Coherent Structures Generated by Inhomogeneities in Oscillatory Media

Abstract

We investigate the effect of spatially localized inhomogeneities on a spatially homogeneous oscillation in a reaction-diffusion system. In dimension up to 2, we find sources and contact defects, that is, the inhomogeneity may either send out phase waves or act as a weak sink. We show that small inhomogeneities cannot act as sources in more than 2 space dimensions. We also derive asymptotics for wavenumbers and group velocities in the far field. The results are established rigorously for radially symmetric inhomogeneities in reaction-diffusion systems, and for arbitrary inhomogeneities in a modulation equation approximation.Keywords: reaction-diffusion, phase diffusion, eikonal equation, coherent structures, target patterns Running head: Inhomogeneities in oscillatory mediaCorresponding author: Arnd Scheel 1 Oscillatory reaction-diffusion systems IntroductionWe are interested in patterns that arise in dissipative, spatially extended systems far from equilibrium. The arguably simplest non-equilibrium pattern in a dynamical system is a periodic orbit. Periodic orbits are ubiquitous in dynamical systems, a fact which is partly justified by their robustness. Indeed, when studying ordinary differential equations or partial differential equations posed on bounded domains, periodic orbits are typically robust: the trivial Floquet multiplier associated with the phase of the oscillation is algebraically simple, and for any small perturbation of the system, one will find a nearby periodic orbit, with similar frequency. 1Spatially extended, large systems of oscillators have attracted attention in the physical and mathematical literature in many contexts. A classical prominent example is the BelousovZhabotinsky reaction, a reaction-diffusion system where the chemical concentrations undergo a relaxation-type oscillation which can be sustained for many cycles; see for example [25,9]. Other examples include biological systems such as cardiac tissue [24], neural systems [23], and ecological systems [6].When studying such large systems, in unbounded or in large domains, two interrelated issues complicate the concept of a robust oscillation.First, robustness turns out to be a delicate issue on a technical level. In large domains, the fixed point problem for the Poincare map is ill-conditioned due to clusters of eigenvalues of the linearization near the neutral phase mode; in unbounded domains, the neutral phase mode is even embedded into a continuum of spectrum; see Section 1.3, below. The spatial diffusive coupling, is responsible for this lack of separation between slow phase modes and the fast normal modes near the periodic orbit, since it covers a full band of possible exponential relaxation rates. As a consequence, it is often not obvious if periodic orbits are robust under changes of system parameters! Second, periodic orbits come in very different spatial flavors: spatially homogeneous oscillations, plane waves, target patterns, and spiral waves, to name but a few. A perturbation theory for spatially extended system...

Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

“…Thus, different members of the periodic travelling wave family are selected by obstacles of different sizes. The particular case of very small obstacles has been studied in detail by Kollár & Scheel (2007).…”

Deformation of striped patterns by inhomogeneities

“…In fact, the Laplacian is invertible for suitable weights γ in Kondratiev spaces in R 3 . We refer the reader to [6], where dynamical systems methods were used to analyze the complex-coefficient Ginzburg-Landau equation in space dimensions 1, 2, 3, exhibiting a related dependence of far field corrections on the dimension. Note however that the analysis there uses exponential localization of g(x) and is not easily extended beyond radial symmetry.…”

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