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In this paper a fractional order mathematical model is constructed to study the dynamics of corona virus in Oman. The model consists of a system of eight non-linear fractional order differential equations in Caputo sense. Existence and uniqueness as well as the stability analysis of the solution of the model are given. The stability analysis is in the frame of Ulam-Hyers and generalized Ulam-Hyers criteria. Numerical simulations are given to support the theoretical results. Many informations on the dynamics of COVID -19 in Oman were obtained using this model. Also many informations on the qualitative behaviour of the model were obtained.

Highlights • Potential advantages of Singular Spectrum Analysis (SSA) for forecasting the number of daily confirmed cases, deaths, and recoveries caused by COVID-19 until 29 October 2020. • Proposing an algorithm to calculate the optimal parameters of SSA. • The results of V-SSA and R-SSA are compared to those from ARIMA, ARFIMA, Exponential Smoothing, TBATS, and NNAR. • SSA technique is found to be viable option for forecasting the number of daily confirmed cases, deaths, and recoveries caused by COVID-19 based on the number of times that it outperforms the competing models. • Providing 40 days ahead point forecasts for top ten countries affected by COVID-19., Coronavirus disease 2019 (COVID-19) is a pandemic that has affected all countries in the world. The aim of this study is to examine the potential advantages of Singular Spectrum Analysis (SSA) for forecasting the number of daily confirmed cases, deaths, and recoveries caused by COVID-19, which are the three main variables of interest. This paper contributes to the literature on forecasting COVID-19 pandemic in several ways. Firstly, an algorithm is proposed to calculate the optimal parameters of SSA including window length and the number of leading components. Secondly, the results of two forecasting approaches in the SSA, namely vector and recurrent forecasting, are compared to those from other commonly used time series forecasting techniques. These include Autoregressive Integrated Moving Average (ARIMA), Fractional ARIMA (ARFIMA), Exponential Smoothing, TBATS, and Neural Network Autoregression (NNAR). Thirdly, the best forecasting model is chosen based on the accuracy measure Root Mean Squared Error (RMSE), and it is applied to forecast 40 days ahead. These forecasts can help us to predict the future behaviour of this disease and make better decisions. The dataset of Center for Systems Science and Engineering (CSSE) at Johns Hopkins University is adopted to forecast the number of daily confirmed cases, deaths, and recoveries for top ten affected countries until October 29, 2020. The findings of this investigation show that no single model can provide the best model for any of the countries and forecasting horizons considered here. However, the SSA technique is found to be viable option for forecasting the number of daily confirmed cases, deaths, and recoveries caused by COVID-19 based on the number of times that it outperforms the competing models.

Most of the nations with deplorable health conditions lack rapid COVID-19diagnostic test due to limited testing kits and laboratories. The un-diagnosticmild cases (who show no critical sign and symptoms) play the role as a route that spread the infection unknowingly to healthy individuals. In this paper, we present a fractional order SIR model incorporating individual with mild cases as a compartment to become SMIR model. The existence of the solutions of the model is investigated by solving the fractional Gronwall's inequality using the Laplace transform approach. The equilibrium solutions (DFE & Endemic) are found to be locally asymptotically stable, and subsequently the basic reproduction number is obtained. Also the global stability analysis is carried out by constructing Lyapunov function. Lastly, numerical simulations that support analytic solution follow. It was also shown that when the rate of infection of the mild cases increases, there is equivalent increase in the overall population of infected individuals. Hence to curtail the spread of the disease there is need to take care of the Mild cases as well.

