Is fractional-order chaos theory the new tool to model chaotic pandemics as Covid-19?

The deadly outbreak of the second wave of Covid-19, especially in worst hit lower-middle-income countries like India, and the drastic rise of another growing epidemic of Mucormycosis, call for an efficient mathematical tool to model pandemics, analyse their course of outbreak and help in adopting quicker control strategies to converge to an infection-free equilibrium. This review paper on prominent pandemics reveals that their dispersion is chaotic in nature having long-range memory effects and features which the existing integer-order models fail to capture. This paper thus puts forward the use of fractional-order (FO) chaos theory that has memory capacity and hereditary properties, as a potential tool to model the pandemics with more accuracy and closeness to their real physical dynamics. We investigate eight FO models of Bombay plague, Cancer and Covid-19 pandemics through phase portraits, time series, Lyapunov exponents and bifurcation analysis. FO controllers (FOCs) on the concepts of fuzzy logic, adaptive sliding mode and active backstepping control are designed to stabilise chaos. Also, FOCs based on adaptive sliding mode and active backstepping synchronisation are designed to synchronise a chaotic epidemic with a non-chaotic one, to mitigate the unpredictability due to chaos during transmission. It is found that severity and complexity of the models increase as the memory fades, indicating that FO can be used as a crucial parameter to analyse the progression of a pandemic. To sum it up, this paper will help researchers to have an overview of using fractional calculus in modelling pandemics more precisely and also to approximate, choose, stabilise and synchronise the chaos control parameter that will eliminate the extreme sensitivity and irregularity of the models.

Collective behavior in a two-layer neuronal network with time-varying chemical connections that are controlled by a Petri net.

In this paper, we propose and study a two-layer network composed of a Petri net in the first layer and a ring of coupled Hindmarsh-Rose neurons in the second layer. Petri nets are appropriate platforms not only for describing sequential processes but also for modeling information circulation in complex systems. Networks of neurons, on the other hand, are commonly used to study synchronization and other forms of collective behavior. Thus, merging both frameworks into a single model promises fascinating new insights into neuronal collective behavior that is subject to changes in network connectivity. In our case, the Petri net in the first layer manages the existence of excitatory and inhibitory links among the neurons in the second layer, thereby making the chemical connections time-varying. We focus on the emergence of different types of collective behavior in the model, such as synchronization, chimeras, and solitary states, by considering different inhibitory and excitatory tokens in the Petri net. We find that the existence of only inhibitory or excitatory tokens disturbs the synchronization of electrically coupled neurons and leads toward chimera and solitary states.

The discrete fractional duffing system: Chaos, 0-1 test, C0 complexity, entropy, and control.

In this paper, we study the dynamics and control of a Caputo fractional difference form of the Duffing map. We use phase plots, bifurcation diagrams, and Lyapunov exponents to establish the existence of chaos over a wide range of fractional orders and examine the nature of the dynamics. Also, we present the 0-1 test to detect chaos and C0 complexity, which is an alternative nonlinear statistical measure that can quantify the regularity of a time series. In addition, we measure the approximate entropy to see the performance of our numerical results. Through phase plots and bifurcation diagrams, it is shown that the proposed fractional map exhibits a range of different dynamical behaviors including chaos and coexisting attractors. A one-dimensional feedback stabilization controller is proposed. The asymptotic convergence of the proposed controller is established by means of the stability theory of linear fractional order discrete-time systems. Simulation results have been carried out to illustrate the findings of the study.

Effect of the policy and consumption delay on the amplitude and length of business cycle.

In this paper, the amplitude and the length of the business cycle are investigated. It is the first time the length of the business cycle based on the Goodwin model (one classical business cycle model) is discussed. The effect of the time delay of the economic policy and consumption on the amplitude and the length of the business cycle is studied. Meanwhile, the memory property of making economic policy is also considered. The theoretical amplitude of the business cycle is obtained by multiple-scale methods. The transitions of the amplitude induced by memory property and time delay are analyzed. How the economic parameter and random excitation affect the length of the business cycle is proposed. Based on the results, we can find that the time delay of both economic policy and consumption can induce the transitions. Moreover, the memory property of economic policy will change the critical value of the parameters when the transitions occur. In one typical induced investment function, the length of the business cycle is determined only by the autonomous investment and consumption. However, the length of the business cycle is not mainly affected by the autonomous investment and consumption in some other typical induced investment function. This states that the type of induced investment function has a very important role in determining the length of the business cycle.

