1,668 research outputs found
Traveling Wave Solutions for Planar Lattice Differential Systems with Applications to Neural Networks
AbstractWe obtain some existence results for traveling wave fronts and slowly oscillatory spatially periodic traveling waves of planar lattice differential systems with delay. Our approach is via Schauder's fixed-point theorem for the existence of traveling wave fronts and via S1-degree and equivarant bifurcation theory for the existence of periodic traveling waves. As examples, the obtained abstract results will be applied to a model arising from neural networks and explicit conditions for traveling wave fronts and global continuation of periodic waves will be obtained
Dynamics of Patterns
Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction
The Kuramoto model: A simple paradigm for synchronization phenomena
Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included
Morse theory on spaces of braids and Lagrangian dynamics
In the first half of the paper we construct a Morse-type theory on certain
spaces of braid diagrams. We define a topological invariant of closed positive
braids which is correlated with the existence of invariant sets of parabolic
flows defined on discretized braid spaces. Parabolic flows, a type of
one-dimensional lattice dynamics, evolve singular braid diagrams in such a way
as to decrease their topological complexity; algebraic lengths decrease
monotonically. This topological invariant is derived from a Morse-Conley
homotopy index and provides a gloablization of `lap number' techniques used in
scalar parabolic PDEs.
In the second half of the paper we apply this technology to second order
Lagrangians via a discrete formulation of the variational problem. This
culminates in a very general forcing theorem for the existence of infinitely
many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification
of two proofs and one definition; 55 pages, 20 figure
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