Overview

This handbook is the second volume 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 modelling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behaviour. 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 18 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.

Full Product Details

Author:   B. Fiedler (Freie Universität Berlin, Institut für Mathematik I, Berlin, Germany)
Publisher:   Elsevier Science & Technology
Imprint:   North-Holland
Volume:   2
Dimensions:   Width: 15.90cm , Height: 7.10cm , Length: 24.10cm
Weight:   2.150kg
ISBN:  

9780444501684

ISBN 10:   0444501681
Pages:   1098
Publication Date:   21 February 2002
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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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 Schrödinger 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. Poláčik). 17.Global attractors in partial differential equations (G. Raugel). 18. Stability of travelling waves (B. Sandstede).

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