The dynamics of periodic solutions of non-stationary combustion along the strip

Physical objects, describes the distribution of self-oscillating system is the
combustion front propagates in a homogeneous medium. Experimental studies conducted
Merzhanov, Borovinskaya showed that the stationary mode of propagation of the front becomes
unstable if the activation energy of the exothermic reaction exceeds a certain value. In the
instability of the stationary mode combustion front, staying flat, moving in an oscillatory mode. In
works of Aldushin, Zeldovich, Malomed unsteady regimes of propagation of the combustion front
was effectively described the phenomenological equation for the coordinate points of the front of
the report in a system where the front of the average rests. In the simplest case - a singularly
perturbed parabolic equation of Van der Pole type (1). Among the most interesting of unsteady
combustion modes distinguish the spin waves are found in particular in the combustion of gasfree systems. In contrast to the self-oscillating unsteady combustion, when the reaction front is
flat spin modes in running the centers arise reactions pervert the front and leaving a spiral track
on the surface of the burned-out cylinder. Study of spin combustion modes are devoted works of
Kolesov, Rozov, Samoilenko, Belan. In this paper the phenomenological equation of propagation
of nonstationary combustion regimes along the strip described by a parabolic equation of Van
der Pole type with small diffusion and Neumann boundary conditions. The above equation
allows analytically investigate the distribution of the curvature of the waves along the front
in problems with various boundary conditions. This problem can be represented as a system of
coupled ordinary and parabolic equations, and therefore according to Henry, is soluble in the
phase space E. We consider the problem of existence, asymptotic form and of the stability of
spatially inhomogeneous periodic solutions bifurcating from a spatially homogeneous - phase of
the waves and the zero solution - standing waves. To solve this problem using methods of nonlinear
mechanics, such as the method of central manifolds and Galerkin method. Shows the theorem
on the existence of asymptotic stability of the form and the first self-similar cycle bifurcating
from losing the stability of phase waves of the original problem in a small neighborhood of the bifurcation parameter. Through numerical calculations and finite-dimensional approximation of
the Galerkin held bifurcation analysis of self-similar regimes by removing the critical parameter in
the region. The values of the bifurcation parameter in which a spatially inhomogeneous periodic
solutions bifurcating from a spatially homogeneous and neutral solutions become resistant.
Mechanism of finding of stability this solutions is described.
Keywords: combustion, bifurcation, periodic solutions, orbital stability, auto-model circles,
parabolic equation.