Fluctuation of similarity (FLUS) to detect transitions between distinct dynamical regimes in short time series

Recently a method which employs computing of fluctuations in a measure of nonlinear similarity based on local recurrence properties in a univariate time series, was introduced to identify distinct dynamical regimes and transitions between them in a short time series [1]. Here we present the details of the analytical relationships between the 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 employ it to identify various types of dynamical transitions in different nonlinear models. Also, we present testbed examples for the new method's robustness against the presence of noise and missing values in the time series. Furthermore, we use this method to analyze time series from the field of social dynamics, where we present an analysis of the US crime record's time series from the year 1975 to 1993. Using this method, we have found that dynamical complexity in robberies was influenced by the unemployment rate till late 1980's. We have also observed a dynamical transition in homicide and robbery rates in the late 1980's and early 1990's, leading to increase in the dynamical complexity of these rates.