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In this paper, we present and analyze a prey-predator system, in which prey species can be infected with some disease. The model presented in this paper is motivated from D. Mukherjee’s model in which he has considered an SI model for the prey species. There are substantial evidences that infected individuals have the ability to recover from the disease if vaccinated/ treated properly. In this regard, Mukherjee’s model is modified by considering SIS model for prey species. Theoretical and numerical simulations show that the recovery of infected prey species plays a crucial role in eliminating the limit cycle oscillations and thus making the interior equilibrium point stable. The possibility of Hopf bifurcation around non zero equilibrium point using the recovery rate as a bifurcation parameter, is discussed. Further, the model is extended by incorporating the harvesting of predator population. A monitory agency has been introduced which monitors the exploitation of resources by implementing certain taxes for each unit biomass of the predator population. The main purpose of the present research is to explore the effect of recovery rate of prey on the dynamics of the system and to optimize the total economical net profits from harvesting of predator species, taking taxation as control parameter.

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.

Randomness is a common phenomenon in nonlinear systems. And conditions to reach synchronization are more complex and difficult when chaotic systems have random parameters. So in this paper, an adaptive scheme for synchronization of chaotic system with random parameters by using the fuzzy impulsive method and combining the properties of Wiener process and Ito differential is investigated. The main concepts of this paper are applying fuzzy method to approximate the nonlinear part of system, then using Ito differential to study the Wiener process of random parameters of chaotic system, and realizing synchronization under fuzzy impulsive method. The stability is analyzed by Lyapunov stability theorem. At the end of the paper, numerical simulation is presented to illustrate the effectiveness of the results obtained in this paper.

A periodic predator-prey model has been introduced in [5] to study the effect of water level on persistence or extinction of fish populations living in an artificial lake. By using the continuation theorem of Mawhin’s coincidence degree theory, the authors give sufficient conditions for the existence of at least one positive periodic solution. In this paper we study the problem in the general case. We begin by analyzing the invariance, permanence, non-persistence and the globally asymptotic stability for the system. Most interestingly, under additional conditions, we find that the periodic solution obtained in [5] is unique. Finally, in order to make the model system more realistic, we consider the special case when the periodicity in [5] is replaced by almost periodicity. We obtain conditions for existence, uniqueness and stability of a positive almost periodic solution. The methods used in this paper will be comparison theorems and Lyapunov functions. An example is employed to illustrate our result.

The present work is devoted to giving new insights into the extended Malkus-Robbins (EMR) dynamo. Firstly, based on differential geometry method, i.e. Kosambi-Cartan-Chern (KCC) theory, the paper investigates the Jacobi stability of the equilibrium and periodic orbit by the eigenvalues of the deviation curvature tensor. The deviation vector is applied to analyze the trajectory behaviors near the equilibrium and periodic orbit. Secondly, the zero-zero-Hopf bifurcation is investigated. The paper obtains the conditions that two periodic solutions appear at the bifurcation point and discusses their stability. Finally, on the global dynamics, the ultimate bound sets of the system are estimated. Numerical simulations are given to verify and visualize the corresponding theoretical results.

Yilmaz et al. (Fluct. Noise Lett. 17, 1830002, 2018) investigated the stochastic phenomenological bifurcations of a generalized Chua circuit driven by Skew-Gaussian distributed noise. They proved it is possible to decrease the number of scrolls by properly choosing the stochastic excitation, manipulating the skewness and noise intensity parameters. Based on the latter, this paper proposes an extension of skew-gaussian noise based on the family of Scale Mixtures of Skew-normal (SMSN) distributions, which includes the skew- t , the skew-gaussian, and the gaussian noises as particular cases. The Lorenz, Generalized Lorenz, Proto–Lorenzand Rossler attractors driven by skew- t distributed noise are considered. Results show that the chaotic regime’s behavior is influenced by the freedom parameter degrees of skew- t noise, increasing the noise variance. This paper concludes that noise intensity increases by rescaling the skew- t distribution at zero mean, rather than by increasing the asymmetry parameter.

