C. Letellier, I. Leyva & I. Sendiña-Nadal
Dynamical complexity measure to distinguish organized from disorganized dynamics
Physical Review E, 101, 022204, 2020. online

- Abstract
We propose a metric to characterize the complex behavior of a dynamical system and to distinguish between organized and disorganized complexity. The approach combines two quantities that separately assess the degree of unpredictability of the dynamics and the lack of describability of the structure in the Poincaré plane constructed from a given time series. As for the former, we use the permutation entropy Sp, while for the latter, we introduce an indicator, the structurality Δ, which accounts for the fraction of visited points in the Poincaré plane. The complexity measure thus defined as the sum of those two components is validated by classifying in the (Sp,Δ) space the complexity of several benchmark dissipative and conservative dynamical systems. As an application, we show how the metric can be used as a powerful biomarker for different cardiac pathologies and to distinguish the dynamical complexity of two electrochemical dissolutions.

C. Letellier
Assessing synchronizability provided by coupling variable from the algebraic structure of dynamical systems
Physical Review E, 101, 042215, 2020. Online

- Abstract
Synchronization is a very generic phenomenon which can be encountered in a large variety of coupled dynamical systems. Being able to synchronize chaotic systems is strongly dependent on the nature of their coupling. Few attempts to explain such a dependency using observability and/or controllability were not fully satisfactory and synchronizability yet remained unexplained. Synchronizability can be defined as the range of coupling parameter values for which two nearly identical systems are fully synchronized. Our objective is here to investigate whether synchronizability can be related to the main rotation necessarily required for structuring any type of attractor, that is, whether synchronizability is significantly improved when the coupling variable is one of the variables involved in the main rotation or not. We thus propose a semi-analytic procedure from a single isolated system, to discard the worst variable for fully synchronizing two (nearly) identical copies of that system.

C. E. Gonzalez, C. Lainscsek, T. J. Sejnowski & C. Letellier
Assessing observability of chaotic systems using Delay Differential Analysis
Chaos, accepted Preprint

- Abstract
Observability can determine which recorded variables of a given system are optimal for discriminating its different states. Quantifying observability requires knowledge of the equations governing the dynamics. These equations are often unknown when experimental data are considered. Consequently, we propose an approach for numerically assessing observability using Delay Differential Analysis (DDA). Given a time series, DDA uses a delay differential equation for approximating the measured data. The lower the least squares error between the predicted and recorded data, the higher the observability. We thus rank the variables of several chaotic systems according to their corresponding least square error to assess observability. The performance of our approach is evaluated by comparison with the ranking provided by the symbolic observability coefficients as well as with two other data-based approaches using reservoir computing and singular value decomposition of the reconstructed space. We investigate the robustness of our approach against noise contamination.

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