DYNAMIC BEHAVIORS OF 2(INFINITY) ATTRACTOR AND Q-PHASE TRANSITIONS ATBIFURCATIONS IN LOGISTIC MAP

Dynamic behaviours of the 2(infinity) attractor at the accumulation of period doubling in the logistic map are studied by the sum of the loc al expansion rates S-n(x(1)) of nearby orbits. The variance [[S-n(x)]( 2)] and algebraic exponent beta(n)(x(1)) = S-n(x(1))/ln(n) exhibits se lf-similar structures. The critical bifurcations such as intermittency , band merging and crisis-sudden widening of the chaotic attractor are studied in terms of a q-weighted average Lambda(q), (-infinity < q < infinity) of the coarse-grained local expansion rates Lambda of nearby orbitals.