Substantial evidence supports that financial returns time series exhibit abnormal properties including leptokurtosis, volatility clustering as well as intermittent jumps and leverage effects between returns and volatility processes. This paper studies a heavy-tailed stochastic volatility (SV) model with jumps components and leverage effects, and the Student's-t distribution is employed to describe the error innovations (SVJLt). Since the existence of high-dimensionality of the latent variables and the special structure of Hessian matrix of the stochastic volatility density, we develop an efficient Markov chain Monte Carlo (MCMC) posterior simulator exploiting the adaptive importance sampling technique based on band and sparse matrix routine rather than the conventional Kalman filter to estimate the new model. And the precision sampler is exploited due to the band structure of the inverse covariance matrix of the state variables. The model comparisons of returns volatility are conducted utilizing the observed-data based deviance information criterion (DIC) and the cross-entropy (CE) based marginal likelihood estimation. The effectiveness of the proposed model and the methodology are illustrated with applications in stock returns volatility forecast. Through employing several loss functions for evaluation, the empirical studies suggest strong evidence in heavy tailed distributions, jumps features and leverage effects simultaneously.

Atangana and Baleanu gave a derivative with fractional order to search the valuable inquiries. We present a novel technique for investigating fractional differential equations including the Atangana-Baleanu fractional derivative in this paper. We give some examples for this method for equations of fractional initial value problems.

In this paper, we are interested in the study of a stochastic viral infection model with immune impairment driven by Levy noise. First we prove the existence of a unique global solution to the model. By means of the Lyapunov method we study the stability of the equilibria. We present sufficient conditions for the extinction and persistence in mean. Furthermore, we present some numerical results to support the theoretical work.

Very little research is available in the field of 4-D autonomous conservative chaotic systems. This paper presents five new 4-D autonomous conservative chaotic systems having non-hyperbolic equilibria with various characteristics. The proposed systems have different numbers of non-hyperbolic equilibrium points. One of the new systems has four non-hyperbolic equilibria points along with lines of equilibria. Hence, this system may belong to the category of hidden attractors chaotic system. The first, second, fourth and fifth type of the systems exhibit coexistence of chaotic flow, whereas the third type of the system exhibits coexistence of chaotic flows with quasi-periodic behaviour. The chaotic behaviours of the proposed systems are verified by using phase portrait plot, Poincare map, local Lyapunov spectrum, bifurcation diagram and frequency spectrum plots. The conservative nature of the proposed systems is proved by finding the sum of finite-time local Lyapunov exponents, finite-time local Lyapunov dimensions and divergence of the vector field. The sum of the finite-time local Lyapunov exponents and divergence of the vector field are equal to zero, and local Lyapunov dimension is equal to the order of the system confirm the conservative nature of the new chaotic systems.

The paper is devoted to investigate three different points including the importance, usefulness of the Bode diagram in calculus including classical and fractional on one hand. On the other hand to answer and disprove the statements made about fractional derivatives with continuous kernels. And finally to show researchers what we see and we do not see in a commutative world. To achieve this, we considered first the Caputo–Fabrizio derivative and used its Laplace transform to obtain a transfer function. We represented the Bode, Nichols, and the Nyquist diagrams of the corresponding transfer function. We in order to assess the effect of exponential decay filter used in Caputo–Fabrizio derivative, compare the transfer function associate to the Laplace transform of the classical derivative and that of Caputo–Fabrizio, we obtained surprisingly a great revelation, the Caputo–Fabrizio kernel provide better information than first derivative according to the diagram. In this case, we concluded that, it was not appropriate to study the Bode diagram of transfer function of Caputo–Fabrizio derivative rather, it is mathematically and practically correct to see the effect of the kernel on the first derivative as it is well-established mathematical operators. The Caputo–Fabrizio kernel Bode diagram shows that, the kernel is low past filter which is very good in signal point of view. We consider the Mittag–Leffler kernel and its corresponding Laplace transform and find out that due to the fractional order, the corresponding transfer function does not exist therefore the Bode diagram cannot be presented as there is no so far a mathematical formula that help to find transfer function of such nature. It is therefore an opened problem, how can we construct exactly a transfer function with the following term ( i w ) α ( i w ) α + b for instance? We proved that fractional derivative with continous kernel are best to model real world problems, as they do not inforce a non-singular model to become singular due to the singularity of the kernel. We show that, by considering initial time to be slightly above the origin then the Riemann–Liouville and Caputo-power derivatives are fractional derivatives with continuous kernel. We considered some interesting chaotic models and presented their numerical solutions in different ways to show what we see or do not see if a commutative world. To end, we presented the terms to be followed to provide a new fractional derivative.

