Dynamics of Spatially Distributed Chains of Coupled Systems of Equations in a Two-Dimensional Domain

The local dynamics of coupled identical nonlinear systems of second-order differential equations in a two-dimensional domain is studied. The main assumption is that the number of such equations is quite large. This makes it possible to move to a system with two continuous spatial variables. Critical cases in the problem of stability of the equilibrium state are highlighted. They all are of infinite dimension, i.e., the infinitely many roots of the characteristic equation for the linearized problem tend to the imaginary axis as the natural small parameter tends to zero. Special nonlinear partial differential equations are constructed whose nonlocal dynamics describes the behavior of the initial system in a neighborhood of the equilibrium state, which plays the role of a normal form. It should especially be noted that the constructed partial differential systems contain four spatial variables with boundary conditions for each of them.