Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions

Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity zeta > 1 at both their pitchfork and tangent bifurcations. These functions have the form of q-exponentials as proposed in Tsallis' generalization of statistical mechanics. We determine the q-indices that characterize these universality classes and perform for the first time the calculation of the q-generalized Lyapunov coefficient lambda(q). The pitchfork and the left-hand side of the tangent bifurcations display weak insensitivity to initial conditions, while the right-hand side of the tangent bifurcations presents a "super-strong" (faster than exponential) sensitivity to initial conditions. We corroborate our analytical results with a priori numerical calculations.