In this paper, we discussed the input-to-state stability of a class of memristive Cohen–Grossberg-type neural networks with variable time delays. Based on a nonsmooth analysis and set-valued maps, some novel sufficient conditions are obtained for the input-to-state stability of such networks, which include some known results as particular cases. Especially, when the input is zero, it reduced to asymptotical stability of the state. Finally, an illustrative example is presented to illustrate the feasibility and effectiveness of our results.

In this paper we have taken a delayed predator-prey system with stage structure among prey species. We assume that there is cooperation among the mature and immature preys to ensure immature prey’s existence and there is a maturation delay for immature to be mature for the preys. With this model the local stability has been discussed in presence of delay. By taking maturation delay as the key parameter, the condition for local stability has been obtained by constructing a Lyapunov functional. By taking the maturation delay as the bifurcation parameter the necessary conditions for Hopf-bifurcation have been discussed both analytically and numerically. The length of delay has been estimated to preserve the stability. Also, Lyapunov exponents have been evaluated numerically for different cases.

Multistability is common feature resulting in nonlinear dynamical systems, and its characteristic can be generally depicted by investigating basin structures of initial conditions for give parameter settings. In this paper, we explore the formation of extinction states according to the change of strength of competition levels in the toy model for three species. Through the linear stability analysis, we find that the extinction state can be stable which is persistent. For specific conditions between intensities of two different competitions, we also found that the extinction state can be either bistable or tristable. In each case, the final state of the system can be characterized sensitively depending on initial conditions. To validate our results, we investigate basin structures of parameters for interspecific competition associated to a strength of intraspecific competition. In addition, we found that coexistence becomes robust as intraspecific competition is intensified relatively to the interspecific competition level. We hope our results can be a chance to suggest the emergence of the multistability according to complex competition structures on systems of many populations.

Variable-order differential operators can be employed as a powerful tool to modeling nonlinear fractional differential equations and chaotical systems. In this paper, we propose a new generalize numerical schemes for simulating variable-order fractional differential operators with power-law, exponential-law and Mittag-Leffler kernel. The numerical schemes are based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. These schemes were applied to simulate the chaotic financial system and memcapacitor-based circuit chaotic oscillator. Numerical examples are presented to show the applicability and efficiency of this novel method.

This paper presents detailed procedure for determining the formula for calculation Tribonacci sequence numbers with arbitrary initial numbers Ta,b,c,(n). Initial solution is based on the concept of damped oscillations of Lucas type series with initial numbers T3,1,3(n). Afterwards coefficient θ3 has been determined which reduces Lucas type Tribonacci series to Tribonacci sequence T0,0,1(n). Determined relation had to be corrected with a phase shift ω3. With known relations of unitary series T0,0,1(n) with remaining two equations of Tribonacci series sequence T1,0,0(n) and T0,1,0(n), Binet type equation of Tribonacci sequence that has initial numbers Ta,b,c(n) is obtained.

In this paper, we use fast oscillating integrated Ornstein–Uhlenbeck (abbreviated as O-U) processes to pathwisely approximate Wiener processes. In physics, such approximation process is known as a colored noise approximation, and is suitable for dealing with stochastic flow problems. Our first result shows that if the drift term of a stochastic differential equation (abbreviated as SDE) satisfies usual Lipschitz constrains and a linear growth condition, then the solution of the SDE can be almost surely approximated with a polynomial rate. Next, we explore the O-U process approximation on the random manifold of stochastic evolution equations with linear multiplicative noise. Our second result shows that if the stochastic evolution equation further satisfies a uniformly hyperbolic condition, then the corresponding random manifold approximation also converges almost surely, with a polynomial rate.

