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IMA Workshop
Pattern Formation in Continuous and Coupled Systems
May 11-15, 1998


Martin Golubitsky, Chair
Dan Luss
Steve Strogatz


Pattern formation in spatially extended systems has been observed recently in a number of experimental systems. These include surface reactions on catalytic ribbons and homogeneous chemical reactions in gels. These patterns do not resemble the spatially periodic structure of Turing patterns but have spatially localized patterns and spatially differentiated dynamics. Moreover, these patterns appear in both PDE systems and in systems of coupled cells. The basis of these new patterns appears to be in non-local non-diffusive coupling. In another direction regular patterns have been observed in the time-averages of spatio-temporally chaotic regimes in the Faraday surface wave experiment and in rotating convection experiments. It is the purpose of this workshop to delve into the mathematical origin of these new types of pattern and the kinds of (numerical and physical) experiments that may exhibit such patterns.

We describe briefly some of the mathematical issues. In recent years, starting with the work of Field and the examples of Guckenheimer and Holmes, it has been seen that the dynamics of systems with symmetry has different structurally stable dynamics than does systems without symmetry. These differences include heteroclinic cycles, drifting along group orbits, symmetric chaos, instant chaos and issues of forced symmetry breaking.

Heteroclinic cycles give one mathematical formulation for intermittency and can be formed in two different ways. The first is through standard bifurcations where saddle-sink connections are formed in invariant (symmetry) planes. The second method involves system symmetry breaking where a perturbation parameter in the equations leads to equations with less symmetry. In this case an orbitally stable group orbit of equilibria is perturbed to a dynamically invariant manifold with less symmetry and more complicated dynamics. In another direction symmetry of attractors has been shown to lead to group invariant time-averages and patterns in time-averages in experiments with spatio-temporal chaos.

Click on the titles to find abstracts and/or links to presentation materials

W. Miller, F. Dulles,
M. Golubitsky
Welcome and Orientation
Harry L. Swinney,
University of Texas
Meandering and resonant spirals in a reaction-diffusion system
H. H. Rotermund,
Fritz-Haber Institut
Pattern formation in a surface reactions with global coupling
Bjorn Sandstede,
Ohio State University
Meandering and Drifting of Spiral Waves
Michael J. Gorman,
University of Houston
Pattern Formation in Cellular Flames
Bernard Matkowsky,
Northwestern University
Coupled KS-CGL and Coupled Burgers-CGL Equations for
Flames Governed by a Sequential Reaction
IMA Tea  
Edgar Knobloch,
University of California Berkeley
A New Mechanism for Bursting
Herre van der Zant,
T. U. Delft
Dynamics of Josephson-junction arrays
Kurt Wiesenfeld,
Georgia Institute of Technology
Josephson Arrays: Puzzles and Prospects
Guenter Ahlers,
University of California Santa Barbara
Pattern formation in Rayleigh-Bénard convection at small Prandtl numbers and with rotation about a vertical axis
Eberhard Bodenschatz,
Cornell University
Bistability with Spiral Defect Chaos and Rolls
Alastair Rucklidge,
University of Cambridge
Bifurcations of Periodic Orbits with Spatio-Temporal Symmetries
Jeroen Lamb,
University of Warwick
Bifurcations of Discrete Rotating Waves
Rajarshi Roy,
Georgia Institute of Technology
Temporal Patterns in Coupled Laser Systems: Coherence, Chaos and Communication
Peter Ashwin,
University of Surrey
Chaotic Intermittency of Patterns in Symmetric Systems
Katarina Krischer,
Fritz-Haber Institute
Pattern Formation at the Electrode/Electrolyte Interface
H.G. Purwins,
Universitat Münster
Pattern Formation in Lateral Gas Discharge Systems: Quasiparticles and Other Structures in Reaction-Diffusion-Systems
F. H. Busse,
Universitat Bayreuth
Formation of Structures Far from Criticality in Isotropic Fluid Layers and the Emergence of Patterns in Chaotic States
Ronald Imbihl,
Institut für Physikalische Chemie
Pattern Formation in Systems with Memory Effects
Hermann Riecke,
Northwestern University
Localized Patterns Related to Experiments in Liquid-Crystal Convection and Granular Media
Jerry Gollub,
Haverford College
Ordered Patterns and Spatiotemporal Chaos in Parametrically Forced Surface Waves
Mary Silber,
Northwestern University
Superlattice Turing Patterns: Why Not?
Moshe Sheintuch,
Pattern Selection in a Diffusion-Reaction System With Global or Long-Range Interaction
Michael Field,
University of Houston
Dynamics of symmetrically coupled systems of identical cells
Stephan van Gils,
Universiteit Twente
The Homoclinic Twist Bifurcation in Josephson Junction Series Arrays
Gabriela M. Gomes,
University of Porto
Sliced planforms: a representation of three-dimensional symmetries in thin domains
Dieter Armbruster,
Arizona State University
Multi-frequency oscillations in (almost) symmetrically coupled oscillators
Workshop Dinner  
Dwight Barkley,
University of Warwick
Bifurcations, Computations, and Open questions in Excitable Media
Paul Umbanhowar,
University of Pennsylvania
Wave patterns in shaken sand
Ian Stewart,
University of Warwick
Belousov and Zhabotinskii meet Bessel and Robin


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1997-1998 Emerging Applications of Dynamical Systems