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vol. 2 ISBN 9780444501684
Description
This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from
interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers.
The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others.
While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to name
just a few, are ubiquitous dynamical concepts throughout the articles.
Table of Contents
A. Finite-Dimensional Methods
1. Mechanisms of phase-locking and frequency control in pairs of coupled neural oscillators (N. Kopell, G.B. Ermentrout).
2. Invariant manifolds and Lagrangian dynamics in the ocean and
atmosphere (C. Jones, S. Winkler).
3. Geometric singular perturbation analysis of neuronal dynamics (J.E. Rubin, D. Terman).
B. Numerics
4. Numerical continuation, and computation of normal forms (W.-J. Beyn, A. Champneys, E. Doedel, W. Govaerts,Y.A. Kuznetsov, B. Sandstede).
5. Set oriented numerical methods for dynamical systems (M. Dellnitz, O. Junge).
6. Numerics and exponential smallness (V. Gelfreich).
7. Shadowability of chaotic dynamical systems (C. Grebogi, L. Poon, T. Sauer, J.A. Yorke, D. Auerbach).
8. Numerical analysis of dynamical systems (J. Guckenheimer).
C. Topological Methods
9. Conley index (K. Mischaikow, M. Mrozek).
10. Functional differential equations (R.D. Nussbaum).
D. Partial Differential Equations
11. Navier--Stokes equations and dynamical systems (C. Bardos, B. Nicolaenko).
12. The nonlinear Schroedinger equation as both a PDE and a
dynamical system (D. Cai, D.W. McLaughlin, K.T.R. McLaughlin).
13. Pattern formation in gradient systems (P.C. Fife).
14. Blow-up in nonlinear heat equations from the dynamical systems point of view (M. Fila, H. Matano).
15. The Ginzburg--Landau equation in its role as a modulation
equation (A. Mielke).
16. Parabolic equations:
asymptotic behavior and dynamics on invariant manifolds (P. Pola ik).
17.Global attractors in partial differential equations (G. Raugel).
18. Stability of travelling waves (B. Sandstede).
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vol. 1B ISBN 9780444520555
Description
This second half of Volume 1 of this Handbook follows Volume 1A, which was published in 2002. The contents of these two tightly integrated parts taken together come close to a realization of the program formulated in the introductory survey "Principal Structures" of Volume 1A.
The present volume contains surveys on subjects in four areas of dynamical systems: Hyperbolic dynamics, parabolic dynamics, ergodic theory and infinite-dimensional dynamical systems (partial differential equations).
Table of Contents
Preface
List of Contributors
Contents of Volume 1A
1. Partially Hyperbolic Dynamical Systems (B. Hasselblatt and Ya. Pesin)
2. Smooth Ergodic Theory and Nonuniformly Hypoerbolic Dynamics (L. Barreira and Ya. Pesin, with an Appendix by O. Sarig)
3. Stochastic-Like Behaviour in Nonuniformly Expanding Maps (S. Luzzatto)
4. Homoclinic Bifurcations, Dominated Splitting, and Robust Transivity (E.R. Pujals and M. Sambarino)
5. Random Dynamics (Yu. Kifer, P.-D. Liu)
6. An Introduction to Veech Surfaces (P. Hubert and T.A. Schmidt)
7. Ergodic Theory of Translation Surfaces (H. Masur)
8. On the Lyapunov Exponents of the Kontsevich-Zorich Cocycle (G. Forni)
9. Counting Problems in Moduli Space (A. Eskin)
10. On the Interplay Between Measurable and Topological Dynamics (E. Glasner and B. Weiss)
11. Spectral Properties and Combinatorial Constructions in Ergodic Theory (A. Katok and J.-P. Thouvenot)
12. Combinatorial and Diophantine Applications of Ergodic Theory (V. Bergelson, with Appendix A by A. Leibman and Appendix B by A. Quas and M. Wierdl)
13. Pointwise Ergodic Theorems for Actions of Groups (A. Nevo)
14. Global Attractors in PDE (A.V. Babin)
15. Hamiltonian PDEs (S.B. Kuksin, with an Appendix by D. Bambusi)
16. Extended Hamiltonian Systems (M.I. Weinstein)
Author Index of Volume 1A
Subject Index of Volume 1A
Author Index
Subject Index
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vol. 3 ISBN 9780444531414
Description
In this volume, the authors present a collection of surveys on various aspects of the theory of bifurcations of differentiable dynamical systems and related topics. By selecting these subjects, they focus on those developments from which research will be active in the coming years. The surveys are intended to educate the reader on the recent literature on the following subjects: transversality and generic properties like the various forms of the so-called Kupka-Smale theorem, the Closing Lemma and generic local bifurcations of functions (so-called catastrophe theory) and generic local bifurcations in 1-parameter families of dynamical systems, and notions of structural stability and moduli.
Table of Contents
1. Introduction, 2. Complex linearization, 3. KAM Theory for circle and annulus maps, 4. KAM Theory for flows, 5. Further developments in KAM Theory, 6. Quasi-periodic bifurcations: dissipative setting, 7. Quasi-periodic bifurcation theory in other settings, 8. Further Hamiltonian KAM Theory, 9. Whitney smooth bundles of KAM tori, 10. Conclusion
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vol. 1A ISBN 9780444826695
Description
Volumes 1A and 1B.
These volumes give a comprehensive survey of dynamics written by specialists in the various subfields of dynamical systems. The presentation attains coherence through a major introductory survey by the editors that organizes the entire subject, and by ample cross-references between individual surveys.
The volumes are a valuable resource for dynamicists seeking to acquaint themselves with other specialties in the field, and to mathematicians active in other branches of mathematics who wish to learn about contemporary ideas and results dynamics. Assuming only general mathematical knowledge the surveys lead the reader towards the current state of research in dynamics.
Volume 1B will appear 2005.
Table of Contents
Volume 1A.
Principal structures (B. Hasselblatt, A. Katok).
Entropy, Isomorphism and Equivalence (J.-P. Thouvenot).
Hyperbolic dynamics (B. Hasselblatt).
Invariant measures for hyperbolic dynamical systems (N. Chernov).
Periodic orbits and zeta functions (M. Pollicott).
Hyperbolic dynamics and Riemannian geometry (G. Knieper).
Topological Methods in Dynamics (J. Franks, M. Misiurewicz).
One-Dimensional Maps (M. Jakobson, G. wiatek).
Ergodic theory and dynamics of G-spaces (R. Feres, A. Katok).
Symbolic and algebraic dynamical systems (D. Lind, K. Schmidt).
Homogeneous flows, applications to number theory, and related topics (D. Kleinbock, N. Shah, A. Starkov).
Random transformations in ergodic theory (A. Furman).
Rational billiards and flat structures (H. Masur, S. Tabachnikov).
Variational methods for Hamiltonian systems (P.H. Rabinowitz).
Pseudoholomorphic curves and dynamics in three dimensions (H. Hofer, K. Wysocki, E. Zehnder).
by "Nielsen BookData"