This paper systematically studies the structure of the financial kNN (k-nearest neighbor) network. First, we use the eigenvalues and eigenvectors of the financial correlation matrix to analyze the structure of the network. We find that the degree is related to the average correlation coefficient, and furthermore, it also has a relationship between the components of the eigenvector corresponding to the maximum eigenvalue. We apply existing research to confirm that the community structure of the kNN network can be used to cluster financial time series. Finally, empirical studies based on financial markets in three countries show that there is a high correlation between the community structure and dimensions. Therefore, this study shows that the structure of the financial kNN network is related to the properties of the correlation matrix, and it extracts a meaningful correlation structure.

In this paper, we present a novel explicit analytical solution for the normalized state equations of mutually-coupled simple chaotic systems. A generalized analytical solution is obtained for a class of simple nonlinear electronic circuits with two different nonlinear elements. The synchronization dynamics of the circuit systems were studied using the analytical solutions. the analytical results thus obtained have been validated through numerical simulation results. Further, we provide a sufficient condition for synchronization in mutually-coupled, second-order simple chaotic systems through an analysis on the eigenvalues of the difference system. The bifurcation of the eigenvalues of the difference system as functions of the coupling parameter in each of the piecewise-linear regions, revealing the existence of stable synchronized states is presented. The stability of synchronized states are studied using the {\emph{Master Stability Function}}. Finally, the electronic circuit experimental results confirming the phenomenon of complete synchronization in each of the circuit system is presented., 12 pages, 6 figures. arXiv admin note: text overlap with arXiv:1611.04289

This paper focuses on the dynamic behavior of phytoplankton–zooplankton system with cell size and time delay. Remarkably, the existence of cell size and time delay make the dynamic behavior of the system more close to the real-world situation, essentially different from those in the existing related literature. To analyze the dynamic behavior of the system, the positiveness and boundedness of the solution are first derived. Then, the asymptotic stability of the coexistent equilibrium and Hopf bifurcation are studied by analyzing the associated characteristic equation. Once more, the direction of Hopf bifurcation and the stability of bifurcated periodic solution are determined. Finally, the theoretical results are illustrated by a numerical example.

This paper is concerned with the dynamics and oscillations of a new class of Lotka-Volterra recurrent neutral networks. We obtain results on the existence and uniqueness of the pseudo almost periodic solution under some sufficient and proper conditions. In addition, the asymptotic and exponential stability of the pseudo almost periodic solutions are investigated. Finally, two numerical examples with their simulations are presented to support our theoretical results.

In this paper, we study the bifurcation of limit cycles in two special near-Hamiltonian polynomial planer systems which their corresponding Hamiltonian systems have a heteroclinic loop connecting a hyperbolic saddle and a cusp of order two. In these systems, we will compute the asymptotic expansions of corresponding first order Melnikov functions near the loop and the center to analyze the number of limit cycles. Moreover, in the first system, by using the Chebychev criterion, we study the Poincare bifurcation.

This paper investigates the mean first-passage times (MFPTs) of a delayed tristable system driven by correlated multiplicative and additive noises. The results suggest that the correlation between the multiplicative and additive noises can induce symmetry-breaking in the delayed tristable system. The noise-induced dynamics, such as the noise enhanced stability (NES) and the resonant activation (RA), can be observed with considering the combined influences of correlated noises and intermediate stable state. The time delay plays an important role in the MFPTs. For example, with respect to the middle well, the increase of time delay results in the weakening of the stability of the two lateral wells; thus, all the MFPTs are decreased notably. Moreover, a law of MFPTs is established for three different potential wells. That is, the MFPT T(s1 → s3) (between the left and right wells) is equal to the sum of T(s1 → s2) (from left well to middle one) and T(s2 → s3) (from middle well to right one). However, this change regulation can be first broken with an increase in time delay, and then restored with the increase of correlation between noises.

This paper is concerned with the fractional differential equations of Sobolev type with boundary conditions in a Banach space. With the help of properties of Hilfer fractional calculus, the theory of propagation family as well as the theory of the measure of noncompactness and the fixed point methods, we obtain the existence results of mild solutions for Sobolev type fractional evolution differential equations involving Hilfer fractional derivative. Finally, two examples are presented to illustrate the main result.

