Reversible chemical reactions model with fractional difference operator: Dynamical analysis and synchronization.

Chemical reactions form the basis of life and understanding the different patterns and unpredictable changes in the reactions are noteworthy in real life situations. The article aims at constructing a mathematical model of two step reversible chemical reactions with a Caputo fractional difference operator. The reversible reaction involves the breakdown of an ester compound in the presence of water followed by the formation of fatty acid salts from the reaction of a free fatty acid with alkali hydroxide, such as NaOH. Using bifurcation diagrams, the chaotic response exhibited by the system is illustrated for state variables with identical fractional order and variables with non-identical fractional orders. The changes in periodic states of the system are investigated for each state variables with time varying plots and maximum Lyapunov exponents using the Jacobian matrix method are presented in support of the bifurcation diagrams. The synchronization of the subsystems of the proposed system is achieved with nonlinear control functions. Numerical simulations are presented to provide comparison of commensurate and incommensurate order models. Understanding the nature of these processes has significant applications in the production of bio-fuels from vegetable oils and animal fats by a transesterification reaction.

Greenhouse gas emissions: A rapid submerge of the world.

The investigation of worldwide climate change is a noticeable exploration topic in the field of sciences. Outflow of greenhouse gases in the environment is the main reason behind the worldwide environmental change. Greenhouse gases retain heat from the sun and prompt the earth to become more sultry, resulting in global warming. In this article, a model based technique is proposed to forecast the future climate dynamics globally. Using past data on annual greenhouse gas emissions and per capita greenhouse gas emissions, the fractal curves are generated and a forecast model called the autoregressive integrated moving average model has been employed to anticipate the future scenario in relation to climate change and its impact on sea-level rise. It is necessary to forecast the climate conditions before the situations become acute. Policy measures aimed at lowering CO and other greenhouse gas emissions, or at least slowing down their development, will have a substantial effect on future warming of the earth.

An exploration of fractal-based prognostic model and comparative analysis for second wave of COVID-19 diffusion.

The coronavirus disease 2019 (COVID-19) pandemic has fatalized 216 countries across the world and has claimed the lives of millions of people globally. Researches are being carried out worldwide by scientists to understand the nature of this catastrophic virus and find a potential vaccine for it. The most possible efforts have been taken to present this paper as a form of contribution to the understanding of this lethal virus in the first and second wave. This paper presents a unique technique for the methodical comparison of disastrous virus dissemination in two waves amid five most infested countries and the death rate of the virus in order to attain a clear view on the behaviour of the spread of the disease. For this study, the data set of the number of deaths per day and the number of infected cases per day of the most affected countries, the USA, Brazil, Russia, India, and the UK, have been considered in the first and second waves. The correlation fractal dimension has been estimated for the prescribed data sets of COVID-19, and the rate of death has been compared based on the correlation fractal dimension estimate curve. The statistical tool, analysis of variance, has also been used to support the performance of the proposed method. Further, the prediction of the daily death rate has been demonstrated through the autoregressive moving average model. In addition, this study also emphasis a feasible reconstruction of the death rate based on the fractal interpolation function. Subsequently, the normal probability plot is portrayed for the original data and the predicted data, derived through the fractal interpolation function to estimate the accuracy of the prediction. Finally, this paper neatly summarized with the comparison and prediction of epidemic curve of the first and second waves of COVID-19 pandemic to visualize the transmission rate in the both times.

Learning dynamical systems in noise using convolutional neural networks.

The problem of distinguishing deterministic chaos from non-chaotic dynamics has been an area of active research in time series analysis. Since noise contamination is unavoidable, it renders deterministic chaotic dynamics corrupted by noise to appear in close resemblance to stochastic dynamics. As a result, the problem of distinguishing noise-corrupted chaotic dynamics from randomness based on observations without access to the measurements of the state variables is difficult. We propose a new angle to tackle this problem by formulating it as a multi-class classification task. The task of classification involves allocating the observations/measurements to the unknown state variables in order to find the nature of these unobserved internal state variables. We employ signal and image processing based methods to characterize the different system dynamics. A deep learning technique using a state-of-the-art image classifier known as the Convolutional Neural Network (CNN) is designed to learn the dynamics. The time series are transformed into textured images of spectrogram and unthresholded recurrence plot (UTRP) for learning stochastic and deterministic chaotic dynamical systems in noise. We have designed a CNN that learns the dynamics of systems from the joint representation of the textured patterns from these images, thereby solving the problem as a pattern recognition task. The robustness and scalability of our approach is evaluated at different noise levels. Our approach demonstrates the advantage of applying the dynamical properties of chaotic systems in the form of joint representation of UTRP images along with spectrogram to improve learning dynamical systems in colored noise.