Prediction of bifurcations by varying critical parameters of COVID-19.

Coronavirus disease 2019 is a recent strong challenge for the world. In this paper, an epidemiology model is investigated as a model for the development of COVID-19. The propagation of COVID-19 through various sub-groups of society is studied. Some critical parameters, such as the background of mortality without considering the disease state and the speed of moving people from infected to resistance, affect the conditions of society. In this paper, early warning indicators are used to predict the bifurcation points in the system. In the interaction of various sub-groups of society, each sub-group can have various parameters. Six cases of the sub-groups interactions are studied. By coupling these sub-groups, various dynamics of the whole society are investigated.

Entropy Analysis and Neural Network-Based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors.

Today, four-dimensional chaotic systems are attracting considerable attention because of their special characteristics. This paper presents a non-equilibrium four-dimensional chaotic system with hidden attractors and investigates its dynamical behavior using a bifurcation diagram, as well as three well-known entropy measures, such as approximate entropy, sample entropy, and Fuzzy entropy. In order to stabilize the proposed chaotic system, an adaptive radial-basis function neural network (RBF-NN)-based control method is proposed to represent the model of the uncertain nonlinear dynamics of the system. The Lyapunov direct method-based stability analysis of the proposed approach guarantees that all of the closed-loop signals are semi-globally uniformly ultimately bounded. Also, adaptive learning laws are proposed to tune the weight coefficients of the RBF-NN. The proposed adaptive control approach requires neither the prior information about the uncertain dynamics nor the parameters value of the considered system. Results of simulation validate the performance of the proposed control method.

Chaotic Map with No Fixed Points: Entropy, Implementation and Control.

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents' diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map.

A Giga-Stable Oscillator with Hidden and Self-Excited Attractors: A Megastable Oscillator Forced by His Twin.

In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer-layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors' images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters.

On Chaos in the Fractional-Order Discrete-Time Unified System and Its Control Synchronization.

In this paper, we propose a fractional map based on the integer-order unified map. The chaotic behavior of the proposed map is analyzed by means of bifurcations plots, and experimental bounds are placed on the parameters and fractional order. Different control laws are proposed to force the states to zero asymptotically and to achieve the complete synchronization of a pair of fractional unified maps with identical or nonidentical parameters. Numerical results are used throughout the paper to illustrate the findings.

A New Chaotic System with Stable Equilibrium: Entropy Analysis, Parameter Estimation, and Circuit Design.

In this paper, we introduce a new, three-dimensional chaotic system with one stable equilibrium. This system is a multistable dynamic system in which the strange attractor is hidden. We investigate its dynamic properties through equilibrium analysis, a bifurcation diagram and Lyapunov exponents. Such multistable systems are important in engineering. We perform an entropy analysis, parameter estimation and circuit design using this new system to show its feasibility and ability to be used in engineering applications.

The Co-existence of Different Synchronization Types in Fractional-order Discrete-time Chaotic Systems with Non-identical Dimensions and Orders.

This paper is concerned with the co-existence of different synchronization types for fractional-order discrete-time chaotic systems with different dimensions. In particular, we show that through appropriate nonlinear control, projective synchronization (PS), full state hybrid projective synchronization (FSHPS), and generalized synchronization (GS) can be achieved simultaneously. A second nonlinear control scheme is developed whereby inverse full state hybrid projective synchronization (IFSHPS) and inverse generalized synchronization (IGS) are shown to co-exist. Numerical examples are presented to confirm the findings.

Fractional Form of a Chaotic Map without Fixed Points: Chaos, Entropy and Control.

In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method.

A memristive hyperchaotic system without equilibrium.