Fractal dimension is the most important parameter for surface characterization. In this paper, four methods used to estimate the fractal dimensions of surface profiles and their applications in machined surfaces are studied. These methods are first evaluated using surface profiles created by Weierstrass–Mandelbrot function from the three aspects of fitting accuracy, calculation accuracy and calculation stability, and then applied to the machined rough surfaces. By comparing the results of the four methods, it is found that none of the methods is particularly prominent in all of the three aspects. However, the three point sinuosity method is found to be relatively the most suitable and reliable method among the four tested methods for extracting fractal dimensions of both generated and measured rough surface profiles.

In this paper, the stochastic kinetics in a time-delayed foraging colony system under non-Gaussian noise were investigated. Using delay Fokker–Planck approach, the stationary probability distribution (SPD), the normalized variance β2, skewness β3 and kurtosis β4 of the state variable are obtained, respectively. The effects of the time delayed feedback and non-Gaussian noise on the SPD are analyzed theoretically. The numerical simulations about the SPD are obtained and in good agreement with the approximate theoretical results. Furthermore, the impacts of the time delayed feedback and non-Gaussian noise on the β2, β3 and β4 are discussed, respectively. It is found that the curves in β2, β3 and β4 exhibit an optimum strength of feedback where β2, β3 and β4 have a maximum. This maximum indicates the large deviations in β2, β3 and β4. From the above findings, it is easy for us to have a further understanding of the roles of the time delayed feedback and non-Gaussian noise in the foraging colonies system.

The present paper deals with an eco-epidemiological prey–predator model with delay. It is assumed that infection floats in predator species only. Both the susceptible and infected predator species are subjected to harvesting at different harvesting rates. Differential predation rates for susceptible and infected predators are considered. It is shown that the time delay can even destabilize the otherwise globally stable non-zero equilibrium state. It is observed that coexistence of all the three species is possible through periodic solutions due to Hopf bifurcation. With the help of normal form theory and central manifold arguments, stability of bifurcating periodic orbits is determined. Numerical simulations have been carried out to justify the theoretical results obtained.

In this paper, we develop a range of efficient and fast fractional difference schemes for the approximation of Caputo time-fractional subdiffusion-reaction equations. The classical time derivative is replaced with the Caputo fractional derivative operator. The experimental results justify that the numerical solution of the proposed methods compares favourably with the exact solution. Experimental results give a clear indication that dynamical models with non-integer order can yield a better spatial pattern when compared with their classical counterparts.

This paper investigates a new impulsive stochastic chemostat model with nonlinear perturbation in a polluted environment. We present the analysis and the criteria of the extinction of the microorganisms, and establish sufficient conditions for the existence of a unique ergodic stationary distribution of the model via Lyapunov functions method. The results show that both stochastic noise and impulsive toxicant input have great effects on the survival and extinction of the microorganisms. Moreover, we provide a series of numerical simulations to illustrate the analytical results.

In this paper, we deal with Abel equations: d x d y = A ( x ) y 2 + B ( x ) y 3 , where A(x) and B(x) are real polynomials. If a solution y = φ ( x ) of the above equations satisfies that φ ( 0 ) = φ ( 1 ) , then we say that it is a periodic solution. If a periodic solution is isolated, then we call it a limit cycle. If a limit cycle y = φ ( x ) is a rational function but not a polynomial, then we call it a nontrivial rational limit cycle. Firstly, we study the existence of nontrivial rational limit cycles. We prove that there exist Abel equations, which have at least two nontrivial rational limit cycles, and there also exists other Abel equations, which have at least one nontrivial rational limit cycle and one non-rational limit cycle. Secondly, we discuss the relation between the existence of nontrivial rational limit cycle and the degrees of A(x) and B(x). Finally we show that the multiplicity of a nontrivial rational limit cycle can be unbounded.

