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Organisms may respond to local stimuli that benefit or threaten their fitness. The adaptive movement behaviour may allow individuals to adjust their speed to maximise the chances of being in comfort zones, where death risk is minimal. We investigate spatial cyclic models where the rock-paper-scissors game rules describe the nonhierarchical dominance. We assume that organisms of one out of the species can control the mobility rate in response to the information obtained from scanning the environment. Running a series of stochastic simulations, we quantify the effects of the movement strategy on the spatial patterns and population dynamics. Our findings show that the ability to change mobility to adapt to environmental clues is not reflected in an advantage in cyclic spatial games. The adaptive movement provokes a delay in the spatial domains occupied by the species in the spiral waves, making the group more vulnerable to the advance of the dominant species and less efficient in taking territory from the dominated species. Our outcomes also show that the effects of adaptive movement behaviour accentuate whether most individuals have a long-range neighbourhood perception. Our results may be helpful for biologists and data scientists to comprehend the dynamics of ecosystems where adaptive processes are fundamental., Comment: 8 pages, 7 figures

In the present paper, discussion of the canonical quantization of a weakly nonideal Bose gas at zero temperature within the framework of the Bogolyubov approach is continued. Contrary to the previous paper on this subject, here the two-body interaction potential is considered in the general form. It is shown that in such a case consideration of the first nonlinear correction also leads to the automatic particle number conservation without any additional assumptions or modification of the resulting effective Hamiltonian., 30 pages. Supplementary material: the standard Maxima .wxmx file is automatically unpacked by the arXiv upload software, so the .xml file, which can also be processed by Maxima, is uploaded instead. v2: major revision of the text, added references, minor corrections in ancillary files. v3: typos corrected

In this paper, we firstly extend the Fourier neural operator (FNO) to discovery the soliton mapping between two function spaces, where one is the fractional-order index space $\{\epsilon|\epsilon\in (0, 1)\}$ in the fractional integrable nonlinear wave equations while another denotes the solitonic solution function space. To be specific, the fractional nonlinear Schr\"{o}dinger (fNLS), fractional Korteweg-de Vries (fKdV), fractional modified Korteweg-de Vries (fmKdV) and fractional sine-Gordon (fsineG) equations proposed recently are studied in this paper. We present the train and evaluate progress by recording the train and test loss. To illustrate the accuracies, the data-driven solitons are also compared to the exact solutions. Moreover, we consider the influences of several critical factors (e.g., activation functions containing Relu$(x)$, Sigmoid$(x)$, Swish$(x)$ and $x\tanh(x)$, depths of fully connected layer) on the performance of the FNO algorithm. We also use a new activation function, namely, $x\tanh(x)$, which is not used in the field of deep learning. The results obtained in this paper may be useful to further understand the neural networks in the fractional integrable nonlinear wave systems and the mappings between two spaces., Comment: 17 pages, 20 figures

Highlights • A compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention policies. • Short-term projection results have been illustrated for Bangladesh and India. • All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. • Optimal control problem for fractional system has been designed and the existence of unique solution has also been showed by using Picard-Lindelof technique., When the entire world is eagerly waiting for a safe, effective and widely available COVID-19 vaccine, unprecedented spikes of new cases are evident in numerous countries. To gain a deeper understanding about the future dynamics of COVID-19, a compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention strategies. Model parameters have been calibrated using sophisticated trust-region-reflective algorithm and short-term projection results have been illustrated for Bangladesh and India. Control reproduction numbers (Rc) have been calculated in order to get insights about the current epidemic scenario in the above-mentioned countries. Forecasting results depict that the aforesaid countries are having downward trends in daily COVID-19 cases. Nevertheless, as the pandemic is not over in any country, it is highly recommended to use efficacious face coverings and maintain strict physical distancing in public gatherings. All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. In addition, optimal control strategies for fractional system have been designed and the existence of unique solution has also been showed using Picard-Lindelof technique. Finally, unconditional stability of the fractional numerical technique has been proved.

A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models., This is a preprint of a paper whose final and definite form is published by 'Chaos Solitons Fractals' (ISSN: 0960-0779). Paper Submitted 04/June/2020; Revised 22/July/2020 and 02/Aug/2020; Accepted 04/Jan/2021

Most of the widely populated countries across the globe have been observing vicious spread and detrimental effects of pandemic COVID-19 since its inception on December 19. Therefore to restrict the spreading of pandemic COVID-19, various researches are going on in both medical and administrative sectors. The focus has been given in this research keeping an administrative point of view in mind. In this paper a dynamic model of infected population due to spreading of pandemic COVID-19 considering both intra and inter zone mobilization factors with rate of detection has been proposed. Few factors related to intra zone mobilization; inter zone mobilization and rate of detection are the key points in the proposed model. Various remedial steps are taken into consideration in the form of operating procedures. Further such operating procedures are applied over the model in standalone or hybridized mode and responses are reported in this paper in a case-studies manner. Further zone-wise increase in infected population due to the spreading of pandemic COVID-19 has been studied and reported in this paper. Also the proposed model has been applied over the real world data considering three states of India and the predicted responses are compared with real data and reported with bar chart representation in this paper.

