear Schr(\ddot o)dinger equations. This self-contained monograph should be useful to mathematicians interested in the geometric theory of multi-and infinite-dimensional dynamical systems, as well as to the applied scientist who wishes to analyze resonances in physical problems."
Review
"An extensive bibliography and the many examples make this clearly-written book an excellent introduction to these techniques for identifying chaos in perturbations of systems with resonance." Applied Mechanics Reviews, Vol. 53/4, April 2000 "Haller makes a point of wanting to see dynamical systems theory fulfil "its long-standing promise to solve real-life problems". His book, through a wealth of detailed examples, delivers on this promise, and is certain ti become a standard text in this area. In particular, it is an excellent introduction to this research area, and contains a wealth of bibliographical and historical detail. Matthew Nicol, Bulletin of the LMS, No. 162, Vol. 33/3, May 2001
Synopsis
Resonances are ubiquitous in dynamical systems with many degrees of freedom. They have the basic effect of introducing slow-fast behavior in an evolutionary system which, coupled with instabilities, can result in highly irregular behavior. This book gives a unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, a general finite dimensional theory of homoclinic jumping is developed and illustrated with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context. Previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds are described. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics. The theory is further used to study resonances in Hamiltonian systems with applications to molecular dynamics and rigid body motion. The final chapter contains an infinite dimensional extension of the finite dimensional theory, with application to the perturbed nonlinear Schrodinger equation and coupled NLS equations.
Synopsis
A unified treatment of resonant problems with special emphasis on the recently discovered phenomenon of homoclinic jumping. After a survey of the necessary background, the book develops a general finite dimensional theory of homoclinic jumping, illustrating it with examples. The main mechanism of chaos near resonances is discussed in both the dissipative and the Hamiltonian context, incorporating previously unpublished new results on universal homoclinic bifurcations near resonances, as well as on multi-pulse Silnikov manifolds. The results are applied to a variety of different problems, which include applications from beam oscillations, surface wave dynamics, nonlinear optics, atmospheric science and fluid mechanics.
Description
Includes bibliographical references (p. [401]-420) and indexes.
Table of Contents
Preface
x 1 Concepts from dynamical systems
x 2 Chaotic jumping near resonances: Finite-dimensional systems
x 3 Chaos due to Resonances in Physical Systems
x 4 Resonances in Hamiltonian systems
x 5 Chaotic jumping near resonances: Infinite-dimensional systems
x Appendices
x References
x Index