Characteristic polynomials for a cycle of non-linear discrete systems with time delays

We study a method associated with constructing of delayed feedback for local stabilization of periodic orbits of nonlinear discrete systems. An alternative approach to the construction of characteristic polynomial for the delay system linearized in the neighborhood of $T$-cycle is suggested. It is proven that our new alternative approach is equivalent to the standard one, however, it allows us to produce directly new forms of polynomials. These forms are convenient in applications to the problems of chaos control and allow us to apply methods of geometric complex function theory. This article is an extension of the results, which received D. Dmitrishin, P. Haglstein, A. Khamitova and A. Stokolos to the vector case. Refs 6. Fig 1.