This paper presents a new chaotic system that exhibits a two wing (2W) chaotic attractor in its integer order dynamics, three-wing (3W) and four-wing (4W) chaotic attractors in its fractional-order (FO) dynamics, and an eight-wing (8W) attractor in its nonautonomous fractional dynamics. An interesting feature of the proposed system is that two distinct periodic orbits coexist with a strange attractor that gradually evolves into a 4W attractor. The asymmetry, dissimilarity and topological structure of this proposed system with respect to those available in literature, manifest increased irregularity, which in turn indicate more chaos. Besides, the authors have drawn its comparison with various well-known fractional-order chaotic systems (FOCS)s to prove its enhanced features in terms of higher Lyapunov Exponent, fractional order orbital velocities, bandwidth, density, range of dynamical behaviour, etc. A control scheme is proposed to enable switching hybrid synchronisation between the 8W nonautonomous FOCS and the 4W autonomous FOCS, using the former as master and the latter as slave. This work throws light on the potential practical applicability of the proposed system by designing a circuit using minimum circuit components possible, thus signifying the objectives of the paper are finally achieved.

This paper is a continue work of [4,5]. In the previous two papers, we studied the Cauchy problem of the multi-dimensional compressible Euler equations with time-depending damping term - μ ( 1 + t ) λ ρ u , where λ≥0 and μ > 0 are constants. We have showed that, for all λ≥0 and μ>0, the smooth solution to the Cauchy problem exists globally or blows up in finite time. In the present paper, instead of the Cauchy problem we consider the initial-boundary value problem in the half space ℝ d + with space dimension d = 2,3. With the help of the special structure of the equations and the fluid vorticity, we overcome the difficulty arisen from the boundary effect. We prove that there exists a global smooth solution for 0 ≤ λ

In this paper, we construct a new two-dimensional convergent scheme to solve Cauchy problem of following two-dimensional scalar conservation law { ∂ t u + ∂ x f ( u ) + ∂ y g ( u ) = 0 , u ( x , y , 0 ) = u 0 ( x , y ) . In which initial data can be unbounded. Although the existence and uniqueness of the weak entropy solution are obtained, little is known about how to investigate two-dimensional or higher dimensional conservation law by the schemes based on wave interaction of 2D Riemann solutions and their estimation. So we construct such scheme in our paper and get some new results.

This review paper aims at answering a basic question on the sign of Lyapunov exponents. A few recent papers reported hyperchaotic system having the sign of Lyapunov exponents as (+, +, −, −). Such (+, +, −, −) sign of Lyapunov exponents is in contradiction with the well known (+, +, 0, −) sign of Lyapunov exponents for a 4-D hyperchaotic system. This paper thus discusses various issues related to Lyapunov exponents and proves that the reported sign of (+, +, −, −) Lyapunov exponents is actually (+, +, 0, −) or (+, 0, −, −). This clarification is very important and essential since Lyapunov exponents are the only quantitative measure for the existence of hyperchaos. Three different algorithms are used for calculating the Laypunov exponents to prove the actual sign of Lyapunov exponents for a hyperchaotic system.

The paper explores an eco-epidemiological model with weak Allee in predator, and the disease in the prey population. We consider a predator-prey model with type II functional response. The curiosity of this paper is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We perform the local and global stability analysis of the equilibrium points and the Hopf bifurcation analysis around the endemic equilibrium point. Further we pay attention to the chaotic dynamics which is produced by disease. Our numerical simulations reveal that the three species eco-epidemiological system without weak-Allee induced chaos from stable focus for increasing the force of infection, whereas in the presence of the weak-Allee effect, it exhibits stable solution. We conclude that chaotic dynamics can be controlled by the Allee parameter as well as the competition coefficients. We apply basic tools of non-linear dynamics such as Poincare section and maximum Lyapunov exponent to identify chaotic behavior of the system.

In a published paper of Journal of Chaos, Solitons and Fractals, some miss prints were found. One is about the calculation of the solution and also about the expression of the solution using Green’s function of the fractional differential equation studied in this paper; The others are for the properties for Green’s function, these miss prints affected the deriving of the main results in Khan’s paper, some corrections for Khan’s paper should be needed. In this paper, we make some corrections and give the correct proof proceeding for the results, a new example is given to validate part of the proven results.

