Archive of Issues
Russia RostovonDon; Vladikavkaz
Section  Mathematics 
Title  Bifurcations in a Rayleigh reactiondiffusion system 
Author(s)  Kazarnikov A.V.^{ab}, Revina S.V.^{ab} 
Affiliations  Southern Federal University^{a}, Southern Mathematical Institute, Vladikavkaz Scientific Center of the Russian Academy of Sciences^{b} 
Abstract  We consider a reactiondiffusion system with a cubic nonlinear term, which is a special case of the FitzhughNagumo system and an infinitedimensional version of the classical Rayleigh system. We assume that the spatial variable belongs to an interval, supplemented with Neumann boundary conditions. It is wellknown that in that specific case there exists a spatiallyhomogeneous oscillatory regime, which coincides with the timeperiodic solution of the classical Rayleigh system. We show that there exists a countable set of critical values of the control parameter, where each critical value corresponds to the branching of new spatiallyinhomogeneous autooscillatory or stationary regimes. These regimes are stable with respect to small perturbations from some infinitedimensional invariant subspaces of the system under study. This, in particular, explains the convergence of numerical solution to zero, periodic or stationary solution, which is observed for some specific initial conditions and control parameter values. We construct the asymptotics for branching solutions by using LyapunovSchmidt reduction. We find explicitly the first terms of asymptotic expansions and study the formulas for general terms of asymptotics. It is shown that a soft loss of stability occurs in invariant subspaces. We study numerically the evolution of secondary regimes due to the increase of control parameter values and observe that the secondary periodic solutions are transformed into stationary ones as the control parameter value increases. Next, the amplitude of stationary solutions continues to grow and the solution asymptotically converges to the square wave regime. 
Keywords  reactiondiffusion systems, pattern formation, LyapunovSchmidt reduction 
UDC  517.955.8 
MSC  35K57 
DOI  10.20537/vm170402 
Received  20 May 2017 
Language  Russian 
Citation  Kazarnikov A.V., Revina S.V. Bifurcations in a Rayleigh reactiondiffusion system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 4, pp. 499514. 
References 

Full text 