The present paper is devoted to the development of a kind of spectral meshless radial point interpolation (SMRPI) technique for solving fractional cable equation in one and two dimensional cases. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved the scheme is unconditional stable with respect to the time variable in H1 and convergent by the order of convergence O ( δ t γ ) , 0

Models with uncertain values of their parameters are of a practical significance in different fields of science and technology. Uncertainty in the competitive models can greatly affect their dynamical behavior and, therefore, the analysis of the effects of uncertain terms in such models is very important in both theory and application. In this paper, we extend the existing N-species impulsive competitive models to the uncertainty case. The existence of a unique strictly positive and robustly exponentially stable almost periodic solution is investigated for the model. The main results are obtained by using Lyapunov-type functions and a comparison principle.

In this paper, stability analysis is applied to a discrete Lotka–Volterra cooperative system with the periodic boundary conditions, then Turing pattern formation conditions can be derived, theory analysis and numerical simulation show that turing patterns can be realized. In addition, we also pay attention on what reason or what system environment to result into the current state patterns, which can be reduced to estimate or identify the system parameter. A regularization method is applied to parameter inversion, and numerical simulation can verify the effectiveness of the algorithm.

This paper introduces an efficient fractional differential transform that is called “conformable fractional partial differential transform (CFPDT)” and its properties for solving linear and nonlinear conformable space-time fractional partial differential equations (CSTFPDEs). Moreover, a CFPDT is more practical and helpful for solving abroad CSTFPDEs. Analytical solutions to linear Navier–Stokes equation and nonlinear gas dynamic equations in sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the accuracy and efficiency of the proposed transform.

In this paper, we study a stochastic tuberculosis model with antibiotic resistance. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the positive solutions to the model. Moreover, we obtain sufficient conditions for extinction of the disease. The existence of a stationary distribution implies stochastic weak stability.

This paper proposes the definition of stretched exponential stability. First, a non-Debye decay is found in the nonlinear Hausdorff dynamical systems by using the Lyapunov direct method, which leads to the stretched exponential stability. It is worthy of noting that the classical exponential stability is a special case of the proposed stretched exponential stability, when the stretched parameter is in its limiting case equal to 1. Therefore the stretched exponential has more flexibility and applicability to the real-world problems than the classical exponential stability. Second, a fractal comparison principle is applied to obtain stability criteria for the proposed systems. Examples are presented to illustrate the applicability of the proposed concept.

This paper considers a fractional order Rosenzweig-MacArthur (R-M) model incorporating a prey refuge. The model is constructed and analyzed in detail. The existence, uniqueness, non-negativity and boundedness of the solutions as well as the local and global asymptotic stability of the equilibrium points are studied. Sufficient conditions for the stability and the occurrence of Hopf bifurcation for the fractional order R-M model are demonstrated. The resolution of the paradox of enrichment is investigated. The impact of fractional order and the prey refuge effects on the stability of the system are also studied both theoretically and by using numerical simulations.

In this paper we consider Hausdorff dimension of the sets of Li-Yorke pairs for some chaotic dynamical systems including A-coupled expanding systems. We prove that Li-Yorke pairs of A-coupled-expanding system under some conditions have full Hausdorff dimension in the invariant set. Moreover we give a generalization of the result of [5] which is on the Hausdorff dimension of Li-Yorke pairs of dynamical systems topologically conjugate to the full shift and have a self-similar invariant set, to the case of the dynamical systems topologically semi-conjugate to some kinds of subshifts. Furthermore Hausdorff dimension of “chaotic invariant set” for some kinds of A-coupled-expanding maps is shown.

The study on network coherence and eigentime identity has gained much interest. In this paper, the first-order network coherence is characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, while the eigentime identity is quantified by the sum of reciprocals of all nonzero normalized Laplacian eigenvalues. We construct a family of weighted fractal networks with the weight factor r (0 1 s r ≤ 1 ; The second law is that the scaling obeys ( ln N n ) 2 N n (i.e., the quotient of the square logarithm of the network size and the network size), when r = 1 s ; The third law is that the scaling obeys ln N n N n (i.e., the quotient of the logarithm of the network size and the network size), when 0 r 1 s . Thus, the scaling of the first-order coherence of weighted fractal networks decreases with the decreasing of r, when 0 Then, all nonzero normalized Laplacian eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity obey two laws according to the range of the weight factor. The first law is that the scaling obeys ln Nn (i.e., the logarithm of the network size), when 0 r ≠ 1 s ; The second law is that the scaling obeys Nnln Nn (i.e., the product of network size and its logarithm), when r = 1 s .

