Diffusive stability of oscillations in reaction diffusion systems

Thierry Gallay

Description

Diffusive stability of oscillations in reaction-diffusion systems Thierry Gallay Universite de Grenoble I Institut Fourier, UMR CNRS 5582 BP 74 38402 Saint-Martin-d'Heres, France Arnd Scheel University of Minnesota School of Mathematics 206 Church St. S.E. Minneapolis, MN 55455, USA Abstract We study nonlinear stability of spatially homogeneous oscillations in reaction-diffusion sys- tems. Assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase, we prove that spatially localized perturbations decay algebraically with the diffusive rate t?n/2 in space dimension n. We also compute the leading order term in the asymptotic expansion of the solution, and show that it corresponds to a spatially localized modulation of the phase. Our approach is based on a normal form transformation in the kinetics ODE which partially decouples the phase equation, at the expense of making the whole system quasilinear. Stability is then obtained by a global fixed point argument in temporally weighted Sobolev spaces. Corresponding author: Arnd Scheel Keywords: periodic solutions, diffusive stability, normal forms, quasilinear parabolic systems

• reaction- diffusion systems


• spatially distributed


• nonlinear heat equation


• floquet exponent


• u?


• any finite-size


• stability analysis


• linear time-periodic

Sujets

• Scheel
• Taylor
• Floquet theory

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

Diusiv e stability of oscillations

Thierry Gallay UniversitedeGrenobleI Institut Fourier, UMR CNRS 5582 BP 74 38402Saint-Martin-d’Heres,France

in

Abstract

reaction-diusion

systems

Arnd Scheel University of Minnesota School of Mathematics 206 Church St. S.E. Minneapolis, MN 55455, USA

Westudynonlinearstabilityofspatiallyhomogeneousoscillationsinreaction-diusionsys- tems. Assuming absence of unstable linear modes and linear diusiv e behavior for the neutral phase,weprovethatspatiallylocalizedperturbationsdecayalgebraicallywiththediusiverate t  n/2 in space dimensionn also compute the leading order term in the asymptotic expansion. We of the solution, and show that it corresponds to a spatially localized modulation of the phase. Our approach is based on a normal form transformation in the kinetics ODE which partially decouples the phase equation, at the expense of making the whole system quasilinear. Stability isthenobtainedbyaglobalxedpointargumentintemporallyweightedSobolevspaces.

Corresponding author:

Arnd Scheel

Keywords:periodic solutions, diusiv

e stability, normal forms, quasilinear parabolic systems

1 Introduction and main results

Synchronization of spatially distributed dissipative oscillators has been observed in a wide variety of physicalsystems.Wementionsynchronizationinyeastcellpopulations[12],reies[4],coupledlaser arrays[24],andspatiallyhomogeneousoscillationsinreaction-diusionsystemssuchastheBelousov- Zhabotinsky reaction [30] and the NO+CO-reaction on a Pt(100) surface [29]. Synchronization strikes us most when the system size is large or the coupling strength is weak. Both situations relate innaturalwaystotheregimeoflargeReynoldsnumberinuidexperiments,whereoneexpects turbulent, incoherent rather than laminar, synchronized behavior. Still, one nds synchronization as a quite common, universal phenomenon, even in very large systems.

The aim of this article is to elucidate the robustness of spatially homogeneous temporal oscilla- tions in spatially extended systems, under most general assumptions, without detailed knowledge of internal oscillator dynamics or coupling mechanisms. In fact, quantitative models are very rarely available for the systems mentioned above. Instead, we make phenomenological assumptions, related to the existence of oscillations and the absence of strongly unstable modes. These assumptions typi- cally guarantee asymptotic stability of a spatially homogeneous oscillation in anynite-sizesystem, when equipped with compatible (say, Neumann) boundary conditions. The results in this article are concerned withinnite-size, reaction-diusion systems, u t =D
u+f(u) u=u(t x)∈R N  x∈R n  t0(1.1) with positive coupling matrixD∈ M NN (R),D=D T >0, and smooth kineticsf∈C ∞ (R N R N ). In this spatially continuous setup, working in the whole spaceR n is an idealization which corresponds to the limit of small coupling matrix and/or large domain size. We will brie y comment on the relationbetweenourresultsinthewholespaceandthestabilityoftemporaloscillationsinnite domains, below.

Tobespecic,wemakethefollowingassumptionsonthekineticsfand the coupling matrixD.

Hypothesis 1.1 (Oscillatory kinetics)We suppose that the ODEu t =f(u)possesses a periodic solutionu  (t) =u  (t+T)with minimal periodT >0.

In particular, to avoid trivial situations, we assume that the periodic orbit is not reduced to a single equilibrium. As is well-known, this is possible only ifN if2, i.e. the system (1.1) does not reduce to a scalar equation.

In addition to existence we will make a number of assumptions on the Floquet exponents of the linearized equation u t =D
u+f 0 (u  (t))u (1.2) whichisformallyequivalenttothefamilyofordinarydierentialequations u t = k 2 Du+f 0 (u  (t))u  k∈R n .(1.3) Foreachxedkwe denote byF k (t s) the two-parameter evolution operator associated to the linear time-periodic system (1.3), so thatu(t) =F k (t s)u(s) for anyts. The asymptotic behavior

2