The paper addresses the mathematical analysis of the fishery model in the context of the fractional derivative operator. We use the Caputo–Fabrizio derivative in the investigations. We first prove the fishery model is biologically well definite by proposing the existence and the uniqueness of its solution. The main objective of this paper is to study the dynamics of the predator and the prey in the fishery model when the fractional-order derivative is used. Notably, we analyze the impact of the fractional-order derivative on the dynamics of the fishery model explicitly. To answer this issue, we introduce a new numerical scheme based on the discretization of the fractional integral associated with the Caputo–Fabrizio derivative. The numerical simulations of the solutions of the fractional model are intended to illustrate the numerical scheme presented in our paper. We finish by analyzing the local and global asymptotic stability of the equilibrium points using the Jacobian matrix and the Lyapunov direct method. The Lyapunov function is constructed using standard construction. We notice here that the solutions of the fractional fishery model with different values of the orders describes a cycle when we depict them in three dimensional spaces in time, and furthermore the marine reserves ensure the sustainability of fractional system.

In this paper we report for the first time on the binary generalized synchronization, when for the certain values of the coupling strength two unidirectionally coupled dynamical systems generating the aperiodic binary sequences are in the generalized synchronization regime. The presence of the binary generalized synchronization has been revealed with the help of both the auxiliary system approach and the largest conditional Lyapunov exponent calculation. The mechanism resulting in the binary generalized synchronization has been explained. The finding discussed in this paper gives a strong potential for new applications under many relevant circumstances.

This paper is concerned with a singular limit for the one-dimensional compressible radiation hydrodynamics model. The singular limit we consider corresponds to the physical problem of letting the Bouguer number infinite while keeping the Boltzmann number constant. In the case when the corresponding Euler system admits a contact discontinuity wave, Wang and Xie (2011) [12] recently verified this singular limit and proved that the solution of the compressible radiation hydrodynamics model converges to the strong contact discontinuity wave in the L∞-norm away from the discontinuity line at a rate of e 1 4 , as the reciprocal of the Bouguer number tends to zero. In this paper, Wang and Xie's convergence rate is improved to e 7 8 by introducing a new a priori assumption and some refined energy estimates. Moreover, it is shown that the radiation flux q tends to zero in the L∞-norm away from the discontinuity line, at a convergence rate as the reciprocal of the Bouguer number tends to zero.

In this paper, we consider a time fractional-order derivative for a diffusive mussel–algae model. The existence of pattern formation was the subject of interest of many previous research works in the case of the diffusive mussel–algae model. Examples include the Turing instability, Hopf bifurcation, Turing-Hopf bifurcation, and others. The presence of the time–fractional–order derivative never been investigated in this model. Next to it ecological relevant, it can generate some important patterns. One of these patterns is produced by the presence of the Turing-Hopf bifurcation. Therefore, our main interest is to analyze the effect of the time fractional–order derivative on the spatiotemporal behavior of the solution, which never been achieved for the mussel-algae model. Besides, Turing–Hopf was studied exclusively on the classical reaction-diffusion systems, where it was also considered for the diffusive mussel-algae model. Thus, our paper puts the fist steps on proving the existence of this type of codimension bifurcation on the diffusive systems with time fractional–order–derivative systems. Further, a suitable numerical simulations are used for confirming the theoretical obtained results.

This paper investigates the dynamic behavior of a diffusive plankton model with defensive and offensive effects in two cases. For the single compartment model, we first derive the sufficient conditions for the stability and Hopf bifurcation of coexisting equilibrium, which implies that the changes of defense and offense can cause oscillation of planktonic population. Then the properties of Hopf bifurcation are discussed by center manifold theorem. For the spatially extended model, we obtain the sufficient conditions for Turing instability and Hopf bifurcation. It is observed that spatial patterns put in place, under the interaction of diffusion, defense and offense. Finally, some numerical simulations are carried out to support the analytical results.

This paper is concerned with an operational matrix method based on the shifted Legendre cardinal functions for solving the nonlinear variable-order time fractional Schrodinger equation. The variable-order fractional derivative operator is defined in the Atangana–Baleanu–Caputo sense. Through the way, a new operational matrix of variable-order fractional derivative is derived for the shifted Legendre cardinal functions and used in the established method. More precisely, the unknown solution is separated into the real and imaginary parts, and then these parts are expanded in terms of the shifted Legendre cardinal functions with undetermined coefficients. These expansions are substituted into the main equation and the generated operational matrix is utilized to extract a system of nonlinear algebraic equations. Thereafter, the yielded system is solved to obtain an approximate solution for the problem. The precision of the established approach is examined through various types of test examples. Numerical simulations confirm that the suggested approach is high accurate in providing satisfactory results.