In this paper, we are working on a pandemic of novel coronavirus (COVID-19). COVID-19 is an infectious disease, it creates severe damage in the lungs. COVID-19 causes illness in humans and has killed many people in the entire world. However, this virus is reported as a pandemic by the World Health Organization (WHO) and all countries are trying to control and lockdown all places. The main objective of this work is to solve the five different tasks such as I) Predicting the spread of coronavirus across regions. II) Analyzing the growth rates and the types of mitigation across countries. III) Predicting how the epidemic will end. IV) Analyzing the transmission rate of the virus. V) Correlating the coronavirus and weather conditions. The advantage of doing these tasks to minimize the virus spread by various mitigation, how well the mitigations are working, how many cases have been prevented by this mitigations, an idea about the number of patients that will recover from the infection with old medication, understand how much time will it take to for this pandemic to end, we will be able to understand and analyze how fast or slow the virus is spreading among regions and the infected patient to reduce the spread based clear understanding of the correlation between the spread and weather conditions. In this paper, we propose a novel Support Vector Regression method to analysis five different tasks related to novel coronavirus. In this work, instead of simple regression line we use the supported vectors also to get better classification accuracy. Our approach is evaluated and compared with other well-known regression models on standard available datasets. The promising results demonstrate its superiority in both efficiency and accuracy.

Highlights • Reviewed on various information related to application of Machine Learning and Artificial Intelligent for tackling Covid-19 pandemic. • Challenges on computational system with real world problem discusses suggestions conveying researchers on model design, medical experts, and policymakers for tackling the Covid-19 pandemic and ahead. • Targeting on treatment, medication, screening, prediction, forecasting, contact tracing, and drug/vaccination process for the Covid-19 pandemic., Background and objective During the recent global urgency, scientists, clinicians, and healthcare experts around the globe keep on searching for a new technology to support in tackling the Covid-19 pandemic. The evidence of Machine Learning (ML) and Artificial Intelligence (AI) application on the previous epidemic encourage researchers by giving a new angle to fight against the novel Coronavirus outbreak. This paper aims to comprehensively review the role of AI and ML as one significant method in the arena of screening, predicting, forecasting, contact tracing, and drug development for SARS-CoV-2 and its related epidemic. Method A selective assessment of information on the research article was executed on the databases related to the application of ML and AI technology on Covid-19. Rapid and critical analysis of the three crucial parameters, i.e., abstract, methodology, and the conclusion was done to relate to the model's possibilities for tackling the SARS-CoV-2 epidemic. Result This paper addresses on recent studies that apply ML and AI technology towards augmenting the researchers on multiple angles. It also addresses a few errors and challenges while using such algorithms in real-world problems. The paper also discusses suggestions conveying researchers on model design, medical experts, and policymakers in the current situation while tackling the Covid-19 pandemic and ahead. Conclusion The ongoing development in AI and ML has significantly improved treatment, medication, screening, prediction, forecasting, contact tracing, and drug/vaccine development process for the Covid-19 pandemic and reduce the human intervention in medical practice. However, most of the models are not deployed enough to show their real-world operation, but they are still up to the mark to tackle the SARS-CoV-2 epidemic.

Chaotic scattering in three dimensions has not received as much attention as in two dimensions so far. In this paper, we deal with a three-dimensional open Hamiltonian system whose Wada basin boundaries become non Wada when the critical energy value is surpassed in the absence of dissipation. In particular, we study here the dissipation effects on this topological change, which has no analogy in two dimensions. Hence, we find that non-Wada basins, expected in the absence of dissipation, transform themselves into partially Wada basins when a weak dissipation reduces the system energy below the critical energy. We provide numerical evidence of the emergence of the Wada points on the basin boundaries under weak dissipation. According to the paper findings, Wada basins are typically driven, enhanced and, consequently, structurally stable under weak dissipation in three-dimensional open Hamiltonian systems., Comment: 18 pages, 9 figures

The paper concerns envelope soliton propagating through the nonlinear transverse dust grain displacement in dusty plasma crystals. The work found its motivation from the recent paper of the authors [7]. To reach the goal of this study we establish the linear dispersion and the nonlinear Schrödinger equation (NLSE) by employing the quasi discrete approximation [7,13–15]. Contrary to ref. [7], it is observed one additional regime of the formation of the dark soliton while choosing the gap frequency value (ωg≈150sec−) which is close to the experimental value given by Motcheyo et al. [7], Tsopgue et al. To corroborate the prediction made analytically to group velocity, was confirmed by numerical and experimental studies. As, it is established that in general manner the modulated waves describing by the NLSE can be stable or unstable to the perturbation, the modulation instability of steady state have been done. Despite, that the bandwidth of the model has been reduced compared to [7,13–15], we can say without hesitation that this work will be a capital contribution in dusty plasma crystal.

We describe in this paper an analysis of the spatial evolution of coronavirus pandemic around the world by using a particular type of unsupervised neural network, which is called self-organizing maps. Based on the clustering abilities of self-organizing maps we are able to spatially group together countries that are similar according to their coronavirus cases, in this way being able to analyze which countries are behaving similarly and thus can benefit by using similar strategies in dealing with the spread of the virus. Publicly available datasets of coronavirus cases around the globe from the last months have been used in the analysis. Interesting conclusions have been obtained, that could be helpful in deciding the best strategies in dealing with this virus. Most of the previous papers dealing with data of the Coronavirus have viewed the problem on temporal aspect, which is also important, but this is mainly concerned with the forecast of the numeric information. However, we believe that the spatial aspect is also important, so in this view the main contribution of this paper is the use of unsupervised self-organizing maps for grouping together similar countries in their fight against the Coronavirus pandemic, and thus proposing that strategies for similar countries could be established accordingly.