Fractional q-deformed chaotic maps: A weight function approach.

The fractional derivative holds long-time memory effects or non-locality. It successfully depicts the dynamical systems with long-range interactions. However, it becomes challenging to investigate chaos in the deformed fractional discrete-time systems. This study turns to fractional quantum calculus on the time scale and reports chaos in fractional q-deformed maps. The discrete memory kernels are used, and a weight function approach is proposed for fractional modeling. Rich q-deformed dynamics are demonstrated, which shows the methodology's efficiency.

Can hyperchaotic maps with high complexity produce multistability?

In this paper, we investigate the dynamical behavior in an M-dimensional nonlinear hyperchaotic model (M-NHM), where the occurrence of multistability can be observed. Four types of coexisting attractors including single limit cycle, cluster of limit cycles, single hyperchaotic attractor, and cluster of hyperchaotic attractors can be found, which are unusual behaviors in discrete chaotic systems. Furthermore, the coexistence of asymmetric and symmetric properties can be distinguished for a given set of parameters. In the endeavor of chaotification, this work introduces a simple controller on the M-NHM, which can add one more loop in each iteration, to overcome the chaos degradation in the multistability regions.

Complexity in congestive heart failure: A time-frequency approach.

Reconstruction of phase space is an effective method to quantify the dynamics of a signal or a time series. Various phase space reconstruction techniques have been investigated. However, there are some issues on the optimal reconstructions and the best possible choice of the reconstruction parameters. This research introduces the idea of gradient cross recurrence (GCR) and mean gradient cross recurrence density which shows that reconstructions in time frequency domain preserve more information about the dynamics than the optimal reconstructions in time domain. This analysis is further extended to ECG signals of normal and congestive heart failure patients. By using another newly introduced measure-gradient cross recurrence period density entropy, two classes of aforesaid ECG signals can be classified with a proper threshold. This analysis can be applied to quantifying and distinguishing biomedical and other nonlinear signals.

Analysis of rattleback chaotic oscillations.

Rattleback is a canoe-shaped object, already known from ancient times, exhibiting a nontrivial rotational behaviour. Although its shape looks symmetric, its kinematic behaviour seems to be asymmetric. When spun in one direction it normally rotates, but when it is spun in the other direction it stops rotating and oscillates until it finally starts rotating in the other direction. It has already been reported that those oscillations demonstrate chaotic characteristics. In this paper, rattleback's chaotic dynamics are studied by applying Kane's model for different sets of (experimentally decided) parameters, which correspond to three different experimental prototypes made of wax, gypsum, and lead-solder. The emerging chaotic behaviour in all three cases has been studied and evaluated by the related time-series analysis and the calculation of the strange attractors' invariant parameters.

Generalized variable projective synchronization of time delayed systems.

We study generalized variable projective synchronization between two unified time delayed systems with constant and modulated time delays. A novel Krasovskii-Lyapunov functional is constructed and a generalized sufficient condition for synchronization is derived analytically using the Lyapunov stability theory and adaptive techniques. The proposed scheme is valid for a system of n-numbers of first order delay differential equations. Finally, a new neural oscillator is considered as a numerical example to show the effectiveness of the proposed scheme.

Framework of fractals in data analysis: theory and interpretation.

This special issue is a compilation of pioneering research articles that explore the robustness of fractal theories to address and analyse the complexity of real-time data under the topic "Framework of Fractals in Data Analysis: Theory and Interpretation".

