Stability analysis of approximate periodic solutions of the spin combustion equation

Most of the processes that take place in the world are non-linear. The dynamics of distributed oscillatory systems is usually modeled by systems of differential equations in partial derivatives with certain boundary conditions. Such a system also contains various parameters that characterize the properties of a real object. The research subject is a nonlinear parabolic equation and the boundary value problem corresponding to it, which describes the phenomenological equation. That is the spin combustion equation. The spin combustion equation is solvable by using the Galerkin's and Poincare's methods. The form of the periodic solution is constructed by using a two-mode approximation and center manifold method. The equation is associated with non-stationary processes of combustion front propagation: thermal conductivity of the connection between adjacent sections of the front and self-oscillatory instability of a flat front, which is stabilized due to nonlinear effects. Thermal layers adjacent to the reaction zone interact with each other. The areas are characterized not only by the temperature and speed of advance, but also by the temperature distribution along the entire reaction site. The Laplace operator expresses the nonlocality of the connection. The equation depends not only on the phase variables, but also on the parameters. The behavior of such a dynamic equation and its solution is subject to a qualitative change with an infinitesimal change in its parameters. There is the process of bifurcation. Passing through the bifurcation parameter leads to a change in the number of solutions, their stability, and the transformation of trajectories. The problem is in studying stable similar equations in such physical processes as optics, radiophysics, and combustion theory. The relevance of the work depends on the fact that the nonlinear parabolic equation describing the process of spin combustion is underexplored. And also in the construction of periodic forms of the solution, their approximation and the analysis of the stability of branching periodic solutions of the equation, which correspond to solutions of the traveling wave type, with small changes in the parameter. The method to be used combines Galerkin’s and Poincare's methods, on the basis of which the form of an approximate solution for a system of functions is constructed like $\cos k\theta$. The solution can be represented as the sum of $2\pi$-periodic functions, which allows one to find the values of the coefficients of the initial and two branching periodic solutions of the system.