Highlights • New COVID-19 mathematical model with lock-down effect. • New fractal-fractional differentiation. • New fractal-fractional integration. • Numerical scheme based on Newton polynomial., Countries around the world are implementing lock-down measures in a bid to flatten the curve of the new deadly COVID-19 disease. Our paper does not claim to have found the cure for COVID-19, neither does it claim that the suggested model have taken into account all the complexities around the spread of the disease. Nonetheless, the fundamental question asked in this paper is to know if within the conditions taken into account in this suggested model, the integral lock-down is effective in saving human lives. To answer this question, a mathematical model was suggested taking into account the possibility of transmission of COVID-19 from dead bodies to humans and the effect of lock-down. Three cases were considered. The first case suggested that there is transmission from dead to the living (medical staffs as they perform postmortem procedures on corpses, and direct contacts with during burial ceremonies). This case has no equilibrium points except for disease free equilibrium, a clear indication that care must be taken when dealing with corpses due to corona-19. In the second case we removed the transmission rate from dead bodies. This case showed an equilibrium point, although the number of deaths, carriers and infected grew exponentially up to a certain stability level. In the last case, we incorporated a lock-down and social distancing effect, using the next generation matrix. We could achieve a zero reproduction number, with number of deaths, infected and carriers decaying very rapidly. This is a clear indication that if lock-down recommendations are observed the threat of COVID-19 can be reduced to zero in few months.While our mathematical model agrees with the effectiveness of the lock-down, it is important to mention damaging effects of inadequate testing. The long waiting period of few days before confirmation of status, can only lead to more infections. The asymptomatic tested person could be positive and spread the infection, or could contact the virus in days after testing and will spread the disease further, after being given a false result. Testing kit that with immediate results are needed for more efficient measures. We used Italy’s Data to guide the construction of the mathematical model. To include non-locality into mathematical formulas, differential and integral operators were suggested. Properties and numerical approximations were presented in details. Finally, the suggested differential and integral operators were applied to the model.

In the previous paper [Yamada, Chaos, Solitons $\&$ Fractals, {\bf 109},99(2018)], we investigated localization properties of one-dimensional disordered electronic system with long-range correlation generated by modified Bernoulli (MB) map. In the present paper, we report localization properties of phonon in disordered harmonic chains generated by the MB map. Here we show that Lyapunov exponent becomes positive definite for almost all frequencies $\omega$ except $\omega=0$, and the $B-$dependence changes to exponential decrease for $B > 2 $, where $B$ is a correlation parameter of the MB map. The distribution of the Lyapunov exponent of the phonon amplitude has a slow convergence, different from that of uncorrelated disordered systems obeying a normal central-limit theorem. Moreover, we calculate the phonon dynamics in the MB chains. We show that the time-dependence of spread in the phonon amplitude and energy wave packet changes from that in the disordered chain to that in the periodic one, as the correlation parameter $B$ increases., Comment: 10 pages, 11 figures

This paper investigates the drive-response finite-time anti-synchronization for memristive bidirectional associative memory neural networks (MBAMNNs). Firstly, a class of MBAMNNs with mixed probabilistic time-varying delays and stochastic perturbations is first formulated and analyzed in this paper. Secondly, an nonlinear control law is constructed and utilized to guarantee drive-response finite-time anti-synchronization of the neural networks. Thirdly, by employing some inequality technique and constructing an appropriate Lyapunov function, some anti-synchronization criteria are derived. Finally, a number simulation is provided to demonstrate the effectiveness of the proposed mechanism.

It has been shown that natural interval extensions (NIE) can be used to calculate the largest positive Lyapunov exponent (LLE). However, the elaboration of NIE are not always possible for some dynamical systems, such as those modelled by simple equations or by Simulink-type blocks. In this paper, we use rounding mode of floating-point numbers to compute the LLE. We have exhibited how to produce two pseudo-orbits by means of different rounding modes; these pseudo-orbits are used to calculate the Lower Bound Error (LBE). The LLE is the slope of the line gotten from the logarithm of the LBE, which is estimated by means of a recursive least square algorithm (RLS). The main contribution of this paper is to develop a procedure to compute the LLE based on the LBE without using the NIE. Additionally, with the aid of RLS the number of required points has been decreased. Eight numerical examples are given to show the effectiveness of the proposed technique.

In this paper, we develop a joint radar-communication waveform based on the unstable basis function. Basis functions of a chaotic waveform are uncertain, vary continuously and hence the use of a traditional matched filter is not practical. Corron et al. [1,2] demonstrated a chaotic circuit for which an analytic matched filter could be constructed, solving the signal detection problem. The Corron circuit produced a narrow band signal, which is not optimal for spread spectrum communications and radar applications. In this paper, we adapt Corron’s concept of a chaotic signal composed of a linear combination of basis functions and show how the basis functions are used to design joint radar-communications systems that still have simple analytic matched filters. The form of the matched filter makes it useful for both the radar and communications systems where it is necessary to minimize weight or power consumption.

