Averaging Method in the Problem of Constructing Self-Oscillatory Solutions of Distributed Kinetic Systems

An averaging method is developed for two-component distributed kinetic systems with weak diffusion in a bounded one-dimensional domain with impermeability conditions at its boundary. Transformations of the considered system are constructed that make it possible to choose one fast variable and a countable number of slow variables. Theorems are proved about correspondence of stationary and periodic solutions and invariant tori of the averaged equations of slow variables to spatially inhomogeneous periodic solutions and invariant tori of the original equations of the same stability type, respectively. Algorithms for constructing periodic solutions (cycles) and invariant tori of the original equations in the form of expansions in powers of a small parameter are proposed, which ensure the construction of asymptotic formulas for the indicated self-oscillatory objects. Convergence conditions for the corresponding expansions are formulated.