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Abstract In many problems it is meaningless not to have dependence on the past. Time delays occur so often, in almost every situation, that to ignore them is to ignore reality [41]. Dynamical systems with time delay have been attracting the attention of researchers of many elds, see [55, 58, 71, 33, 59, 17, 9, 46, 44, 57]. One of the principal mathematical instruments of modern nonlinear dynamics is the dierence equation with continuous argument given in the form x(t) = f(x(t 1)); t 2 [0; T]: (1) The rst eld of applications of these equations is delay dierential equa- tions theory. The second point of application is partial dierential equations theory. There are many classes of problems for partial dierential equations that can be reduced directly to continuous time dierence equations. Lastly, dierence equations with continuous argument are suitable for simulating such phenomena as fractal structures, cascade process of emergence of co- herent structures[63]. I. |