Convergence to the Critical Attractor at Infinite and Tangent Bifurcation Points

The dynamics of the convergence to the critical attractor for the logistic map is investigated. At the border of chaos, when the Liapunov exponent is zero, the use of the non-extensive statistical mechanics formalism allows to define a weak sensitivity or insensitivity to initial conditions. Using this formalism we analyse how a set of initial conditions spread all over the phase space converges to the critical attractor in the case of infinite bifurcation and tangent bifurcation points. We show that the phenomena is governed in both cases by a power-law regime but the critical exponents depend on the type of bifurcation and may also depend on the numerical experiment set-up. Differences and similarities between the two cases are also discussed.Comment: 5 figures, accepted for publication in Int. J. of Bifurcation and Chaos (July 2006