Analysis of local dynamics of difference and close to them differential-difference equations

We consider the local dynamics of a class nonlinear difference equations which is important for applications. Using the perturbation theory methods we built the sets of singularly perturbed differential-difference equations close to the original difference equations to some extent. We show that the critical cases in the problem of stability of a null balance state have infinite dimension. We offer the method to set special non-linear boundary-value problems that do not contain small parameters. They play the role of normal forms. Their nonlocal dynamics describes the structure of solutions to original equations in a small neighborhood of a balance state. We show that the dynamic properties of difference and close to them differential-difference equations considerably differ.