11,246 research outputs found
Nonlinear metastability for a parabolic system of reaction-diffusion equations
We consider a system of reaction-diffusion equations in a bounded interval of
the real line, with emphasis on the metastable dynamics, whereby the
time-dependent solution approaches its steady state in an asymptotically
exponentially long time interval as the viscosity coefficient
goes to zero. To rigorous describe such behavior, we analyze the dynamics of
solutions in a neighborhood of a one-parameter family of approximate steady
states, and we derive an ODE for the position of the internal interfaces.Comment: This paper has been withdrawn by the author due to an error in
Theorem 1.1. Please refer to the paper "Slow dynamics in reaction-diffusion
systems
Resonantly Forced Inhomogeneous Reaction-Diffusion Systems
The dynamics of spatiotemporal patterns in oscillatory reaction-diffusion
systems subject to periodic forcing with a spatially random forcing amplitude
field are investigated. Quenched disorder is studied using the resonantly
forced complex Ginzburg-Landau equation in the 3:1 resonance regime. Front
roughening and spontaneous nucleation of target patterns are observed and
characterized. Time dependent spatially varying forcing fields are studied in
the 3:1 forced FitzHugh-Nagumo system. The periodic variation of the spatially
random forcing amplitude breaks the symmetry among the three quasi-homogeneous
states of the system, making the three types of fronts separating phases
inequivalent. The resulting inequality in the front velocities leads to the
formation of ``compound fronts'' with velocities lying between those of the
individual component fronts, and ``pulses'' which are analogous structures
arising from the combination of three fronts. Spiral wave dynamics is studied
in systems with compound fronts.Comment: 14 pages, 19 figures, to be published in CHAOS. This replacement has
some minor changes in layout for purposes of neatnes
Spatiotemporal pattern formation in a three-variable CO oxidation reaction model
The spatiotemporal pattern formation is studied in the catalytic carbon
monoxide oxidation reaction that takes into account the diffusion processes
over the Pt(110) surface, which may contain structurally different areas. These
areas are formed during CO-induced transition from a reconstructed phase with
geometry of the overlayer to a bulk-like () phase with
square atomic arrangement. Despite the CO oxidation reaction being
non-autocatalytic, we have shown that the analytic conditions of the existence
of the Turing and the Hopf bifurcations can be satisfied in such systems. Thus,
the system may lose its stability in two ways --- either through the Hopf
bifurcation leading to the formation of temporal patterns in the system or
through the Turing bifurcation leading to the formation of regular spatial
patterns. At a simultaneous implementation of both scenarios, spatiotemporal
patterns for CO and oxygen coverages are obtained in the system.Comment: 11 pages, 6 figures, 1 tabl
Frozen spatial chaos induced by boundaries
We show that rather simple but non-trivial boundary conditions could induce
the appearance of spatial chaos (that is stationary, stable, but spatially
disordered configurations) in extended dynamical systems with very simple
dynamics. We exemplify the phenomenon with a nonlinear reaction-diffusion
equation in a two-dimensional undulated domain. Concepts from the theory of
dynamical systems, and a transverse-single-mode approximation are used to
describe the spatially chaotic structures.Comment: 9 pages, 6 figures, submitted for publication; for related work visit
http://www.imedea.uib.es/~victo
Spatial organization in cyclic Lotka-Volterra systems
We study the evolution of a system of interacting species which mimics
the dynamics of a cyclic food chain. On a one-dimensional lattice with N<5
species, spatial inhomogeneities develop spontaneously in initially homogeneous
systems. The arising spatial patterns form a mosaic of single-species domains
with algebraically growing size, , where
(1/2) and 1/3 for N=3 with sequential (parallel) dynamics and N=4,
respectively. The domain distribution also exhibits a self-similar spatial
structure which is characterized by an additional length scale, , with and 2/3 for N=3 and 4, respectively. For
, the system quickly reaches a frozen state with non interacting
neighboring species. We investigate the time distribution of the number of
mutations of a site using scaling arguments as well as an exact solution for
N=3. Some possible extensions of the system are analyzed.Comment: 18 pages, 10 figures, revtex, also available from
http://arnold.uchicago.edu/~ebn
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