A new memristive system is presented in this paper. The peculiarity of the model is that it does not display any equilibria and exhibits periodic, chaotic, and also hyperchaotic dynamics in a particular range of the parameters space. The behavior of the proposed system is investigated through numerical simulations, such as phase portraits, Lyapunov exponents, and Poincaré sections, and circuital implementation confirmed the hyperchaotic dynamic.

Assessing sigmoidal function on memristive maps.

Memristors offer a crucial element for constructing discrete maps that have garnered significant attention in complex dynamics and various potential applications. In this study, we have integrated memristive and sigmoidal function to propose innovative mapping techniques. Our research confirms that the amalgamation of memristor and sigmoidal functions represents a promising approach for creating both 2D and 3D maps. Particularly noteworthy are the chaotic maps featuring multiple sigmoidal functions and multiple memristors, as highlighted in our findings. Specifically focusing on the novel STMM1 map, we delve into its dynamics and assess its feasibility. Intriguingly, the introduction of sigmoidal functions leads to alterations in the quantity of fixed points and the symmetry of the map.

Human papillomavirus type distribution in invasive cervical cancer and high-grade cervical intraepithelial neoplasia across 5 countries in Asia.

OBJECTIVE: Independent, prospective, multicenter, hospital-based cross-sectional studies were conducted across 5 countries in Asia, namely, Malaysia, Vietnam, Singapore, South Korea, and the Philippines. The objectives of these studies were to evaluate the prevalence of human papillomavirus (HPV) types (high risk and others including coinfections) in women with invasive cervical cancer (ICC) and high-grade precancerous lesions. METHODS: Women older than 21 years with a histologic diagnosis of ICC and cervical intraepithelial neoplasia [CIN 2 or 3 and adenocarcinoma in situ (AIS)] were enrolled. Cervical specimens were reviewed by histopathologists to confirm the presence of ICC or CIN 2/3/AIS lesion and tested with short PCR fragment 10-DNA enzyme immunoassay-line probe assay for 14 oncogenic HPV types and 11 non-oncogenic HPV types. The prevalence of HPV 16, HPV 18, and other high-risk HPV types in ICC [including squamous cell carcinoma (SCC) and adenocarcinoma/adenosquamous carcinoma (ADC/ASC)] and CIN 2/3/AIS was estimated. RESULTS: In the 5 Asian countries, diagnosis of ICC was confirmed in 500 women [SCC (n = 392) and ADC/ASC (n = 108)], and CIN 2/3/AIS, in 411 women. Human papillomavirus DNA was detected in 93.8% to 97.0% (84.5% for the Philippines) of confirmed ICC cases [94.0%-98.7% of SCC; 87.0%-94.3% (50.0% for the Philippines) of ADC/ASC] and in 93.7% to 100.0% of CIN 2/3/AIS. The most common types observed among ICC cases were HPV 16 (36.8%-61.3%), HPV 18 (12.9%-35.4%), HPV 52 (5.4%-10.3%), and HPV 45 (1.5%-17.2%), whereas among CIN 2/3/AIS cases, HPV 16 (29.7%-46.6%) was the most commonly observed type followed by HPV 52 (17.0%-66.7%) and HPV 58 (8.6%-16.0%). CONCLUSIONS: This article presents the data on the HPV prevalence, HPV type distribution, and their role in cervical carcinogenesis in 5 Asian countries. These data are of relevance to public health authorities for evaluating the existing and future cervical cancer prevention strategies including HPV-DNA testing-based screening and HPV vaccination in these Asian populations.

Robustness to noise in synchronization of network motifs: experimental results.

In this work, we experimentally investigate the robustness to noise of synchronization in all the four-nodes network motifs. The experimental setup consists of four Chua's circuits diffusively coupled in order to implement the six different undirected network motifs that can be obtained with four nodes. In this experimental setup, synchronization in the presence of noise injected in one of the network nodes is investigated and network motifs are compared in terms of the synchronization error obtained. The analysis has been then extended to some selected case studies of networks with five and six nodes. Numerical simulations have been also performed and results in agreement with experiments have been obtained. A correlation between node degree and robustness to noise has been found also in these networks.

