29 Pages
English
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Diffusive stability of oscillations in reaction diffusion systems

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29 Pages
English

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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


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Exrait

Di usiv e stability of oscillations
Thierry Gallay UniversitedeGrenobleI Institut Fourier, UMR CNRS 5582 BP 74 38402Saint-Martin-dHeres,France
in
Abstract
reaction-di usion
systems
Arnd Scheel University of Minnesota School of Mathematics 206 Church St. S.E. Minneapolis, MN 55455, USA
Westudynonlinearstabilityofspatiallyhomogeneousoscillationsinreaction-di usionsys-tems. Assuming absence of unstable linear modes and linear di usiv e behavior for the neutral phase,weprovethatspatiallylocalizedperturbationsdecayalgebraicallywiththedi usiverate t n/2in 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, di usiv
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-di usionsystemssuchastheBelousov-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-di usion systems, ut=Du+f(u) u=u(t x)RN xRn t0(1.1) with positive coupling matrixD∈ MNN(R),D=DT>0, and smooth kineticsfC(RNRN). In this spatially continuous setup, working in the whole spaceRnis 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 ODEut=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 ut=Du+f0(u(t))u (1.2) whichisformallyequivalenttothefamilyofordinarydi erentialequations ut= k2Du+f0(u(t))u  kRn.(1.3) Foreach xedkwe denote byFk(t s) the two-parameter evolution operator associated to the linear time-periodic system (1.3), so thatu(t) =Fk(t s)u(s) for anyts. The asymptotic behavior
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