Volume 9, no. 2Pages 16 - 28

The Onset of Auto-Oscillations in Rayleigh System with Diffusion

A.V. Kazarnikov, S.V. Revina
A reaction-diffusion system with cubic nonlinear term, which is the infinite-dimensional case of classical Rayleigh oscillator, is considered in the present paper. Spatial variable belongs to a bounded m-dimensional domain D, supposed that Dirichlet or Neumann conditions are set on the boundary. Critical values of control parameter, corresponding to monotonous and oscillatory instability are found. Asymptotic approximations of patterns, branching from zero uniform solution due to oscillatory instability are found. Asymptotic approximations are valid for different types of boundary conditions. It is shown that soft loss of stability takes place in the system. By developing an abstract scheme and applying Lyapunov-Schmidt method, formulas for consecutive terms of asymptotic expansion are found. It was found that all terms of asymptotic expansion are odd trigonometric polynomials in time. Several applications of abstract scheme to one-dimensional domain are shown. In this case, branching solutions have certain symmetries. It is shown that the n-th term of asymptotic contains eigenfunctions of Laplace operator with indexes less or equal to n in the case of Diriclet boundary conditions or less or equal to (n+1)/2 otherwise.
Full text
Rayleigh equation; Lyapunov - Schmidt reduction; self-oscillations; reaction-diffusion systems.
1. Fitzhugh R. Impulses and Physiological States in Theoretical Models of Nerve Membrane. Biophysical Journal, 1961, no. 1, pp. 257-278.
2. Nagumo J.M., Arimoto S., Yoshizawa S. An Active Pulse Transmission Line Simulating Nerve Axon. Proceedings of the IRE, 1962, vol. 50, pp. 2061-2070.
3. Tuwankotta J.M. Studies on Rayleigh Equation. Integral, 2000, vol. 5, no. 1, pp. 1-9.
4. Glyzin S.D., Kolesov A.Yu., Rozov N.Kh. Finite-Dimensional Models of Diffusion Chaos. Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 5, pp. 816-830. DOI: 10.1134/S0965542510050076
5. Bashkirtseva I.A., Ryashko L.B., Slepoykhina E.S. [Splitting Bifurcation of Stochastic Cycles in FitzHugh - Nagumo Model]. Nelineynaya dinamika [Nonlinear Dynamics], 2013, vol. 9, no. 2, pp. 295-307. (in Russian)
6. Kazarnikov A.V., Revina S.V. [Hopf Bifurcation in Spatially Distributed Rayleigh Equation]. Dep. VINITI, 2013, vol. 83, no. 242-В2013. (in Russian)
7. Yudovich 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, no. 3, pp. 450-459.
8. 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.
9. 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.
10. 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