Highlights • In the presence of numerous research articles, extracting best-suited articles is timeconsuming and manually impractical. The objective of this paper is to extract the activity and trends of coronavirus related research articles using machine learning approaches to help the research community for future exploration concerning COVID-19 prevention and treatment techniques. • The COVID-19 open research dataset (CORD-19) is used for experiments, whereas several target-tasks along with explanations are defined for classification, based on domain knowledge. • Clustering techniques are used to create the different clusters of available articles, and later the task assignment is performed using parallel one-class support vector machines (OCSVMs). These defined tasks describes the behaviour of clusters to accomplish targetclass guided mining. • Experiments with original and reduced features validate the performance of the approach. It is evident that the k-means clustering algorithm, followed by parallel OCSVMs, outperforms other methods for both original and reduced feature space., The novel coronavirus disease 2019 (COVID-19) began as an outbreak from epicentre Wuhan, People’s Republic of China in late December 2019, and till June 27, 2020 it caused 9,904,906 infections and 496,866 deaths worldwide. The world health organization (WHO) already declared this disease a pandemic. Researchers from various domains are putting their efforts to curb the spread of coronavirus via means of medical treatment and data analytics. In recent years, several research articles have been published in the field of coronavirus caused diseases like severe acute respiratory syndrome (SARS), middle east respiratory syndrome (MERS) and COVID-19. In the presence of numerous research articles, extracting best-suited articles is time-consuming and manually impractical. The objective of this paper is to extract the activity and trends of coronavirus related research articles using machine learning approaches to help the research community for future exploration concerning COVID-19 prevention and treatment techniques. The COVID-19 open research dataset (CORD-19) is used for experiments, whereas several target-tasks along with explanations are defined for classification, based on domain knowledge. Clustering techniques are used to create the different clusters of available articles, and later the task assignment is performed using parallel one-class support vector machines (OCSVMs). These defined tasks describes the behaviour of clusters to accomplish target-class guided mining. Experiments with original and reduced features validate the performance of the approach. It is evident that the k-means clustering algorithm, followed by parallel OCSVMs, outperforms other methods for both original and reduced feature space.

Highlights • A stochastic staged progression AIDS model with staged treatment and second-order perturbation is studied. • The existence and uniqueness of the ergodic stationary distribution is obtained under the condition of. • If, we obtain that the AIDS epidemic will go to extinction in long-term., Focusing on deterministic AIDS model proposed by Hyman (2000) and the detailed data from the World Health Organization (WHO), there are three stages of AIDS process which are described as Acute infection period, Asymptomatic phase and AIDS stage. Our paper is therefore concerned with a stochastic staged progression AIDS model with staged treatment. In view of the complexity of random disturbances, we reasonably take second-order perturbation into consideration for realistic sense. By means of our creative transformation technique and stochastic Lyapunov method, a critical value R0H>1 is firstly obtained for the existence and uniqueness of ergodic stationary distribution to the stochastic system. Not only does it respectively reveal the corresponding dynamical effects of the linear and second-order perturbations to the model, but the unified form of second-order and linear fluctuations is derived. Next, some sufficient conditions about extinction of stochastic system are established in view of the basic reproduction number R0. Finally, some examples and numerical simulations are introduced to illustrate our analytical results. In addition, some advantages of our new method and theory are highlighted by comparison with other existing results at the end of this paper.

Highlights • We applied support vector machine for COVID-19 data from twelve countries • We tested different structures of nonlinearity using five Kernel functions • Empirical results indicated overfitting for the Gaussian Kernel • The linear Kernel had the worst in-sample but the best out-of-sample performance • We provided a visual analysis of the hyperparameters’ impacts on forecasting error, In this paper, we applied support vector regression to predict the number of COVID-19 cases for the 12 most-affected countries, testing for different structures of nonlinearity using Kernel functions and analyzing the sensitivity of the models’ predictive performance to different hyperparameters settings using 3-D interpolated surfaces. In our experiment, the model that incorporates the highest degree of nonlinearity (Gaussian Kernel) had the best in-sample performance, but also yielded the worst out-of-sample predictions, a typical example of overfitting in a machine learning model. On the other hand, the linear Kernel function performed badly in-sample but generated the best out-of-sample forecasts. The findings of this paper provide an empirical assessment of fundamental concepts in data analysis and evidence the need for caution when applying machine learning models to support real-world decision making, notably with respect to the challenges arising from the COVID-19 pandemics.

The traditional antenna calibration for time division duplex multiple input multiple output (TDD-MIMO) systems assume that free-space channel keeps constant during calibration, which is unreasonable under the high-speed rail and other time-varying channel scenarios for time variability can cause calibration error. This paper analyzes the performance of traditional antenna calibration methods, and then proposes an antenna calibration method based on Wiener channel prediction for time-varying TDD-MIMO system. Through theoretical derivation, we get the capacity formula of the TDD-MIMO system using traditional antenna calibration and antenna calibration based on channel prediction. The theoretical analysis and simulation results both indicate that the performance degradation of antenna calibration can be caused by time-varying channel and the prediction algorithm proposed in the paper can well compensate for the performance loss and significantly improve the antenna calibration performance for time-varying TDD-MIMO system.

Highlights • A fully automated, real-time forecasting model for COVID-19 transmission to help frontline health workers and government policy makers. • Use of Artificial intelligence (AI) and Deep Learning to model Infectious diseases without loosing temporal components. • One of the early studies to use LSTM networks to predict the COVID-19 transmission. • We showed the trends of different countries and compared them with Canadian data to predict the future infections., On March 11th 2020, World Health Organization (WHO) declared the 2019 novel corona virus as global pandemic. Corona virus, also known as COVID-19 was first originated in Wuhan, Hubei province in China around December 2019 and spread out all over the world within few weeks. Based on the public datasets provided by John Hopkins university and Canadian health authority, we have developed a forecasting model of COVID-19 outbreak in Canada using state-of-the-art Deep Learning (DL) models. In this novel research, we evaluated the key features to predict the trends and possible stopping time of the current COVID-19 outbreak in Canada and around the world. In this paper we presented the Long short-term memory (LSTM) networks, a deep learning approach to forecast the future COVID-19 cases. Based on the results of our Long short-term memory (LSTM) network, we predicted the possible ending point of this outbreak will be around June 2020. In addition to that, we compared transmission rates of Canada with Italy and USA. Here we also presented the 2, 4, 6, 8, 10, 12 and 14th day predictions for 2 successive days. Our forecasts in this paper is based on the available data until March 31, 2020. To the best of our knowledge, this of the few studies to use LSTM networks to forecast the infectious diseases.

