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 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 qindices 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. PACS numbers: 05.10.Cc, 05.45.Ac, 05.90.+m The nonextensive generalization [1] of the canonical statistical mechanics has raised interest in testing its applicability in several suggested circumstances in a variety of physical systems [1]. A class of problems where this issue has been much studied recently is the dynamical behavior of nonlinear iterated maps under critical conditions [2, 3, 4, 5, 6]. Such is the case of the bifurcation points associated to deterministic chaos in simple nonlinear dissipative maps, like those occurring in the logistic map and its generalization to nonlinearity ζ > 1. These types of critical states provide an exceptional opportunity to examine explicitly the underlying mathematical structure and physical implications of their E-mail addresses: [email protected], [email protected]