The problem of birth of autowaves in parabolic systems with small diffusion

A parabolic reaction-diffusion system with zero Neumann boundary conditions at the end-points of a finite interval is considered under the following basic assumptions. First, the matrix diffusion coefficient in the system is proportional to a small parameter $\varepsilon>0$, and the system itself possesses a spatially homogeneous cycle (independent of the space variable) of amplitude of order $\sqrt\varepsilon$ born by a zero equilibrium at an Andronov–Hopf bifurcation. Second, it is assumed that the matrix diffusion depends on an additional small parameter $\mu\geqslant0$, and for $\mu=0$ there occurs in the stability problem for the homogeneous cycle the critical case of characteristic multiplier 1 of multiplicity 2 without Jordan block. Under these constraints and for independently varied parameters $\varepsilon$ and $\mu$ the problem of the existence and the stability of spatially inhomogeneous auto-oscillations branching from the homogeneous cycle is analysed.
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