Eur. Phys. J. B (2014) 87: 32
https://doi.org/10.1140/epjb/e2014-40882-1

Regular Article

Sums of variables at the onset of chaos

¹ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, 87501 New Mexico, USA
² Centro Atómico Bariloche, Instituto Balseiro and CONICET, 8400 Bariloche, Argentina
³ Centro de Investigación en Complejidad Social, Facultad de Gobierno, Universidad del Desarrollo, Santiago, Chile
⁴ Instituto de Física y Centro de Ciencias de la Complejidad, Universidad Nacional Autónoma de México, Apartado Postal 20-364, México 01000 DF, Mexico

^a e-mail: fuentesm@santafe.edu

Received: 27 September 2013
Published online: 5 February 2014

Abstract

We explain how specific dynamical properties give rise to the limit distribution of sums of deterministic variables at the transition to chaos via the period-doubling route. We study the sums of successive positions generated by an ensemble of initial conditions uniformly distributed in the entire phase space of a unimodal map as represented by the logistic map. We find that these sums acquire their salient, multiscale, features from the repellor preimage structure that dominates the dynamics toward the attractors along the period-doubling cascade. And we explain how these properties transmit from the sums to their distribution. Specifically, we show how the stationary distribution of sums of positions at the Feigebaum point is built up from those associated with the supercycle attractors forming a hierarchical structure with multifractal and discrete scale invariance properties.