This paper presents a purely dynamic economic-environmental growth model with a S-shaped production function and a pollution externality. We first outline the role of the saving rate in inducing/suppressing a dual steady-state, one with lower capital and pollution (the “green” steady state) than the other (the “dirty” steady state). Assuming the saving rate is such that the economy possesses a dual steady state, the second part of the paper exploits global analysis tecniques to study conditions such that the economy is able to escape the dirty steady state and approach the green steady state.

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

This paper illustrates an approach to generate multi-wing attractors in coupled Lorenz systems. In particular, novel four-wing (eight-wing) hyperchaotic attractors are generated by coupling two (three) identical Lorenz systems. The paper shows that the equilibria of the proposed systems have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. In analogy with the original Lorenz system, where the two-wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four-wings (eight-wings) of these attractors are located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.

Comprehensive numerical simulations (reviewed in Dissipative Solitons, Akhmediev and Ankiewicz (Eds.), Springer, Berlin, 2005) of pulse solutions of the cubic–quintic Ginzburg–Landau Equation (CGLE), a canonical equation governing the weakly nonlinear behavior of dissipative systems in a wide variety of disciplines, reveal various intriguing and entirely novel classes of solutions. In particular, there are five new classes of pulse or solitary waves solutions, viz. pulsating, creeping, snake, erupting, and chaotic solitons. In contrast to the regular solitary waves investigated in numerous integrable and non-integrable systems over the last three decades, these dissipative solitons are not stationary in time. Rather, they are spatially confined pulse-type structures whose envelopes exhibit complicated temporal dynamics. The numerical simulations also reveal very interesting bifurcations sequences of these pulses as the parameters of the CGLE are varied. In this paper, we address the issues of central interest in the area, i.e., the conditions for the occurrence of the five categories of dissipative solitons, as well the dependence of both their shape and their stability on the various parameters of the CGLE, viz. the nonlinearity, dispersion, linear and nonlinear gain, loss and spectral filtering parameters. Our predictions on the variation of the soliton amplitudes, widths and periods with the CGLE parameters agree with simulation results. First, we elucidate the Hopf bifurcation mechanism responsible for the various pulsating solitary waves, as well as its absence in Hamiltonian and integrable systems where such structures are absent. Next, we develop and discuss a variational formalism within which to explore the various classes of dissipative solitons. Given the complex dynamics of the various dissipative solutions, this formulation is, of necessity, significantly generalized over all earlier approaches in several crucial ways. Firstly, the starting formulation for the Lagrangian is recent and not well explored. Also, the trial functions have been generalized considerably over conventional ones to keep the shape relatively simple (and the trial function integrable!) while allowing arbitrary temporal variation of the amplitude, width, position, speed and phase of the pulses. In addition, the resulting Euler–Lagrange equations are treated in a completely novel way. Rather than consider the stable fixed points which correspond to the well-known stationary solitons or plain pulses, we use dynamical systems theory to focus on more complex attractors viz. periodic, quasiperiodic, and chaotic ones. Periodic evolution of the trial function parameters on stable periodic attractors yield solitons whose amplitudes and widths are non-stationary or time dependent. In particular, pulsating and snake dissipative solitons may be treated in this manner. Detailed results are presented here for the pulsating solitary waves – their regimes of occurrence, bifurcations, and the parameter dependences of the amplitudes, widths, and periods agree with simulation results. Snakes and chaotic solitons will be addressed in subsequent papers. This overall approach fails only to address the fifth class of dissipative solitons, viz. the exploding or erupting solitons.

This paper deals with the observer-based synchronization in a cascade connection of hyperchaotic discrete-time systems. The paper demonstrates that exact synchronization in finite time is achievable between pairs of drive–response systems using only a scalar synchronizing signal. This “propagated synchronization” starts from the innermost drive–response system pair and propagates toward the outermost drive-system pair. Choosing the drive-system input to be an information signal (encrypted via an arbitrary encryption function) yields a potential application of this architecture in chaos-based communications.

Firstly, we recover an ancient work of Gerber in 1898 as a precursor of the retarded theories, see [Gerber P. Die raumliche und zeitliche Ausbreitung der Gravitation (Space and temporary propagation of gravitation). Z Math Phys 1898;43:93–104]. In that paper Gerber gave an explanation of the anomalous precession of Mercury’s perihelion in terms of a velocity-dependent potential. In the present paper an explanation of the anomalous precession of Mercury’s perihelion is given in terms of a simple retarded potential, which, at first orders, coincides with Gerber’s potential, and which agrees with the author’s previous works [Gine J. On the origin of quantum mechanics. Chaos, Solitons & Fractals 2006;30(3):532–41; Gine J. On the origin of gravitational quantization: the Titius–Bode law. Chaos, Solitons & Fractals 2007;32(2):363–69].

Turing theory of pattern formation is among the most popular theoretical means to account for the variety of spatio-temporal structures observed in Nature and, for this reason, finds applications in many different fields. While Turing patterns have been thoroughly investigated on continuous support and on networks, only a few attempts have been made towards their characterization in systems with higher-order interactions. In this paper, we propose a way to include group interactions in reaction-diffusion systems, and we study their effects on the formation of Turing patterns. To achieve this goal, we rewrite the problem originally studied by Turing in a general form that accounts for a microscropic description of interactions of any order in the form of a hypergraph, and we prove that the interplay between the different orders of interaction may either enhance or repress the emergence of Turing patterns. Our results shed light on the mechanisms of pattern-formation in systems with many-body interactions and pave the way for further extensions of Turing original framework.

