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

TL;DR

This paper analytically investigates the sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps, revealing q-exponential behaviors and calculating generalized Lyapunov coefficients within a nonextensive statistical mechanics framework.

Contribution

It provides exact analytical solutions for the dynamics and sensitivity at bifurcations using the Feigenbaum RG, including the first calculation of the q-generalized Lyapunov coefficient.

Findings

Weak insensitivity to initial conditions at pitchfork and tangent bifurcations

Super-strong sensitivity on the tangent bifurcation's right side

Analytical results confirmed by numerical calculations

Abstract

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 {\em a priori} numerical calculations.