Methods of small and large δ in the nonlinear dynamics -: A comparative analysis

New asymptotic approaches for dynamical systems containing a power nonlinear term x(n) are proposed and analyzed. Two natural limiting cases are studied: n approximate to 1 + delta, delta much less than 1 and n --> infinity. In the first case, the 'small delta method' (S delta M) is used and its applicability for dynamical problems with the nonlinear term sin alpha as well as the usefulness of the S delta M for the problem with small denominators are outlined. For n --> infinity, a new asymptotic approach is proposed (conditionally we call it the 'large delta method' - L delta M). Error estimations lead to the following conclusion: the L delta M may be used, even for small n, whereas the S delta M has a narrow application area. Both of the discussed approaches overlap all values of the parameter n.