The present paper describes the dynamics of the Pine Wilt disease with variable population size. The basic mathematical results for the model are presented. The stability analysis of both the disease-free and endemic cases are presented whenever R 0 lesser or greater than one, respectively. Further, an optimal control problem is formulated and the necessarily involved results are presented. Moreover, the numerical simulation of the optimal control problem with suggested control strategies for the possible eliminations of the infection in the pine trees population is presented.

In the present study, an age-structured SEIR endemic model is considered. The population is assumed to be not constant due to different inflows and outflows that are imbalanced by demographics, and migration etc. An abstract Cauchy problem is formulated from the model in order to show that the model is well-posed. The local and global stability of the disease-free steady state is examined using the basic reproduction number R0. Under certain condition, the endemic steady state is shown to be exist and locally stable. To illustrate the main theorems, sample simulations are presented at the end of the paper.

In this paper, we identify the unknown space-dependent source term in a time-fractional diffusion equation with variable coefficients in a bounded domain where additional data are consider at a fixed time. Using the generalized and revised generalized Tikhonov regularization methods, we construct regularized solutions. Convergence estimates for both methods under an a-priori and a-posteriori regularization parameter choice rules are given, respectively. Numerical example shows that the proposed methods are effective and stable.

There are various types of fractional derivatives in literature. One of the most natural and well-behaved fractional derivatives is recently introduced by the authors Khalil et al. [34], namely the conformable fractional derivative. In this paper, some more results about conformable fractional Laplace transform introduced by Abdeljawad [43] are investigated. The invariant subspace method is developed to get the exact solutions of various conformable time fractional differential equations. Finally, this theory is extended for the coupled system of conformable fractional differential equations, as well.

Chaotic attractors in the classical Lorenz system have long been known as self-excited attractors. This paper, for the first time, reveals a novel hidden chaotic attractor in the classical Lorenz system. Either a self-excited or a hidden chaotic attractor is now possible in the classical Lorenz system depending on values of both system parameters and initial conditions. A systematically exhaustive computer search is employed to directly search for the hidden chaotic attractor with elegant values of both system parameters and initial conditions. Time series of trajectories, Lyapunov exponents, and bifurcations of the hidden chaotic attractor are reported. Basins of attraction of individual equilibria are depicted to verify that the hidden chaotic attractor is found. Dynamic regions of attractors are illustrated to reveal seamless connections between self-excited and hidden chaotic attractors in the classical Lorenz system with wide ranges of parameters.

In this paper, bidirectional associative memory (BAM) neural networks with time-varying delays in leakage terms are investigated. A set of sufficient conditions are obtained for the existence and the exponential stability of pseudo almost periodic solutions for this class of neural networks by applying Banach’s fixed point theorems, and differential inequality techniques. Finally, we present a numerical example and simulation to illustrate the effectiveness of the theoretical results. We extend and improve previously know results.

The present paper uses a relatively new approach and methodology to solve one and two dimensional nonlinear Schrodinger equations numerically. We use the horizontal method of lines and θ-method, θ ∈ [1/2, 1] for time discretization that reduces the problem into an amenable system of ordinary differential equations. The resulting system of ODEs in space subsequently have been solved by quintic polynomial spline scheme. Convergence of the scheme in maximum norm is established rigorously. The convergence orders are O ( k + h x 4 + h y 4 ) and O ( k 2 + h x 4 + h y 4 ) , where k is the temporal grid size and hx and hy are spatial grid sizes, respectively. Matrix stability analysis shows that the method is conditionally stable. The efficacy of proposed approach has been confirmed with four numerical experiments, where comparison is made with some earlier works. It is clear that the results obtained are acceptable and are in good agreement with earlier studies. The present scheme is very simple, effective and convenient for obtaining numerical solution of Schrodinger equation.

