The onset of auto-oscillations in Rayleigh system with diffusion

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 $\mathrm{D}$, 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 $\frac{n+1}{2}$ otherwise.