Target patterns in two-dimensional heterogeneous oscillatory reaction–diffusion systems

Stich, Michael; Mikhailov, Alexander S.

doi:10.1016/j.physd.2006.01.011

Cited by 39 publications

(30 citation statements)

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“…Some of the results on modulation equations in Section 3 appear to be folklore. We refer to [22] and the references therein for related results and an overview of the experiments. We included this a discussion of inhomogeneities in the eikonal approximation since it allows for a comparison with our main technical result, Theorem 1, and for extensions to non-symmetric settings.…”

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confidence: 99%

Coherent Structures Generated by Inhomogeneities in Oscillatory Media

Kollár¹,

Scheel²

2007

SIAM J. Appl. Dyn. Syst.

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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...

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confidence: 99%

Coherent Structures Generated by Inhomogeneities in Oscillatory Media

Kollár¹,

Scheel²

2007

SIAM J. Appl. Dyn. Syst.

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“…Eq. (1) admits a stable target solution, which can be written in the polar coordinate as [8,9,28] W (r, θ, t) = ρ(r)e iφ = ρ(r)e i(−Ω k t+ψ(r)) , (2) ρ(r) and ψ(r) are real functions and dψ/dr → k and ρ → √ 1 − k 2 as r → ∞. It is hence that for large r, target solution (2) asymptotes to plane wave solution [28] …”

Section: Ingoing and Outgoing Target Waves In The Complex Ginzburg-lamentioning

confidence: 99%

“…To model the heterogeneity that could create target waves in CGLE, it is exceedingly convenient to consider ω as a spatial function [9,14] ω(r) = ω 0 r > r 0 ,…”

Section: Ingoing and Outgoing Target Waves In The Complex Ginzburg-lamentioning

confidence: 99%

“…With the phase dynamics approximation, it has been shown that a heterogeneity as a wave source or pacemaker that is capable of creating an spatially extended target pattern requires [9] (β − α) ω > 0. (8) A spatially extended target wave pattern means positive or outward group velocity, i.e., v gr > 0, which we consider throughout this Letter.…”

Section: Ingoing and Outgoing Target Waves In The Complex Ginzburg-lamentioning

confidence: 99%

“…The existence of the great majority of target waves in practice is attributed to a local heterogeneity (e.g., a dust particle, gas bubble, etc.) that plays the role of a pacemaker [2,[6][7][8][9][10][11][12][13][14][15][16], while self-organized target waves are also found both in several models [17][18][19] and in the experiments [20]. Theoretical studies on the heterogeneity sustained target patterns have also been provided [21,22], which makes us understand better the mechanism underlying the formation of target waves.…”

Section: Introductionmentioning

confidence: 99%

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