Global vector field reconstruction from a chaotic experimental signal in copper electrodissolution

Scenarios for the Altamira cave CO2 concentration from 1950 to 2100

A data-driven approach insensitive to the initial conditions was developed to extract governing equations for the concentration of CO2 in the Altamira cave (Spain) and its two main drivers: the outside temperature and the soil moisture. This model was then reformulated in order to use satellite observations and meteorological predictions, as a forcing. The concentration of CO2 inside the cave was then investigated from 1950 to 2100 under various scenarios. It is found that extreme levels of CO2 were reached during the period 1950-1972 due to the massive affluence of visitors. It is demonstrated that it is possible to monitor the CO2 in the cave in real time using satellite information as an external forcing. For the future, it is shown that the maximum values of CO2 will exceed the levels reached during the 1980s and the 1990s when the CO2 introduced by the touristic visits, although intentionally reduced, still enhanced considerably the micro corrosion of walls and pigments.

COVID-19 in Africa: Underreporting, demographic effect, chaotic dynamics, and mitigation strategy impact

The epidemic of COVID-19 has shown different developments in Africa compared to the other continents. Three different approaches were used in this study to analyze this situation. In the first part, basic statistics were performed to estimate the contribution of the elderly people to the total numbers of cases and deaths in comparison to the other continents; Similarly, the health systems capacities were analysed to assess the level of underreporting. In the second part, differential equations were reconstructed from the epidemiological time series of cases and deaths (from the John Hopkins University) to analyse the dynamics of COVID-19 in seventeen countries. In the third part, the time evolution of the contact number was reconstructed since the beginning of the outbreak to investigate the effectiveness of the mitigation strategies. Results were compared to the Oxford stringency index and to the mobility indices of the Google Community Mobility Reports.

Compared to Europe, the analyses show that the lower proportion of elderly people in Africa enables to explain the lower total numbers of cases and deaths by a factor of 5.1 on average (from 1.9 to 7.8). It corresponds to a genuine effect. Nevertheless, COVID-19 numbers are effectively largely underestimated in Africa by a factor of 8.5 on average (from 1.7 to 20. and more) due to the weakness of the health systems at country level. Geographically, the models obtained for the dynamics of cases and deaths reveal very diversified dynamics. The dynamics is chaotic in many contexts, including a situation of bistability rarely observed in dynamical systems. Finally, the contact number directly deduced from the epidemiological observations reveals an effective role of the mitigation strategies on the short term. On the long term, control measures have contributed to maintain the epidemic at a low level although the progressive release of the stringency did not produce a clear increase of the contact number. The arrival of the omicron variant is clearly detected and characterised by a quick increase of interpeople contact, for most of the African countries considered in the analysis.

Can the original equations of a dynamical system be retrieved from observational time series?

The aim of the present work is to investigate the possibility to retrieve the original sets of dynamical equations directly from observational time series when all the system variables are observed. Time series are generated from chosen dynamical systems, and the global modeling technique is applied to obtain optimal models of parsimonious structure from these time series. The obtained models are then compared to the original equations to investigate if the original equations can be retrieved. Twenty-seven systems are considered in the study. The Rössler system is first used to illustrate the procedure and then to test the robustness of the approach under various conditions, varying the initial conditions, time series length, dynamical regimes, subsampling (and resampling), measurement noise, and dynamical perturbations. The other 26 systems (four rational ones included) of various algebraic structures, sizes, and dimensions are then considered to investigate the generality of the approach.

Two chaotic global models for cereal crops cycles observed from satellite in northern Morocco

The dynamics underlying cereal crops in the northern region of Morocco is investigated using a global modelling technique applied to a vegetation index time series derived from satellite measurements, namely, the normalized difference vegetation index from 1982 to 2008. Two three-dimensional chaotic global models of reduced size (14-term and 15-term models) are obtained. The model validation is performed by comparing their horizons of predictability with those provided in previous studies. The attractors produced by the two global models have a complex foliated structure-evidenced in a Poincaré section-rending a topological characterization difficult to perform. Thus, the Kaplan-Yorke dimension is estimated from the synthetic data produced by our global models. Our results suggest that cereal crops in the northern Morocco are governed by a weakly dissipative three-dimensional chaotic dynamics.

Optimization of global model composed of radial basis functions using the term-ranking approach

A term-ranking method is put forward to optimize the global model composed of radial basis functions to improve the predictability of the model. The effectiveness of the proposed method is examined by numerical simulation and experimental data. Numerical simulations indicate that this method can significantly lengthen the prediction time and decrease the Bayesian information criterion of the model. The application to real voice signal shows that the optimized global model can capture more predictable component in chaos-like voice data and simultaneously reduce the predictable component (periodic pitch) in the residual signal.

Validating the physical model of a chaotic system by topological analysis

Topological analysis is employed for the first time to our knowledge as a method of validation for a physical model describing a chaotic system. Topological analysis theory provides both a way to characterize the topological structure of chaotic attractors by means of a set of integer numbers and a method to obtain this set departing from a time series generated by the chaotic system. The validation method proposed here consists of comparing the topological structure of chaotic attractors obtained from time series generated on the one hand by an experimental system and on the other hand by the numerical model under test. This procedure has been applied to an erbium-doped fiber laser subject to pump power sine-wave modulation.

