24,483 research outputs found
Delay-induced patterns in a two-dimensional lattice of coupled oscillators
We show how a variety of stable spatio-temporal periodic patterns can be
created in 2D-lattices of coupled oscillators with non-homogeneous coupling
delays. A "hybrid dispersion relation" is introduced, which allows studying the
stability of time-periodic patterns analytically in the limit of large delay.
The results are illustrated using the FitzHugh-Nagumo coupled neurons as well
as coupled limit cycle (Stuart-Landau) oscillators
Coupled logistic maps and non-linear differential equations
We study the continuum space-time limit of a periodic one dimensional array
of deterministic logistic maps coupled diffusively. First, we analyse this
system in connection with a stochastic one dimensional Kardar-Parisi-Zhang
(KPZ) equation for confined surface fluctuations. We compare the large-scale
and long-time behaviour of space-time correlations in both systems. The dynamic
structure factor of the coupled map lattice (CML) of logistic units in its deep
chaotic regime and the usual d=1 KPZ equation have a similar temporal stretched
exponential relaxation. Conversely, the spatial scaling and, in particular, the
size dependence are very different due to the intrinsic confinement of the
fluctuations in the CML. We discuss the range of values of the non-linear
parameter in the logistic map elements and the elastic coefficient coupling
neighbours on the ring for which the connection with the KPZ-like equation
holds. In the same spirit, we derive a continuum partial differential equation
governing the evolution of the Lyapunov vector and we confirm that its
space-time behaviour becomes the one of KPZ. Finally, we briefly discuss the
interpretation of the continuum limit of the CML as a
Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) non-linear diffusion equation with
an additional KPZ non-linearity and the possibility of developing travelling
wave configurations.Comment: 23 page
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