title: Fluctuation of similarity to detect transitions between distinct dynamical regimes in short time series.
abstract: A method to identify distinct dynamical regimes and transitions between those regimes in a short univariate time series was recently introduced [N. Malik et al., Europhys. Lett. 97, 40009 (2012)], employing the computation of fluctuations in a measure of nonlinear similarity based on local recurrence properties. In this work, we describe the details of the analytical relationships between this newly introduced measure and the well-known concepts of attractor dimensions and Lyapunov exponents. We show that the new measure has linear dependence on the effective dimension of the attractor and it measures the variations in the sum of the Lyapunov spectrum. To illustrate the practical usefulness of the method, we identify various types of dynamical transitions in different nonlinear models. We present testbed examples for the new method's robustness against noise and missing values in the time series. We also use this method to analyze time series of social dynamics, specifically an analysis of the US crime record time series from 1975 to 1993. Using this method, we find that dynamical complexity in robberies was influenced by the unemployment rate until the late 1980s. We have also observed a dynamical transition in homicide and robbery rates in the late 1980s and early 1990s, leading to increase in the dynamical complexity of these rates.
authors.affiliation: Department of Mathematics, CB#3250, University of North Carolina, Chapel Hill, North Carolina 27599, USA.
authors.forename: Nishant
authors.initials: N
authors.lastname: Malik
authors.affiliation: Potsdam Institute for Climate Impact Research, P. O. Box 601203, 14412 Potsdam, Germany.
authors.forename: Norbert
authors.initials: N
authors.lastname: Marwan
authors.affiliation: Potsdam Institute for Climate Impact Research, P. O. Box 601203, 14412 Potsdam, Germany and Department of Physics, East China Normal University - Shanghai 200241, China.
authors.forename: Yong
authors.initials: Y
authors.lastname: Zou
authors.affiliation: Department of Mathematics, CB#3250, University of North Carolina, Chapel Hill, North Carolina 27599, USA and Department of Applied Physical Sciences, CB#3216, University of North Carolina, Chapel Hill, North Carolina 27599, USA.
authors.forename: Peter J
authors.initials: PJ
authors.lastname: Mucha
authors.affiliation: Potsdam Institute for Climate Impact Research, P. O. Box 601203, 14412 Potsdam, Germany and Department of Physics, Humboldt University Berlin, Newtonstr. 15, 12489 Berlin, Germany.
authors.forename: Jürgen
authors.initials: J
authors.lastname: Kurths
mesh_terms: D003415:Crime / Q000706:statistics & numerical data; D016014:Linear Models; D017711:Nonlinear Dynamics; D013995:Time; D014478:Unemployment / Q000706:statistics & numerical data; D014481:United States