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Spring 1998

IMA Workshop:

Pattern Formation in Continuous and Coupled Systems

May 11-15, 1998

Organizers:

Martin Golubitsky, Chair
Dan Luss
Steve Strogatz

Participants

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

SCHEDULE for MONDAY, MAY 11
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  
SCHEDULE for TUESDAY, MAY 12
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
SCHEDULE for WEDNESDAY, MAY 13
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
SCHEDULE for THURSDAY, MAY 14
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,
Technion		 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  
SCHEDULE for FRIDAY, MAY 15
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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