This book summarizes and highlights progress in our understanding of Dynamical Systems during six years of the German Priority Research Program ``Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems''. The program was funded by the Deutsche Forschungsgemeinschaft (DFG) and aimed at combining, focussing, and enhancing research efforts of active groups in the field by cooperation on a federal level. The surveys in the book are addressed to experts and non-experts in the mathematical community alike. In addition they intend to convey the significance of the results for applications far into the neighboring disciplines of Science.
Three fundamental topics in dynamical systems are at the core of our research effort:
Each of these topics is, of course, a highly complex problem area in itself and does not fit naturally into the deplorably traditional confines of any of the disciplines of ergodic theory, analysis, or numerical analysis alone. The necessity of mathematical cooperation between these three disciplines is quite obvious when facing the formidable task of establishing a bidirectional transfer which bridges the gap between deep, detailed theoretical insight and relevant, specific applications. Both analysis and numerical analysis play a key role when it comes to building that bridge. Some steps of our joint bridging efforts are collected in this volume.
Neither our approach nor the presentations in this volume are monolithic. Rather, like composite materials, the contributions are gaining strength and versatility through the broad variety of interwoven concepts and mathematical methodologies which they span.
Fundamental concepts which are present in this volume include bifurcation, homoclinicity, invariant sets and attractors, both in the autonomous and nonautonomous situation. These concepts, at first sight, seem to mostly address large time behavior, most amenable to methodologies of analysis. Their intimate relation to concepts like (nonstrict) hyperbolicity, ergodicity, entropy, stochasticity and control should become quite apparent, however, when browsing through this volume.
The fundamental topic of dimension is similarly ubiquitous throughout our articles. In analysis it figures, for example, as a rigorous reduction from infinite-dimensional settings like partial differential equations, to simpler infinite-, finite- or even low-dimensional model equations, still bearing full relevance to the original equations. But in numerical analysis - including and transcending mere discretization - specific computational realization of such reductions still poses challenges which are addressed here.
Another source of inspiration comes from very refined measure-theoretic and dimensional concepts of ergodic theory which found their way into algorithmic realizations presented here.
By no means do these few hints exhaust the conceptual span of the articles. It would be even more demanding to discuss the rich circle of methods, by which the three fundamental topics of large time behavior, dimension, and measure are tackled. In addition to SBR-measures, Perron-Frobenius type transfer operators, Markov decompositions, Pesin theory, entropy, and Oseledets theorems, we address kneading invariants, fractal geometry and self-similarity, complex analytic structure, the links between billiards and spectral theory, Lyapunov exponents, and dimension estimates. Including Lyapunov-Schmidt and center manifold reductions together with their Shilnikov and Lin variants and their efficient numerical realizations, symmetry and orbit space reductions together with closely related averaging methods, we may continue, numerically, with invariant subspaces, Godunov type discretization schemes for conservation laws with source terms, (compressed) visualization of complicated and complex patterns of dynamics, and present an algorithm, GAIO, which enables us to approximately compute, in low dimensions, objects like SBR-measures and Perron-Frobenius type transfer operators. At which point our cursory excursion through methodologies employed here closes up the circle.
So much for the mathematical aspects. The range of applied issues, mostly from physics but including some topics from the life sciences, can also be summarized at most superficially, at this point. This range comprises such diverse areas as crystallization and dendrite growth, the dynamo effect, and efficient simulation of biomolecules. Fluid dynamics and reacting flows are addressed, including the much studied contexts of Rayleigh-Bénard and Taylor-Couette systems as well as the stability question of three-dimensional surface waves. The Ginzburg-Landau and Swift-Hohenberg equations appear, for example, as do mechanical problems involving friction, population biology, the spread of infectious diseases, and quantum chaos. It is the diversity of these applied fields which well reflects both the diversity and the power of the underlying mathematical approach. Only composite materials enable a bridge to span that far.
The broad scope of our program has manifested itself in many meetings, conferences, and workshops. Suffice it to mention the workshop on ``Entropy'' which was coorganized by Andreas Greven, Gerhard Keller, and Gerald Warnecke at Dresden in June 2000 with the two neighboring DFG Priority Research Programs ``Analysis and Numerics for Conservation Laws'' and ``Interacting Stochastic Systems of High Complexity''. For further information concerning program and participants of the DFG Priority Research Program ``Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems'', including a preprint server, see
For other DFG programs we refer toAt the end of this preface, I would like to thank at least some of the many friends and colleagues who have helped on so many occasions to make this program work. First of all, I would like to mention the members of the scientific committee who have helped initiate the entire program and who have accompanied and shaped the scientific program throughout its funding period: Ludwig Arnold, Hans-Günther Bothe, Peter Deuflhard, Klaus Kirchgässner, and Stefan Müller. The precarious conflict between great expectations and finite funding was expertly balanced by our all-understanding referees Hans Wilhelm Alt, Jürgen Gärtner, François Ledrappier, Wilhelm Niethammer, Albrecht Pietsch, Gerhard Wanner, Harry Yserentant, Eberhard Zeidler, and Eduard Zehnder. The hardships of finite funding as well as any remaining administrative constraints were further alleviated as much as possible, and beyond, by Robert Paul Königs and Bernhard Nunner, representing DFG from its best side. The www-services were designed, constantly expanded and improved with unrivalled expertise and independence by Stefan Liebscher. And Regina Löhr, as an aside to her numerous other secretarial activities and with ever-lasting patience and friendliness, efficiently reduced the administrative burden of the coordinator to occasional emails which consisted of no more than ``OK. BF''. Martin Peters and his team at Springer-Verlag ensured a very smooth cooperation, including efficient assistance with all TeXnicalities. But last, and above all, my thanks as a coordinator of this program go to the authors of this volume and to all participants - principal investigators, PostDocs and students alike - who have realized this program with their contributions, their knowledge, their dedication, and their imagination.
Bernold Fiedler
Berlin, August 2000
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danse@math.fu-berlin.de Sep 18 2000