STRUCTURE OF THE ATTRACTION ZONES OF THE FINAL-STATES IN THE PRESENCEOF DYNAMICAL PERIOD-DOUBLING BIFURCATIONS

The structure of the attraction zones of the final states associated w ith dynamical period doubling bifurcations is investigated. It is foun d that on the ''initial value-transition rate'' plane the attraction z ones of the two possible final states alternate with each other and th at a subdivision of the attraction regions occurs with a decrease in t he transition rate. It is shown that the boundaries of the attraction zones are smeared out because of the effect of noise and in this situa tion the fine structure of the attraction zones is destroyed. As analy tical and numerical calculations have shown, the critical value of the noise variance, corresponding to the boundary between the dynamical ( or predictable) and stochastic (or unpredictable) modes, has a power-l aw dependence on the transition rate with a typical exponent value of one. The existence of ''noise'' invariants is also observed: the integ rated (over all initial values) probability of achieving the final sta te is invariant with respect to the noise level. (C) 1998 American Ins titute of Physics.