Scaling behaviour for recurrence-based measures at the edge of chaos

¹ Department of Physics, Faculty of Science, Ege University - 35100 Izmir, Turkey
² Potsdam Institute for Climate Impact Research (PIK) - 14473 Potsdam, Germany
³ Department of Physics, Humboldt University - 12489 Berlin, Germany
⁴ Institute for Complex Systems and Mathematical Biology, University of Aberdeen - Aberdeen AB24 3UE, UK

Received: 31 July 2015
Accepted: 1 October 2015

Abstract

The study of phase transitions with critical exponents has helped to understand fundamental physical mechanisms. Dynamical systems which go to chaos via period doublings show an equivalent behavior during transitions between different dynamical regimes that can be expressed by critical exponents, known as the Huberman-Rudnick scaling law. This universal law is well studied, e.g., with respect to the Lyapunov exponents. Recurrence plots and related recurrence quantification analysis are popular tools to investigate the regime transitions in dynamical systems. However, the measures are mostly heuristically defined and lack clear theoretical justification. In this letter we link a selection of these heuristical measures with theory by numerically studying their scaling behavior when approaching a phase transition point. We find a promising similarity between the critical exponents to those of the Huberman-Rudnick scaling law, suggesting that the considered measures are able to indicate dynamical phase transition even from the theoretical point of view.