The Keynes model of the business cycle and the problem of diffusion instability

In this paper, we consider a version of the “reaction-diffusion” system, which can be interpreted as a mathematical model of the Keynes business cycle, taking into account spatial factors. The system is considered together with homogeneous Neumann boundary conditions. For such a nonlinear boundary-value problem, bifurcations in a neighborhood of a spatially homogeneous equilibrium state are studied in the near-critical case of zero and a pair of purely imaginary eigenvalues of the stability spectrum. An analysis of bifurcations allows one to obtain sufficient conditions for the existence and stability of spatially homogeneous and spatially inhomogeneous cycles and a spatially inhomogeneous equilibrium state. The analysis of the problem stated is based on the methods of the theory of infinite-dimensional dynamical systems, namely, the method of integral (invariant) manifolds and the method of normal forms. These methods and asymptotic methods of analysis lead to asymptotic formulas for periodic solutions and inhomogeneous equilibria. For such solutions, we also examine their stability.