Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

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Springer Science & Business Media, 2001 - Mathematics - 820 pages
This book summarizes and highlights progress in Dynamical Systems achieved during six years of the German Priority Research Program "Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems," funded by the Deutsche Forschungsgemeinschaft (DFG). The three fundamental topics of large time behavior, dimension, and measure are tackled with by a rich circle of uncompromisingly rigorous mathematical concepts. The range of applied issues comprises such diverse areas as crystallization and dendrite growth, the dynamo effect, efficient simulation of biomolecules, fluid dynamics and reacting flows, mechanical problems involving friction, population biology, the spread of infectious diseases, and quantum chaos. 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.

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Contents

Robustness Numerics and Chaotic Dynamics
1
SelfSimilar Measures
31
Continuation of LowDimensional Invariant Subspaces in Dynamical Systems of Large Dimension
47
On Hybrid Methods for Bifurcation and Center Manifolds for General Operators
73
Dimension Theory of Smooth Dynamical Systems
109
Collision of Control Sets
131
The Algorithms Behind GAIO Set Oriented Numerical Methods for Dynamical Systems
145
Polynomial Skew Products
175
Aspects on Data Analysis and Visualization for Complicated Dynamical Systems
417
An Overview
431
Forced Symmetry Breaking and Relative Periodic Orbits
453
On Dynamics and Bifurcations of Nonlinear Evolution Equations Under Numerical Discretization
469
an Example
501
An Extension of the Thermodynamic Formalism Approach to Selbergs Zeta Function for General Modular Groups
523
Stability and Diffusive Dynamics on Extended Domains
563
Dimension Estimates for Invariant Sets of Dynamical Systems
585

Transfer Operator Approach to Conformational Dynamics in Biomolecular Systems
191
Simulation and Numerical Analysis of Dendritic Growth
225
Bifurcation Phenomena and Dynamo Effect in Electrically Conducting Fluids
253
Cascades of Homoclinic Doubling Bifurcations
271
Existence Bifurcation and Stability of Profiles for Classical and NonClassical Shock Waves
287
Dynamical Systems of Population Dynamics
311
Topological and Measurable Dynamics of Lorenz Maps
333
ThreeDimensional Steady CapillaryGravity Waves
363
Discretization Inflation and Perturbation of Attractors
399
On the Inverse Problem of Fractal Compression
617
Theory and Application
649
MultiPulse Homoclinic Loops in Systems with a Smooth First Integral
691
Quantum Chaos and Quantum Ergodicity
717
Periodic Orbits and Attractors for Autonomous ReactionDiffusion Systems
753
Unconditionally Stable Explicit Schemes for the Approximation of Conservation Laws
775
Color Plates
805
Author Index
819
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