Rotating Wave Solutions to Lattice Dynamical Systems I: The Anti-continuum Limit
TLDR
In this article, a spatially discrete lattice dynamical system of Ginzburg-Landau type was considered and the existence of rotating waves in the anti-continuum limit was proved.Abstract:
Rotating waves are a fascinating feature of a wide array of complex systems, particularly those arising in the study of many chemical and biological processes. With many rigorous mathematical investigations of rotating waves relying on the model exhibiting a continuous Euclidean symmetry, this work is aimed at understanding these nonlinear waves in the absence of such symmetries. Here we will consider a spatially discrete lattice dynamical system of Ginzburg–Landau type and prove the existence of rotating waves in the anti-continuum limit. This result is achieved by providing a link between the work on phase systems stemming from the study of identically coupled oscillators on finite lattices to carefully track the solutions as the size of the lattice grows. It is shown that in the infinite square lattice limit of these phase systems that a rotating wave solution exists, which can be extended to the Ginzburg–Landau system of study here. The results of this work provide a necessary first step in the investigation of rotating waves as solutions to lattice dynamical systems in an effort to understand the dynamics of such solutions outside of the idealized situation where the underlying symmetry of a differential equation can be exploited.read more
Citations
More filters
Journal ArticleDOI
The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)
TL;DR: In this paper, the authors describe the rules of the ring, the ring population, and the need to get off the ring in order to measure the movement of a cyclic clock.
Journal ArticleDOI
Vortex pairs in the discrete nonlinear Schrödinger equation
TL;DR: In this paper, the role of discreteness in the interaction of co-winding and counterwinding vortices in the context of the nonlinear Schrodinger equation is examined.
Journal ArticleDOI
Stability of infinite systems of coupled oscillators via random walks on weighted graphs
TL;DR: In this article, the stability of phase-locked solutions of weakly coupled oscillators was investigated and the main stability result of this work comes from adapting a series of investigations into random walks on infinite weighted graphs.
Journal ArticleDOI
Rotating Wave Solutions to Lattice Dynamical Systems II: Persistence Results
TL;DR: In this paper, the second part of a series of investigations into the dynamics of rotating waves as solutions to lattice dynamical systems is presented, where it is shown that there exists an interval of sufficiently small coupling values for which a rotating wave solution persists.
Journal ArticleDOI
Vortex Pairs in the Discrete Nonlinear Schr\"odinger Equation
TL;DR: In this article, the role of discreteness in the interaction of co-winding and counterwinding vortices in the context of the nonlinear Schr{o}dinger equation is examined.
References
More filters
Journal ArticleDOI
The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)
TL;DR: In this paper, the authors describe the rules of the ring, the ring population, and the need to get off the ring in order to measure the movement of a cyclic clock.
Book
The geometry of biological time
TL;DR: The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways is presented.
Journal ArticleDOI
Spiral Waves of Chemical Activity
TL;DR: The Zhabotinsky-Zaikin reagent propagates waves of chemical activity that include spiral waves, resembling involutes of the circle, that appear, persist, and eventually exclude all concentric ring waves.
Journal ArticleDOI
Propagation and its failure in coupled systems of discrete excitable cells
TL;DR: In this paper, it is shown that propagation fails when coupling is weak, but succeeds if coupling is strong enough, and that propagation is successful when the coupling strength is large enough.