Polynomial search and global modeling: Two algorithms for modeling chaos

On the coupling of intracellular K + ${{\rm{K}}}^{+}$ to glycolytic oscillations in yeast

We have investigated the interplay between glycolytic oscillations and intracellularK + ${{\rm{K}}}^{+}$ concentration in the yeast Saccharomyces cerevisiae. IntracellularK + ${{\rm{K}}}^{+}$ concentration was measured using the fluorophore potassium-binding benzofuranisophthalate (PBFI). We found thatK + ${{\rm{K}}}^{+}$ is an essential ion for the occurrence of glycolytic oscillations and that intracellularK + ${{\rm{K}}}^{+}$ concentration oscillates synchronously with other variables such as nicotinamide adenine dinucleotide hydride (NADH), intracellular adenosine triphosphate (ATP), and mitochondrial membrane potential. We also investigated if glycolysis and intracellularK + ${{\rm{K}}}^{+}$ concentration oscillate in a number of yeast strains with mutations inK + ${{\rm{K}}}^{+}$ transporters in the plasma membrane, mitochondrial membrane and in the vacuolar membrane. Most of these strains are still capable of showing glycolytic oscillations, but two strains are not: (i) a strain with a deletion in the mitochondrial Mdm38pK + ∕ H + ${{\rm{K}}}^{+}\unicode{x02215}{{\rm{H}}}^{+}$ transporter and (ii) a strain with deletion of the late endosomal Nhx1pK + ∕ H + ${{\rm{K}}}^{+}\unicode{x02215}{{\rm{H}}}^{+}$ (Na + ∕ H + ${\text{Na}}^{+}\unicode{x02215}{{\rm{H}}}^{+}$ ) transporter. In these two mutant strains intracellularK + ${{\rm{K}}}^{+}$ concentration seems to be low, indicating that the two transporters may be involved in transport ofK + ${{\rm{K}}}^{+}$ into the cytosol. In the strain, Mdm38pΔ ${\rm{\Delta }}$ oscillations in glycolysis could be restored by addition of theK + ∕ H + ${{\rm{K}}}^{+}\unicode{x02215}{{\rm{H}}}^{+}$ exchange ionophore nigericin. Furthermore, in two nonoscillating mutant strains with a defective V-ATPase and deletion of the Arp1p protein the intracellularK + ${{\rm{K}}}^{+}$ is relatively high, suggesting that the V-ATPase is essential for transport ofK + ${{\rm{K}}}^{+}$ out of the cytosol and that the cytoskeleton may be involved in bindingK + ${{\rm{K}}}^{+}$ to reduce the concentration of free ion in the cytosol. Analyses of the time series of oscillations of NADH, ATP, mitochondrial membrane potential, and potassium concentration using data-driven modeling corroborate the conjecture thatK + ${{\rm{K}}}^{+}$ ion is essential for the emergence of oscillations and support the experimental findings using mutant strains.

Chaos theory in the understanding of COVID-19 pandemic dynamics

The chaos theory, a field of study in mathematics and physics, offers a unique lens through which to understand the dynamics of the COVID-19 pandemic. This theory, which deals with complex systems whose behavior is highly sensitive to initial conditions, can provide insights into the unpredictable and seemingly random nature of the pandemic's spread. In this review, we will discuss some literature data with the aim of showing how chaos theory could provide valuable perspectives in understanding the complex and dynamic nature of the COVID-19 pandemic. In particular, we will emphasize how the chaos theory can help in dissecting the unpredictable, non- linear progression of the disease, the importance of initial conditions, and the complex interactions between various factors influencing its spread. These insights are crucial for developing effective strategies to manage and mitigate the impact of the pandemic.

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.

Earthworm activity and its coupling to soil hydrology: A deterministic analysis

Considering in situ observations, chaos theory was taken as a basis to study the activity of anecic earthworms based on cast production from September 2016 to January 2018 in the Dong Cao watershed (Vietnam). To study this activity, the global modeling technique was used to obtain deterministic models of ordinary differential equations directly from observational time series. The obtained models show that the behavior of earthworms is chaotic; it is coupled to the dynamics of soil water content in a complex (integrative) way and can be interpreted as an habituation/sensitization process. However, this coupling is insufficient to explain the desynchronization of the cast production variations observed at different study sites. The retroaction of earthworm activity on soil is required to explain the spatiotemporal discrepancies.

Assessing observability of chaotic systems using Delay Differential Analysis

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.

Chaos theory applied to the outbreak of COVID-19: an ancillary approach to decision making in pandemic context

While predicting the course of an epidemic is difficult, predicting the course of a pandemic from an emerging virus is even more so. The validity of most predictive models relies on numerous parameters, involving biological and social characteristics often unknown or highly uncertain. Data of the COVID-19 epidemics in China, Japan, South Korea and Italy were used to build up deterministic models without strong assumptions. These models were then applied to other countries to identify the closest scenarios in order to foresee their coming behaviour. The models enabled to predict situations that were confirmed little by little, proving that these tools can be efficient and useful for decision making in a quickly evolving operational context.

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.

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.

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.

Recovering chaotic properties from small data

Physical properties are obviously essential to study a chaotic system that generates discrete-time signals, but recovering chaotic properties of a signal source from small data is a very troublesome work. Existing chaotic models are weak in dealing with such case in that most of them need big data to exploit those properties. In this paper, geometric theory is considered to solve this problem. We build a smooth trajectory from series to implicitly exhibit the chaotic properties with series-nonuniform rational B-spline (S-NURBS) modeling method, which is presented by our team to model slow-changing chaotic time series. As for the part of validation, we reveal how well our model recovers the properties from both the statistical and the chaotic aspects to confirm the effectiveness of the model. Finally a practical chaotic model is built up to recover the chaotic properties contained in the Musa standard dataset, which is used in analyzing software reliability, thereby further proves the high credibility of this model in practical time series. The effectiveness of the S-NURBS modeling leads us to believe that it is really a feasible and worthy research area to study chaotic systems from geometric perspective. For this reason, we reckon that we have opened up a new horizon for chaotic system research.

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.

Series-NonUniform Rational B-Spline (S-NURBS) model: a geometrical interpolation framework for chaotic data

Time series is widely exploited to study the innate character of the complex chaotic system. Existing chaotic models are weak in modeling accuracy because of adopting either error minimization strategy or an acceptable error to end the modeling process. Instead, interpolation can be very useful for solving differential equations with a small modeling error, but it is also very difficult to deal with arbitrary-dimensional series. In this paper, geometric theory is considered to reduce the modeling error, and a high-precision framework called Series-NonUniform Rational B-Spline (S-NURBS) model is developed to deal with arbitrary-dimensional series. The capability of the interpolation framework is proved in the validation part. Besides, we verify its reliability by interpolating Musa dataset. The main improvement of the proposed framework is that we are able to reduce the interpolation error by properly adjusting weights series step by step if more information is given. Meanwhile, these experiments also demonstrate that studying the physical system from a geometric perspective is feasible.