On analytic periodic solutions to nonlinear differential equations with a delay (advancing)

We consider the system of a type of reaction–diffusion in which the diffusion coefficients depend in an arbitrary way on the spatial variables and concentrations, and the reactions are described by homogeneous functions with coefficients that depend in a special way on the spatial variables. It is shown that the system has family of exact solutions expressed through solutions to a system of ordinary differential equations (ODE) with the homogeneous functions in right-hand sides. For a special case of the ODE system we construct the general solution representable by Jacobi higher transcendental functions. It is established that solutions are periodic functions and satisfy non-linear differential equations with delay (advancing) which size is defined by the choice of initial conditions for ODE system. It is shown that these periodic solutions are analytic functions, representable in the neighborhood of each point on the period by the convergent power series.