Made available in DSpace on 2016-10-10T03:51:34Z (GMT). No. of bitstreams: 5 Long memory testing for Fed Funds Futures contracts.pdf: 230580 bytes, checksum: 03ec1431f4d200c4fab72e79cae07e65 (MD5) license_url: 52 bytes, checksum: 3d480ae6c91e310daba2020f8787d6f9 (MD5) license_text: 21716 bytes, checksum: 282d2b1a583fb55b557e8a3be8d5dd05 (MD5) license_rdf: 23930 bytes, checksum: 6b71892b27c4389434057b8b0e86b43e (MD5) license.txt: 1872 bytes, checksum: 9ede5d1aaff3f6277cd24454ee44422e (MD5) Previous issue date: 2008 In this paper, the evaluation of the long memory in returns and volatilities of returns of the daily prices of closing of 6 future generic contracts of Fed Funds negotiated in the Chicago Board of Trade (CBOT) is performed. This evaluation is also made for the spreads between prices of these generic contracts for the evaluation of the transmission of the effects of shock in the interest rates for the various horizons of expectations until 6 months. The study uses the classical R/S analysis for the determination of the Hurst exponent and the bootstrap through moving blocks for the determination of the standard error of the exponent. Long memory for the returns and for the volatility of the returns of the studied future contracts was identified, which suggests that the Monetary Authority of the United States has been able to maintain the stability of the interest rates in spite of shocks, or that the American economy is not significantly affected by shocks. The results of this paper also suggest that the adjustments of the expected interest rates of the Fed Funds occur quickly for the various horizons, not presenting long memory for the returns of the spreads, but for volatility. Sim Publicado

This paper is concerned with the design of a proportional-integral-derivative (PID) controller for synchronization of delayed discrete chaotic models. The evolutionary programming algorithm (EPA) has been considered as a useful technique for finding global optimization solutions for certain complicated functions in recent years. Therefore, in this paper, we attempt to use the EP algorithm in PID control design for deriving optimal or near optimal PID control gains such that a performance index between the master and slave chaotic systems is minimized. A numerical result exemplifies the synchronization procedure.

A digital watermark is a visible, or preferably invisible, identification code that is permanently embedded in digital media, to prove owner authentication and provide protection for documents. Given the interest in watermark generation using chaotic functions a detailed study of one chaotic function for this purpose is performed. In this paper, we present an approach for the generation of watermarks using the logistic map. Using this function, in conjunction with seed management, it is possible to generate chaotic sequences that may be used to create highpass or lowpass digital watermarks. In this paper we provide a detailed study on the generation of optically detectable watermarks and we provide some guidelines on successful chaotic watermark generation using the logistic map, and show using a recently published scheme, how care must be taken in the selection of the function seed.

In a recent paper, a cosmological model based on El Naschie E infinity Cantorian space–time was presented [Iovane G. Varying G, accelerating universe, and other relevant consequences of a stochastic self-similar and fractal universe. Chaos, Solitons & Fractals 2004;20:657–67]. In that work, it was claimed that the present accelerated expansion of the universe can be obtained as the effect of a scaling law on Newtonian cosmology with a certain time-dependent gravitational constant (G). In the present work we show that such a cosmological model actually describes a decelerated universe. Then starting from the scenario presented in that paper, we realize a complementary approach based on an extended Friedmann model. In fact, we apply the same scaling law and a time-dependent gravitational constant, that follows from the observational constraints, to relativistic cosmology, i.e. a (extended) Friedmann’s model. We are able to show that for a matter-dominated flat universe, with the scaling law and a varying G, an accelerated expansion emerges in such a way that the function luminosity distance vs redshift can be made close to the corresponding function that comes from the usual Friedmann’s model supplemented with a cosmological constant, of value ΩΛ ≃ 0.7. Then the measurements of high redshift supernovae, could be interpreted as a consequence of the fractal self-similarity of the G varying relativistic universe.

Oligopoly theory, i.e., the economic theory for competition among the few, goes back to 1838 and Augustin Cournot [7]. See also [11]. Quite early it was suspected to lead to complex dynamic behaviour and chaos. See Rand 1978 [13]. The probably simplest case under which this happens with reasonable economics assumptions was suggested by one of the present authors in 1991, see [9]. It assumes an isoelastic demand function, which always arises when the consumers maximize utility functions of the Cobb-Douglas type, combined with constant marginal costs. The particular layout was a duopoly, the case of only two competitors. The model was shown to produce a period doubling sequence of ip bifurcations ending in chaos for the outputs of each of the two competitors. Later the triopoly case under these assumptions was studied. See [2], [3], and [4] for examples. An interesting fact is that with three competitors the main frame becomes the Neimark-Hopf bifurcation, which provides new and di erent scenarios. The main reason for economists to study increasing numbers of competitors is to nd out whether it is the number of competitors that uniquely decides a road from monopoly over duopoly, oligopoly, and polypoly, to perfect competition, a state where each rm is so small that its actions cannot in uence the market at all. To nd out about this it is of primary interest to know whether the number of competitors stabilizes or destabilizes the equilibrium state. Some authors have questioned the assumption, to which most economists adhered, that increasing numbers of competitors bring stabilization. However, we must be clear about what we compare. If we study increasing numbers of competitors with constant unit production costs, we are not reducing the size of the rms when their number increases. Constant marginal cost means that potentially each rm has in nite capacity, and adding such rms is not what we want for comparison. It is therefore interesting to combine an increased number of rms with decreasing size of each rm, but in order to do so we have to introduce capacity limits. Already Edgeworth [8] insisted on the importance of capacity limits. It is not so easy to nd non-constant marginal cost functions which allow us to solve for the reaction functions for the rms in explicit form, but one of the present authors, see [12], found one type of function, which models the capacity limit by letting marginal cost go to in nity at a nite output. That paper discussed the competition between two duopolists. The objective of the present paper is to nd out the facts when there are three competitors, and we still keep the assumption of capacity limits

