Dynamic behaviours of 2∞ attractor and q-phase transitions at bifurcations in logistic map

Dynamic behaviours of the 2∞ attractor at the accumulation of period doubling in the logistic map are studied by the sum of the local expansion rates Sn(x1) of nearby orbits. The variance [Sn(x)]2 and algebraic exponent ßn(x1) = Sn(x1)/ln(n) exhibits self-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 Λ(q), (− ∞ < q < ∞) of the coarse-grained local expansion rates Λ of nearby orbitals