Complex behavior of COVID-19's mathematical model.

It is almost more than a year that earth has faced a severe worldwide problem called COVID-19. In December 2019, the origin of the epidemic was found in China. After that, this contagious virus was reported almost all over the world with different variants. Besides all the healthcare system attempts, quarantine, and vaccination, it is needed to study the dynamical behavior of this disease specifically. One of the practical tools that may help scientists analyze the dynamical behavior of epidemic disease is mathematical models. Accordingly, here, a novel mathematical system is introduced. Also, the complex behavior of this model is investigated considering different dynamical analyses. The results represent that some range of parameters may lead the model to chaotic behavior. Moreover, comparing the two same bifurcation diagrams with different initial conditions reveals that the model has multi-stability.

Finite-time stabilization of a perturbed chaotic finance model.

Introduction: Robust, stable financial systems significantly improve the growth of an economic system. The stabilization of financial systems poses the following challenges. The state variables' trajectories (i) lie outside the basin of attraction, (ii) have high oscillations, and (iii) converge to the equilibrium state slowly. Objectives: This paper aims to design a controller that develops a robust, stable financial closed-loop system to address the challenges above by (i) attracting all state variables to the origin, (ii) reducing the oscillations, and (iii) increasing the gradient of the convergence. Methods: This paper proposes a detailed mathematical analysis of the steady-state stability, dissipative characteristics, the Lyapunov exponents, bifurcation phenomena, and Poincare maps of chaotic financial dynamic systems. The proposed controller does not cancel the nonlinear terms appearing in the closed-loop. This structure is robust to the smoothly varying system parameters and improves closed-loop efficiency. Further, the controller eradicates the effects of inevitable exogenous disturbances and accomplishes a faster, oscillation-free convergence of the perturbed state variables to the desired steady-state within a finite time. The Lyapunov stability analysis proves the closed-loop global stability. The paper also discusses finite-time stability analysis and describes the controller parameters' effects on the convergence rates. Computer-based simulations endorse the theoretical findings, and the comparative study highlights the benefits. Results: Theoretical analysis proofs and computer simulation results verify that the proposed controller compels the state trajectories, including trajectories outside the basin of attraction, to the origin within finite time without oscillations while being faster than the other controllers discussed in the comparative study section. Conclusions: This article proposes a novel robust, nonlinear finite-time controller for the robust stabilization of the chaotic finance model. It provides an in-depth analysis based on the Lyapunov stability theory and computer simulation results to verify the robust convergence of the state variables to the origin.

Chaos in fractional system with extreme events.

Understanding extreme events attracts scientists due to substantial impacts. In this work, we study the emergence of extreme events in a fractional system derived from a Liénard-type oscillator. The effect of fractional-order derivative on the extreme events has been investigated for both commensurate and incommensurate fractional orders. Especially, such a system displays multistability and coexistence of multiple extreme events.

Complex dynamics of a neuron model with discontinuous magnetic induction and exposed to external radiation.

The last two decades have seen many literatures on the mathematical and computational analysis of neuronal activities resulting in many mathematical models to describe neuron. Many of those models have described the membrane potential of a neuron in terms of the leakage current and the synaptic inputs. Only recently researchers have proposed a new neuron model based on the electromagnetic induction theorem, which considers inner magnetic fluctuation and external electromagnetic radiation as a significant missing part that can participate in neural activity. While the flux coupling of the membrane is considered equivalent to a memductance function of a memristor, standard memductance model of α + 3 ß Ï• 2 has been used in the literatures, but in this paper we propose a new memductance function based on discontinuous flux coupling. Various dynamical properties of the neuron model with discontinuous flux coupling are studied and interestingly the proposed model shows hyperchaotic behavior which was not identified in the literatures. Furthermore, we consider a ring network of the proposed model and investigate whether the chimera state can emerge. The chimera state relates to the state with simultaneously coherence and incoherence in oscillatory networks and has received much attention in recent years.