In this paper, we show that, for any $\beta \in [1,2]$, a given strictly positive real-valued continuous function on $[0,1]$ whose graph has upper box-counting dimension less than or equal to $\beta $ can be decomposed as a product of two real-valued continuous functions on $[0,1]$ whose graphs have upper box-counting dimension equal to $\beta$. We also obtain a formula for the upper box-counting dimension of every element of a ring of polynomials in finite number of continuous functions on $[0,1]$ over the field $\mathbb{R}.$

This paper simulates the non-linear vibration of a gear pair including shaft flexibility. Backlash is included because of the clearance between the teeth and the periodic variation in the stiffness of the mesh produces a parametric excitation. This paper shows the different behaviour that may be observed when the additional degree of freedom is added to the single degree of freedom model previously analysed.

In this talk we present main points of some of Fock papers which were not properly followed when published and review Fock's interpretation of the energy–time uncertainty relation, as well as his ideas on generalization of the concept of physical space published in his last paper.

This paper presents an analysis of the dynamical behaviour of a non-symmetric oscillator with piecewise-linearity. The Chen–Langford (C–L) method is used to obtain the averaged system of the oscillator. Using this method, the local bifurcation and the stability of the steady-state solutions are studied. A Runge–Kutta method, Poincaré map and the largest Lyapunov’s exponent are used to detect the complex dynamical phenomena of the system. It is found that the system with piecewise-linearity exhibits periodic oscillations, period-doubling, period-3 solution and then chaos. When chaos is found, it is detected by examining the phase plane, bifurcation diagram and the largest Lyapunov’s exponent. The results obtained in this paper show that the vibration system with piecewise-linearity do exhibit quite similar dynamical behaviour to the discrete system given by the logistic map.

In this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system trough a finite element reduction, by composing the associated transfer operator with a suitable finite dimensional projection (a discretization scheme) as in the well-known Ulam method. We introduce a general framework based on a list of properties (of the system and of the projection) that need to be verified so that we can take advantage of a so-called ``coarse-fine'' strategy. This strategy is a novel method in which we exploit information coming from a coarser approximation of the system to get useful information on a finer approximation, speeding up the computation. This coarse-fine strategy allows a precise estimation of invariant densities and also allows to estimate rigorously the speed of mixing of the system by the speed of mixing of a coarse approximation of it, which can easily be estimated by the computer. The estimates obtained here are rigourous, i.e., they come with exact error bounds that are guaranteed to hold and take into account both the discretiazation and the approximations induced by finite-precision arithmetic. We apply this framework to several discretization schemes and examples of invariant density computation from previous works, obtaining a remarkable reduction in computation time. We have implemented the numerical methods described here in the Julia programming language, and released our implementation publicly as a Julia package.

This paper focuses on the fixed-time (FXT) and preassigned-time (PAT) synchronization of uncertain singularly perturbed complex networks (SPCNs) with time-varying delay and stochastic disturbances. Firstly, the fault-tolerant quantized state feedback control strategy for the SPCNs systems to reach PAT and FXT synchronization is considered for the first time. Besides, by adopting suitable Lyapunov functional with parameter ε, with an approach of comparison system and mathematical techniques, the FXT synchronization criteria can be obtained. Next, the PAT synchronization can also be derived, which the synchronization time will be pre-designed based on the actual case. Finally, two simulation examples are presented to verify the effectiveness of the obtained results.

A pair of fractal gravity models can be derived from the spatial interaction models based on entropy maximizing principle and allometric scaling law. The models can be expressed as dual form in mathematics and are important for analyzing and predicting spatial flows in network of cities. However, the dual relationship of urban gravity parameters has been an empirical relationship for a long time and lacks theoretical proof. This paper is devoted to proving the duality of fractal gravity models by means of ideas from scaling invariance. The results show that a fractal gravity model can be derived from its dual form. The observational data of interurban spatial flows of Beijing-Tianjin-Hebei region of China are employed to make a case study, lending a further support to the theoretical derivation. A conclusion can be reached that the duality of gravity models rests with scaling symmetry of fractal structure. This work may be helpful for understanding the theoretical essence and application direction of spatial interaction modeling., 19 pages, 1 figure, 6 tables

The recent outbreaks of Ebola encourage researchers to develop mathematical models for simulating the dynamics of Ebola transmission. We continue the study of the models focusing on those with a variable population. Hence, this paper presents a compartmental model consisting of 8-dimensional nonlinear dif- ferential equations with a dynamic population and investigates its basic reproduction number. Moreover, a dimensionless model is introduced for numerical analysis, thus proving the disease-free equilibrium is locally asymptotically stable whenever the threshold condition, known as a basic reproduction number, is less than one. Finally, we use a fractional differential form of the model to sufficiently fit long time-series data of Guinea, Liberia, and Sierra Leone retrieved from the World Health Organization, and the numerical results demonstrate the performance of the model. published

In an earlier paper we had pointed out that Quantum Mechanical type effects are seen at different scales in the macro universe also. In this paper we obtain a rationale for this, which lies in the picture of bound material systems, spanning a Compton wavelength type extent, separated by much larger and relatively much less dense distances.

