Section
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Mathematics
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Title
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Bifurcations in a Rayleigh reaction-diffusion system
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Author(-s)
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Kazarnikov A.V.ab,
Revina S.V.ab
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Affiliations
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Southern Federal Universitya,
Southern Mathematical Institute, Vladikavkaz Scientific Center of the Russian Academy of Sciencesb
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Abstract
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We consider a reaction-diffusion system with a cubic nonlinear term, which is a special case of the Fitzhugh-Nagumo system and an infinite-dimensional version of the classical Rayleigh system. We assume that the spatial variable belongs to an interval, supplemented with Neumann boundary conditions. It is well-known that in that specific case there exists a spatially-homogeneous oscillatory regime, which coincides with the time-periodic 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 spatially-inhomogeneous auto-oscillatory or stationary regimes. These regimes are stable with respect to small perturbations from some infinite-dimensional 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 Lyapunov-Schmidt 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.
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Keywords
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reaction-diffusion systems, pattern formation, Lyapunov-Schmidt reduction
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UDC
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517.955.8
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MSC
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35K57
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DOI
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10.20537/vm170402
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Received
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20 May 2017
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Language
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Russian
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Citation
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Kazarnikov A.V., Revina S.V. Bifurcations in a Rayleigh reaction-diffusion system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 4, pp. 499-514.
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References
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- Turing A.M. The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London. Ser. B, 1952, vol. 237, issue 641, pp. 37-72. DOI: 10.1098/rstb.1952.0012
- Szalai I., De Kepper P. Pattern formation in the ferrocyanide-iodate-sulfite reaction: the control of space scale separation, Chaos, 2008, vol. 18, issue 2, pp. 026105. DOI: 10.1063/1.2912719
- Upadhyay R.K., Roy P., Datta J. Complex dynamics of ecological systems under nonlinear harvesting: Hopf bifurcation and Turing instability, Nonlinear Dynamics, 2014, vol. 79, issue 4, pp. 2251-2270. DOI: 10.1007/s11071-014-1808-0
- Tang X., Song Y., Zhang T. Turing-Hopf bifurcation analysis of a predator-prey model with herd behavior and cross-diffusion, Nonlinear Dynamics, 2016, vol. 86, issue 1, pp. 73-89. DOI: 10.1007/s11071-016-2873-3
- Fitzhugh R. Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1961, vol. 1, issue 6, pp. 445-466. DOI: 10.1016/S0006-3495(61)86902-6
- Nagumo J., Arimoto S., Yoshizawa S. An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 1962, vol. 50, issue 10, pp. 2061-2070. DOI: 10.1109/JRPROC.1962.288235
- Zhang Q., Tian C. Pattern dynamics in a diffusive Rössler model, Nonlinear Dynamics, 2014, vol. 78, issue 2, pp. 1489-1501. DOI: 10.1007/s11071-014-1530-y
- Galaktionov V.A., Svirshchevskii S.R. Exact solutions and invariant subspaces on nonlinear partial differential equations in mechanics and physics, New York: CRC Press, 2006, 528 p. DOI: 10.1201/9781420011623
- Kazarnikov A.V., Revina S.V. The onset of auto-oscillations in Rayleigh system with diffusion, Bulletin of the South Ural State University. Series “Mathematical Modelling, Programming and Computer Software”, 2016, vol. 9, no. 2, pp. 16-28 (in Russian). DOI: 10.14529/mmp160202
- Kazarnikov A.V., Revina S.V. Asymptotics of stationary solutions of Rayleigh reaction-diffusion system, Izvestiya Vysshykh Uchebnykh Zavedenii. Severo-Kavkazskii Region. Seriya: Estestvennye Nauki, 2016, issue 3 (191), pp. 13-19 (in Russian). DOI: 10.18522/0321-3005-2016-3-13-19
- Iudovich V.I. Investigation of auto-oscillations of a continuous medium, occurring at loss of stability of a stationary mode, Journal of Applied Mathematics and Mechanics, 1972, vol. 36, issue 3, pp. 424-432. DOI: 10.1016/0021-8928(72)90055-X
- Kozitskii S.B. Model of three dimensional double-diffusive convection with cells of an arbitrary shape, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2012, issue 4, pp. 46-61 (in Russian). DOI: 10.20537/vm120404
- Revina S.V., Yudovich V.I. Initiation of self-oscillations at loss of stability of spatially-periodic, three-dimensional viscous flows with respect to long-wave perturbations, Fluid Dynamics, 2001, vol. 36, no. 2, pp. 192-203. DOI: 10.1023/A:1019225815875
- Melekhov A.P., Revina S.V. Onset of self-oscillations upon the loss of stability of spatially periodic two-dimensional viscous fluid flows relative to long-wave perturbations, Fluid Dynamics, 2008, vol. 43, no. 2, pp. 203-216. DOI: 10.1134/S0015462808020051
- Revina S.V. Recurrence formulas for long wavelength asymptotics in the problem of shear flow stability, Computational Mathematics and Mathematical Physics, 2013, vol. 53, no. 8, pp. 1207-1220. DOI: 10.1134/S096554251306016X
- Revina S.V. Stability of the Kolmogorov flow and its modifications, Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 6, pp. 995-1012. DOI: 10.1134/S0965542517020130
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