Issue |
Europhys. Lett.
Volume 60, Number 4, November 2002
|
|
---|---|---|
Page(s) | 518 - 524 | |
Section | General | |
DOI | https://doi.org/10.1209/epl/i2002-00249-7 | |
Published online | 01 November 2002 |
Sensitivity to initial conditions at bifurcations in one-dimensional nonlinear maps: Rigorous nonextensive solutions
1
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
Corresponding authors: baldovin@cbpf.br, robledo@fisica.unam.mx
Received:
22
May
2002
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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