Chaos and multi-layer attractors in asymmetric neural networks coupled with discrete fractional memristor.

This article introduces a novel model of asymmetric neural networks combined with fractional difference memristors, which has both theoretical and practical implications in the rapidly evolving field of computational intelligence. The proposed model includes two types of fractional difference memristor elements: one with hyperbolic tangent memductance and the other with periodic memductance and memristor state described by sine functions. The authenticity of the constructed memristor is confirmed through fingerprint verification. The research extensively investigates the dynamics of a coupled neural network model, analyzing its stability at equilibrium states, studying bifurcation diagrams, and calculating the largest Lyapunov exponents. The results suggest that when incorporating sine memristors, the model demonstrates coexisting state variables depending on the initial conditions, revealing the emergence of multi-layer attractors. The article further demonstrates how the memristor state shifts through numerical simulations with varying memductance values. Notably, the study emphasizes the crucial role of memductance (synaptic weight) in determining the complex dynamical characteristics of neural network systems. To support the analytical results and demonstrate the chaotic response of state variables, the article includes appropriate numerical simulations. These simulations effectively validate the presented findings and provide concrete evidence of the system's chaotic behavior.

Color-Gray Multi-Image Hybrid Compression-Encryption Scheme Based on BP Neural Network and Knight Tour.

In the research of multi-image encryption (MIE), the image type and size are important factors that limit the algorithm design. For this reason, the multi-image (MI) hybrid encryption algorithm that can flexibly encrypt color images and grayscale images of various sizes is proposed. Based on this, combining the back propagation (BP) neural network compression technology and the MI hybrid encryption algorithm, an MI hybrid compression-encryption (MIHCE) scheme can be obtained to reduce the pressure of simultaneous transmission and storage of multiple cipher images. Besides, two chaotic maps are used in the scheme design process. By plotting the phase diagrams under different parameter conditions, the rich variation of the behavior of the chaotic maps in the phase space is exhibited. The MIHCE scheme based on the chaotic maps consists of three parts: 1) compressing the MI cube by using the BP neural network; 2) scrambling the compressed MI cube based on the knight tour problem and chaotic sequences; and 3) diffusing the scrambled MI cube. After the MIHCE is completed, the obtained cipher images are stored and transmitted. Subsequently, the security analysis and compression performance analysis prove the feasibility and safety of the designed compression-encryption scheme.

Comment on "Generalized projective synchronization in time-delayed systems: nonlinear observer approach" [Chaos 19, 013102 (2009); 20, 029902 (2010)].

In Chaos 19, 013102 (2009), the author proposed generalized projective synchronization for time delay systems using nonlinear observer and obtained sufficient condition to ensure projective synchronization for modulated time varying delay. There are concerns with the obtained conditions as the result was applicable only to trivial case of time varying delay τ[over dot](1)(t)=dτ(1)(t)/dt<1. In this paper, we note the drawbacks of the proposed sufficient condition. The new improved sufficient condition for ensuring the projective synchronization of time varying delayed systems is presented. The proposed new criteria have been verified by adopting the Ikeda system.

Dynamics of the COVID-19 pandemic: nonlinear approaches on the modelling, prediction and control.

This special issue contains 35 regular articles on the analysis and dynamics of COVID-19 with several applications. Some analyses are on the construction of mathematical models representing the dynamics of COVID-19, and some are on the estimations and predictions of the disease, a few with possible applications. The various contributions report important, timely, and promising results, such as the effects of several waves, deep learning-based COVID-19 classifications, and multivariate time series with applications.

Omicron: a mysterious variant of concern.

Omicron (B.1.1.529), a highly mutated SARS-CoV-2 variant, has emerged in the south of African continent in the November 2021. The spike protein of Omicron has 26 amino acid mutations, which makes it distinct from the other variants of concern. Researches are underway to know the virulence and transmission rate of Omicron variant. In this letter, the seven-day moving average of most affected Omicron variant countries Denmark, Germany, India, Netherlands, South Africa and UK has been investigated and compared with each other. Further, the seven-day average of daily positive Omicron cases of the prescribed countries has been predicted for the months of December 2021, January 2022 and February 2022 using the fractal interpolation method. Results elucidate that the curve of daily positive case follows the same pattern even though the new variant of concern, Omicron added in the existing variants.

