Calculation of coefficients of simplest normal forms of Neimark-Sacker and generalized Neimark-Sacker bifurcations

Abstract

The coefficients of the simplest normal forms of both high-dimensional Neimark-Sacker and generalized Neimark-Sacker bifurcation systems were discussed. On the basis of the simplest normal form theory, using appropriate nonlinear transformations, direct computation and the second complete induction, the central manifold equations were further reduced to the simplest normal forms which only contained two nonlinear terms. Theorems were established for the explicit expression of the simplest normal forms in terms of the coefficients of the conventional normal forms of Neimark-Sacker and generalized Neimark-Sacker bifurcations systems.