Polynomial search and global modeling: Two algorithms for modeling chaos

Global modeling aims to build mathematical models of concise description. Polynomial Model Search (PoMoS) and Global Modeling (GloMo) are two complementary algorithms (freely downloadable at the following address: http://www.cesbio.ups-tlse.fr/us/pomos_et_glomo.html) designed for the modeling of observed dynamical systems based on a small set of time series. Models considered in these algorithms are based on ordinary differential equations built on a polynomial formulation. More specifically, PoMoS aims at finding polynomial formulations from a given set of 1 to N time series, whereas GloMo is designed for single time series and aims to identify the parameters for a selected structure. GloMo also provides basic features to visualize integrated trajectories and to characterize their structure when it is simple enough: One allows for drawing the first return map for a chosen Poincaré section in the reconstructed space; another one computes the Lyapunov exponent along the trajectory. In the present paper, global modeling from single time series is considered. A description of the algorithms is given and three examples are provided. The first example is based on the three variables of the Rössler attractor. The second one comes from an experimental analysis of the copper electrodissolution in phosphoric acid for which a less parsimonious global model was obtained in a previous study. The third example is an exploratory case and concerns the cycle of rainfed wheat under semiarid climatic conditions as observed through a vegetation index derived from a spatial sensor.

Nonuniqueness of global modeling and time scaling

Starting from an observed single time series, it is shown how to reconstruct a global model in the original phase space by using the ansatz library approach. This model is then compared to the underlying dynamical system that describes the initial time series, and the nonuniqueness of the reconstructed model is discussed. This framework is extended by taking an additional time scaling factor in the reconstructed model class under consideration.

Identifying chaos from heart rate: the right task?

Providing a conclusive answer to the question "is this dynamics chaotic?" remains very challenging when experimental data are investigated. We showed that such a task is actually a difficult problem in the case of heart rates. Nevertheless, an appropriate dynamical analysis can discriminate healthy subjects from patients.

Frequently asked questions about global modeling

When a global model is attempted from experimental data, some preprocessing might be required. Therefore it is only natural to wonder what kind of effects the preprocessing might have on the modeling procedure. This concern is manifested in the form of recurrent frequently asked questions, such as "how does the preprocessing affect the underlying dynamics?" This paper aims at providing answers to important questions related to (i) data interpolation, (ii) data smoothing, (iii) data-estimated derivatives, (iv) model structure selection, and (v) model validation. The answers provided will hopefully remove some of those doubts and one shall be more confident not only on global modeling but also on various data analyses which may be also dependent on data preprocessing.

Spatiotemporal system reconstruction using Fourier spectral operators and structure selection techniques

A technique based on trigonometric spectral methods and structure selection is proposed for the reconstruction, from observed time series, of spatiotemporal systems governed by nonlinear partial differential equations of polynomial type with terms of arbitrary derivative order and nonlinearity degree. The system identification using Fourier spectral differentiation operators in conjunction with a structure selection procedure leads to a parsimonious model of the original system by detecting and eliminating the redundant parameters using orthogonal decomposition of the state data. Implementation of the technique is exemplified for a highly stiff reaction-diffusion system governed by the Kuramoto-Sivashinsky equation. Numerical experiments demonstrate the superior performance of the proposed technique in terms of accuracy as well as robustness, even with smaller sets of sampling data.

Modeling global vector fields of chaotic systems from noisy time series with the aid of structure-selection techniques

We address the problem of reconstructing a set of nonlinear differential equations from chaotic time series. A method that combines the implicit Adams integration and the structure-selection technique of an error reduction ratio is proposed for system identification and corresponding parameter estimation of the model. The structure-selection technique identifies the significant terms from a pool of candidates of functional basis and determines the optimal model through orthogonal characteristics on data. The technique with the Adams integration algorithm makes the reconstruction available to data sampled with large time intervals. Numerical experiment on Lorenz and Rossler systems shows that the proposed strategy is effective in global vector field reconstruction from noisy time series.

Modeling continuous processes from data

Experimental and simulated time series are necessarily discretized in time. However, many real and artificial systems are more naturally modeled as continuous-time systems. This paper reviews the major techniques employed to estimate a continuous vector field from a finite discrete time series. We compare the performance of various methods on experimental and artificial time series and explore the connection between continuous (differential) and discrete (difference equation) systems. As part of this process we propose improvements to existing techniques. Our results demonstrate that the continuous-time dynamics of many noisy data sets can be simulated more accurately by modeling the one-step prediction map than by modeling the vector field. We also show that radial basis models provide superior results to global polynomial models.

Nonlinear multivariable modeling and analysis of sleep apnea time series

This paper investigates the modeling and analysis of physiological data recorded from a 49-year-old male and are composed of three time series: blood oxygen saturation, heart rate and respiration. In particular, it is desired to verify if the models estimated from data can distinguish between the dynamics underlying two different breathing patterns (normal breathing and apnea). The estimated models are nonlinear autoregressive, moving average with exogenous inputs (NARMAX) and the regressors used to compose such models are carefully chosen, among hundreds of candidates, by an automatic procedure. The results discussed in this paper suggest that the dynamics underlying the data are nonlinear and basically deterministic. Using estimated models it seems to be possible to quantify the stability of the fixed point in phase space reconstructed using the blood oxygen time series. This, as discussed, could be the basis of an algorithmic monitoring system.