The global dynamical behaviors of a hexagonal centrifugal governor with a spring are expanded in this paper. The high-definition resolution phase diagrams are used to identify how the complex mode-locking behaviors arise with the parameters changed. By calculating the rotation number, it is unfolded that the interesting “Devils staircase” is presented with the parameter changed, meanwhile, the mode-locking structure is organized according to the broken Farey tree sequence. In addition, the fractal structure with self-similarity can also be observed in this system.

This paper presents a novel approach to calculate the affine parameters of fractal encoding, in order to reduce its computational complexity. A simple but efficient approximation of the scaling parameter is derived which satisfies all properties necessary to achieve convergence. It allows us to substitute to the costly process of matrix multiplication with a simple division of two numbers. We have also proposed a modified horizontal-vertical (HV) block partitioning scheme, and some new ways to improve the encoding time and decoded quality, over their conventional counterparts. Experiments on standard images show that our approach yields performance similar to the state-of-the-art fractal based image compression methods, in much less time.

In this paper we present applications of special numbers obtained from a difference equation of degree three. As a particular case of this difference equation of degree three, we obtain the generalized Pell-Fibonacci-Lucas numbers, which were extended to the generalized quaternion algebras. Using properties of these quaternion elements, we can define a set with an interesting algebraic structure, namely, an order on a generalized rational quaternion algebra. Another presented application is in the Coding Theory, since some of these numbers can be used to built cyclic codes with good properties (MDS codes).

In this paper, we constructed new solitary structures for the space-time fractional simplified modified Camassa-Holm (MCH) equation and space-time fractional symmetric regularized long wave (SRLW) equation using the modified extended tanh method. The space-time fractional derivatives are defined in the sense of the new conformable fractional derivative. Further, with the help of Mathematica software, the set of over-determined algebraic equations obtained after reducing the equations to ordinary differentials equations are treated. We finally provide graphical illustrations for some structures.

This paper is concerned with the Diophantine properties of the orbits of real numbers in continued fraction system under the doubling metric. More precisely, let φ be a positive function defined on N . We determine the Lebesgue measure and Hausdorff dimension of the set E ( φ ) = { ( x , y ) ∈ [ 0 , 1 ) × [ 0 , 1 ) : | T n x − y | φ ( n ) for i.m. n } , where T is the Gauss map and “i.m.” stands for “infinitely many”.

Published online: 13 November 2017 Since its inception, the digital currency market is considerably growing, especially in the most recent years. The main purpose of this paper is to investigate, assess and detect chaos, randomness, and multi-scale temporal correlation structure in prices and returns of this specific virtual and speculative market throughout two distinct time periods; namely under a low-level regime period during which prices slowly increased, and during a high and turbulent regime time period whereby they exponentially increased. We found that chaos is only present in prices during both periods, whilst the level of uncertainty in returns has significantly increased during the high-price time period. Furthermore, both prices and returns exhibit long-range correlations and multi-fractality. The fat-tailed probability distributions are the main source of multi-fractality in the time series of prices and returns. Finally, short (long) fluctuations in returns are dominant during low (high) price-regime time period, respectively. Overall, the high-price regime phase has profoundly revealed consistent nonlinear dynamical patterns in the Bitcoin market.

An Apollonian packing is one of the most beautiful circle packings based on an old theorem of Apollonius of Perga. Ford circles are important objects for studying the geometry of numbers and the hyperbolic geometry. In this paper we pursue a research on the Ford sphere packing, which is not only the three dimensional extension of Ford circle packing, but also a degenerated case of the Apollonian sphere packing. We focus on two interesting sequences in Ford sphere packings. One sequence converges slowly to an infinitesimal sphere touching the origin of the horizontal plane. The other sequence converges at fastest rate to an infinitesimal sphere in a particular position on the plane. All these sequences have their counterparts in Ford circle packings and keep similar features. For example, our finding shows that the x-coordinate of one Ford circle sequence converges to the golden ratio gracefully. We define a Ford sphere group to interpret the Ford sphere packing and its sequences finally.