Frontiers of fractals for complex systems: recent advances and future challenges.

This special issue of the European Physical Journal Special Topics titled "Frontiers of Fractals for Complex Systems: Recent Advances and Future Challenges"  is a collection of cutting-edge research proposing the application of fractal features to the dynamics of highly nonlinear complex systems.

Comment on "Homoclinic chaos in strongly dissipative strongly coupled complex dusty plasmas".

In a recent paper [S. Ghosh, Phys. Rev. E 103, 023205 (2021)2470-004510.1103/PhysRevE.103.023205], the transport properties of the two-dimensional weakly nonlinear quasilongitudinal dust lattice mode were investigated in a highly viscous, strongly coupled, and weakly ionized plasma. Based on the computational results, the author predicted strong viscosity induced Shilnikov homoclinic chaos. In this Comment, it is shown that the dynamical system presented in the paper is inconsistent and incorrect because of an error in the expression of ε. As a result, period-2, period-4, chaotic and homoclinic chaotic phenomena do not exist, which were observed in the paper.

The second and third waves in India: when will the pandemic be culminated?

An unprecedented upsurge of COVID-19-positive cases and deaths is currently being witnessed across India. According to WHO, India reported an average of 3.9 lakhs of new cases during the first week of May 2021 which equals 47% of new cases reported globally and 276 daily cases per million population. In this letter, the concept of SIR and fractal interpolation models is applied to predict the number of positive cases in India by approximating the epidemic curve, where the epidemic curve denotes the two-dimensional graphical representation of COVID-19-positive cases in which the abscissa denotes the time, while the ordinate provides the number of positive cases. In order to estimate the epidemic curve, the fractal interpolation method is implemented on the prescribed data set. In particular, the vertical scaling factors of the fractal function are selected from the SIR model. The proposed fractal and SIR model can also be explored for the assessment and modeling of other epidemics to predict the transmission rate. This letter investigates the duration of the second and third waves in India, since the positive cases and death cases of COVID-19 in India have been highly increasing for the past few weeks, and India is in a midst of a catastrophizing second wave. The nation is recording more than 120 million cases of COVID-19, but pandemics are still concentrated in most states. In order to predict the forthcoming trend of the outbreaks, this study implements the SIR and fractal models on daily positive cases of COVID-19 in India and its provinces, namely Delhi, Karnataka, Tamil Nadu, Kerala and Maharashtra.

A new image encryption scheme based on fractional-order hyperchaotic system and multiple image fusion.

A multi-image encryption scheme based on the fractional-order hyperchaotic system is designed in this paper. The chaotic characteristics of this system are analyzed by the phase diagram, Lyapunov exponent and bifurcation diagram. According to the analyses results, an interesting image encryption algorithm is proposed. Multiple grayscale images are fused into a color image using different channels. Then, the color image is scrambled and diffused in order to obtain a more secure cipher image. The pixel confusion operation and diffusion operation are assisted by fractional hyperchaotic system. Experimental simulation and test results indicate that the devised multi-image encryption scheme can effectively encrypt multiple images, which increase the efficiency of image encryption and transmission, and have good security performance.

Can India develop herd immunity against COVID-19?

World Health Organization declared the novel coronavirus disease 2019 (COVID-19) outbreak to be a public health crisis of international concern. Further, it provided advice to the global community that countries should place strong measures to detect disease early, isolate and treat cases, trace contacts and promote "social distancing" measures commensurate with the risk. This study analyses the COVID-19 infection data from the top 15 affected countries in which we observed heterogeneous growth patterns of the virus. Hence, this paper applies multifractal formalism on COVID-19 data with the notion that country-specific infection rates follow a power law growth behaviour. According to the estimated generalized fractal dimension curves, the effects of drastic containment measures on the pandemic in India indicate that a significant reduction of the infection rate as its population is concern. Also, comparison results with other countries demonstrate that India has less death rate or more immunity against COVID-19.