Volume 60, Number 4, November 2002
|Page(s)||518 - 524|
|Published online||01 November 2002|
Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions
Centro Brasileiro de Pesquisas Físicas -
Rua Xavier Sigaud 150
22290-180 Rio de Janeiro, RJ, Brazil
2 Instituto de Física, Universidad Nacional Autónoma de México Apartado Postal 20-364, México 01000 D.F., Mexico
Accepted: 29 August 2002
Using the Feigenbaum renormalization group (RG) transformation we work out exactly the dynamics and the sensitivity to initial conditions for unimodal maps of nonlinearity 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 . 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: 05.10.Cc – Renormalization group methods / 05.45.Ac – Low-dimensional chaos / 05.90.+m – Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems
© EDP Sciences, 2002
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