Normal and quasinormal forms for systems of difference and differential-difference equations

•The local dynamics of difference and singularly perturbed differential-difference equations is investigated.•Special nonlinear partial differential equations are constructed.•Their nonlocal dynamics is shown to determine the local behavior of solutions to the initial system.

The local dynamics of systems of difference and singularly perturbed differential-difference equations is studied in the neighborhood of a zero equilibrium state. Critical cases in the problem of stability of its state of equilibrium have infinite dimension. Special nonlinear evolution equations, which act as normal forms, are set up. It is shown that their dynamics defines the behavior of solutions to the initial system.