Nonstationary Process: Nonstationary Bifurcation Maps, Evolutionary Dynamics

Abstract

The study presented in this paper is one of a series of paperspublished by the authors on nonstationary problems. It addresses itselfto the characterization of the types of dynamical responses and theirranges contained in the time flow of the Duffing nonlinear,nonstationary, dissipative, forced oscillator. A new effective method –a Nonstationary Bifurcation Map (EI-Lu map) – has been introduced bythe authors that allows us to do precisely this. This new technique isby far more advantageous than the customary methods in use: the phaseportrait or Poincaré maps. The latter may give inadequate informationbecause of the overlapping dynamical responses contained within rangesof time. The main feature of nonstationary processes is that thenonstationary responses are transient. The phenomena of the transiencyare presented in detail. Significant cases are those when thenonstationary transmission of the signals crosses differentnonstationary bifurcation boundaries. This is significant because mostof dynamical-biological activities occur in the regions between orderand chaos. It characterizes nonstationary dynamical processes. Thepossibility of constructing responses for arbitrary small nonstationaryinputs may be used as nonstationary perturbations, replacing customaryperturbations of integrable Hamiltonians.