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.

Optimal placement of sensor and actuator for controlling low-dimensional chaotic systems based on global modeling.

Controlling chaos is fundamental in many applications, and for this reason, many techniques have been proposed to address this problem. Here, we propose a strategy based on an optimal placement of the sensor and actuator providing global observability of the state space and global controllability to any desired state. The first of these two conditions enables the derivation of a model of the system by using a global modeling technique. In turn, this permits the use of feedback linearization for designing the control law based on the equations of the obtained model and providing a zero-flat system. The procedure is applied to three case studies, including two piecewise linear circuits, namely, the Carroll circuit and the Chua circuit whose governing equations are approximated by a continuous global model. The sensitivity of the procedure to the time constant of the dynamics is also discussed.

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.

Topological characterization of toroidal chaos: A branched manifold for the Deng toroidal attractor.

When a chaotic attractor is produced by a three-dimensional strongly dissipative system, its ultimate characterization is reached when a branched manifold-a template-can be used to describe the relative organization of the unstable periodic orbits around which it is structured. If topological characterization was completed for many chaotic attractors, the case of toroidal chaos-a chaotic regime based on a toroidal structure-is still challenging. We here investigate the topology of toroidal chaos, first by using an inductive approach, starting from the branched manifold for the Rössler attractor. The driven van der Pol system-in Robert Shaw's form-is used as a realization of that branched manifold. Then, using a deductive approach, the branched manifold for the chaotic attractor produced by the Deng toroidal system is extracted from data.

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.

Observability of laminar bidimensional fluid flows seen as autonomous chaotic systems.

Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Bénard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed.

Using global modeling to unveil hidden couplings in small network motifs.

One of the main tasks in network theory is to infer relations among interacting elements. We propose global modeling as a tool to detect links between nodes and their nature. Various situations using small network motifs are investigated under the assumption that the variable to be measured at each node provides full observability when isolated. Such a choice ensures no intrinsic difficulties for getting a global model in the coupled situation. As a first step toward unveiling the coupling function in larger network motifs, we consider three different scenarios involving Rössler systems diffusively coupled, in a couple or embedded in a network, or parametrically forced. We show that the global modeling is able to determine not only the existence of an interaction but also its functional form, to retrieve the dynamics of the whole system, and to extract the equations governing the single node dynamics as if it was isolated.

Topological characterization versus synchronization for assessing (or not) dynamical equivalence.

Model validation from experimental data is an important and not trivial topic which is too often reduced to a simple visual inspection of the state portrait spanned by the variables of the system. Synchronization was suggested as a possible technique for model validation. By means of a topological analysis, we revisited this concept with the help of an abstract chemical reaction system and data from two electrodissolution experiments conducted by Jack Hudson's group. The fact that it was possible to synchronize topologically different global models led us to conclude that synchronization is not a recommendable technique for model validation. A short historical preamble evokes Jack Hudson's early career in interaction with Otto E. Rössler.

How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment?

Although recent advances in oncology emphasized the role of microenvironment in tumor growth, the role of delays for modeling tumor growth is still uncertain. In this paper, we considered a model, describing the interactions of tumor cells with their microenvironment made of immune cells and host cells, in which we inserted, as suggested by the clinicians, two time delays, one in the interactions between tumor cells and immune cells and, one in the action of immune cells on tumor cells. We showed analytically that the singular point associated with the co-existence of the three cell populations loses its stability via a Hopf bifurcation. We analytically calculated a range of the delays over which tumor cells are inhibited by immune cells and over which a period-1 limit cycle induced by this Hopf bifurcation is observed. By using a global modeling technique, we investigated how the dynamics observed with two delays can be reproduced by a similar model without delays. The effects of these two delays were thus interpreted in terms of interactions between the cell populations.

A chaotic model for the epidemic of Ebola virus disease in West Africa (2013-2016).

An epidemic of Ebola Virus Disease (EVD) broke out in Guinea in December 2013. It was only identified in March 2014 while it had already spread out in Liberia and Sierra Leone. The spill over of the disease became uncontrollable and the epidemic could not be stopped before 2016. The time evolution of this epidemic is revisited here with the global modeling technique which was designed to obtain the deterministic models from single time series. A generalized formulation of this technique for multivariate time series is introduced. It is applied to the epidemic of EVD in West Africa focusing on the period between March 2014 and January 2015, that is, before any detected signs of weakening. Data gathered by the World Health Organization, based on the official publications of the Ministries of Health of the three main countries involved in this epidemic, are considered in our analysis. Two observed time series are used: the daily numbers of infections and deaths. A four-dimensional model producing a very complex dynamical behavior is obtained. The model is tested in order to investigate its skills and drawbacks. Our global analysis clearly helps to distinguish three main stages during the epidemic. A characterization of the obtained attractor is also performed. In particular, the topology of the chaotic attractor is analyzed and a skeleton is obtained for its structure.

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.