Stability of mappings with a spectrum on a unit circle

Nonlinear mappings with neutral fixed points are considered. It is assumed that the linear approximation spectrum is non-resonant and lies on a unit circle. Conditions for asymptotic stability and instability of a fixed point of a mapping under various degrees of the stability degeneration problem are obtained. In the main case they are necessary and sufficient conditions. For mappings in the normal form the Lyapunov and Chetayev polynomial functions have been designed and the instability zones constructed.