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

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Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity ζ>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 λq. 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.