The purpose of the present paper is to present the main applications of a new method for the determination of the fractal structure of plane curves. It is focused on the inverse problem, that is, given a curve in the plane, find its fractal dimension. It is shown that the dynamical approach extends the characterization of a curve as a fractal object introducing the effects of mass density, elastic properties, and transverse geometry. The dynamical dimension characterizes material objects and suggests that biological characterization can be much more complete with the methodology presented here.

Analytical approaches in models of opinion formation have been extensively studied either for an opinion represented as a discrete or a continuous variable. In this paper, we analyze a model which combines both approaches. The state of an agent is represented with an internal continuous variable (the leaning or propensity), that leads to a discrete public opinion: pro, against or neutral. This model can be described by a set of master equations which are a nonlinear coupled system of first order differential equations of hyperbolic type including non-local terms and non-local boundary conditions, which can't be solved analytically. We developed an approximation to tackle this difficulty by deriving a set of master equations for the dynamics of the average leaning of agents with the same opinion, under the hypothesis of a time scale separation in the dynamics of the variables. We show that this simplified model accurately predicts the expected transition between a neutral consensus and a bi-polarized state, and also gives an excellent approximation for the dynamics of the average leaning of agents with the same opinion, even when the time separation scale hypothesis is not completely fulfilled.

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the Riesz-Feller sense. Lastly, using Riccati equation, we study families of fractional PDEs with variable coefficients which allow explicit solutions. Those solutions connect Lie symmetries to fractional PDEs., 23 pages, 5 figures

The study of the dynamics of chemical reactions, and in particular phenomena such as oscillating reactions, has led to the recognition that many dynamical properties of a chemical reaction can be predicted from graph theoretical properties of a certain directed graph, called a Chemical Reaction Network (CRN). In this graph, the edges represent the reactions and the vertices the reacting combinations of chemical substances. In contrast with the classical treatment, in this work, we heavily rely on a recently developed theory of directed graph Laplacians to simplify the traditional treatment of the so-called deficiency zero systems of CRN theory. We show that much of the dynamics of these polynomial systems of differential equations can be understood by analyzing the directed graph Laplacian associated with the system. Beside the more concise mathematical treatment, this leads to considerably stronger results. In particular, (i) we show that our Laplacian deficiency zero theorem is markedly stronger than the traditional one and (ii) we derive simple equations for the locus of the equilibria in all (Laplacian) deficiency zero cases. This paper is written in a way to make the material easily accessible to a mathematical audience. In particular, no knowledge of chemistry or physics is assumed., 27 pages, 5 figures

This is the accepted version of the following article: "A numerical method for solving fractional-order viscoelastic Euler–Bernoulli beams", which has been published in final form at https://www.sciencedirect.com/science/article/pii/S0960077919302863; International audience; In this paper, a new method is presented for solving the constitutive equations of fractional-order viscoelastic Euler-Bernoulli beams. Firstly, the constitutive equation of the Euler-Bernoulli beam is established by analyzing the constitutive relation between the fractional viscoelastic materials. Secondly, the constitutive equation of the beam is transformed into a matrix equation by using a Quasi-Legendre polynomial in the time domain. Then the matrix equation is discretized and solved, and the numerical solution is obtained for the constitutive equation of the beam. Finally, numerical analysis of two different fractional viscoelastic materials is carried out by numerical experiments. Displacements under different external loads are obtained for the polybutadiene beam and butyl B252 beam. With the change of time and position, the change law of displacements is found. And the performance of the two materials is compared and analyzed.

In this paper we analyse random walk on a fractal structure, specifi- cally fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the corresponding probability distribution which is gaussian like in nature, but shows deviation from the standard behaviour. Moments are calculated in terms of Euclidean distance for a von Koch curve. We also analyse Levy distribution on the same fractal structure, where the dimen- sion of the fractal curve shows significant contribution to the distrubution law by modyfying the nature of moments. The appendix gives a short note on Fourier transform on fractal curves., Comment: 15 pages, 4 figures

The concept of mechanical screening is widely applied in solid-state systems. Examples include nucleation of defects in crystalline materials, scars and pleats in curved crystals, wrinkles in strongly confined thin sheets, and cell-rearrangements in biological tissue. Available theories of such screening usually contain a crucial ingredient, which is the existence of an ordered reference state, with respect to which screening elements nucleate to release stresses. In contradistinction, amorphous materials in which a unique reference state does not exist, nevertheless exhibits plastic events that act as screening geometric charges with significant implications on the mechanical response. In a recent paper [Phys. Rev. E 104, 024904] it was proposed that mechanical strains in amorphous solids can be either weakly or strongly screened by the formation of low or high density of plastic events. At low densities the screening effect is reminiscent of the role of dipoles in dielectrics, in only renormalizing the elastic moduli. The effect of high density screening has no immediate electrostatic analog and is expected to change qualitatively the mechanical response, as seen for example in the displacement field. On the basis of experiments and simulations, we show that in granular matter, strong screening results in significant deviation from elasticity theory. The theoretical analysis, which accounts for an emergent inherent length scale, the experimental measurements and the numerical simulations of frictional granular amorphous assemblies are in agreement with each other, and provide a strong support for the novel continuum theory.

In this paper we consider a modified fractional Maxwell model based on the application of Hadamard-type fractional derivatives. The model is physically motivated by the fact that we can take into account at the same time memory effects and the time-dependence of the viscosity coefficient. We obtain an ultra-slow relaxation response whose explicit analytic form is given by the Mittag-Leffler function with a logarithmic argument. We show graphically the main properties of this relaxation response, also with the asymptotic behaviour., Comment: 10 pages, 3 figures