This paper aims to study the critical sectional area (CSA) of surge chamber considering the nonlinearity of head loss of diversion tunnel and steady output of turbine. Firstly, three basic equations for hydropower station with surge chamber are established. Four mathematical models for the derivation of CSA of surge chamber are constructed. Then, the stability of hydropower station with surge chamber is analyzed by Hopf bifurcation. Based on the critical stable state of hydropower station, the formulas for the CSA of surge chamber are derived. Finally, the verification and comparison of different CSAs are conducted. The correctness and rationality of obtained formulas are explained. The results indicate that, under load increase operation condition, both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can reduce the value of CSA of surge chamber and are favorable for the stability of hydropower station. Under load decrease operation condition, the rules are opposite. Under both load increase and load decrease operation conditions, the effect of the nonlinearity of head loss of diversion tunnel on CSA of surge chamber is much more significant than that of the nonlinearity of steady output of turbine. The formula for CSA of surge chamber considering both the nonlinearity of head loss of diversion tunnel and nonlinearity of steady output of turbine can be expressed as an amplification coefficient times of the Thoma formula. That formula has a higher precision than Thoma formula.

In this paper we are established the existence of positive solutions (EPS) and the Hyers-Ulam (HU) stability of a general class of nonlinear Atangana-Baleanu-Caputo (ABC) fractional differential equations (FDEs) with singularity and nonlinear p-Laplacian operator in Banach’s space. To find the solution for the EPS, we use the Guo-Krasnoselskii theorem. The fractional differential equation is converted into an alternative integral structure using the Atangana-Baleanu fractional integral operator. Also, HU-stability is analyzed. We include an example with specific parameters and assumptions to show the results of the proposal.

In this paper, the third order partial differential equation defined by Caputo fractional derivative with Atangana–Baleanu derivative has been investigated. The stability estimates are proved for the exact solution. Difference schemes for Crank–Nicholson finite difference scheme method is constructed. The stability of difference schemes for this problem is shown by Von Neumann method (Fourier analysis method). Numerical results with respect to the exact solution confirm the accuracy and effectiveness of the technique. The reproducing kernel function for the problem has been found.

In this paper, we propose a fractional form of two-dimensional generalized mythical bird, butterfly wings and paradise bird maps involving the fractional conformable derivative of Khalil’s and Atangana’s type, the Liouville–Caputo and Atangana–Baleanu derivatives with constant and variable-order. We obtain new chaotical behaviors considering numerical schemes based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Also, the dynamics of the proposed maps are investigated numerically through phase plots considering combinations of these derivatives and mixed integration methods for each map. The numerical simulations show very strange and new behaviors for the first time in this manuscript.

In this paper, we incorporate a nonlinear incidence rate and a logistic growth rate into a SIR epidemic model for a vector-borne disease with incubation time delay and Holling type II saturated treatment. We compute the basic reproduction number and show that it completely determines the local stability of the disease-free equilibrium. Sufficient conditions for the existence of backward bifurcation and Hopf bifurcation are also established. Furthermore, we determine the direction and stability of the Hopf bifurcation around the endemic equilibrium by means of the center manifold theory. Our study reveals that the model admits a Bogdanov–Takens bifurcation when the time delay and the maximal disease transmission rate are varied. Numerical simulations are presented to illustrate the dynamics of the model and to study the effects caused by varying the treatment rate and delay parameters.

In this paper, we provide an epidemic SIR model with long-range temporal memory. The model is governed by delay differential equations with fractional-order. We assume that the susceptible is obeying the logistic form in which the incidence term is of saturated form with the susceptible. Several theoretical results related to the existence of steady states and the asymptotic stability of the steady states are discussed. We use a suitable Lyapunov functional to formulate a new set of sufficient conditions that guarantee the global stability of the steady states. The occurrence of Hopf bifurcation is captured when the time-delay τ passes through a critical value τ*. Theoretical results are validated numerically by solving the governing system, using the modified Adams-Bashforth-Moulton predictor-corrector scheme. Our findings show that the combination of fractional-order derivative and time-delay in the model improves the dynamics and increases complexity of the model. In some cases, the phase portrait gets stretched as the order of the derivative is reduced.

Latterly, variable carrying capacity in the predator–prey population models has been widely studied to provide more realistic understanding about population dynamics. In the present paper, we discuss the bio-economic harvesting of an intraguild fishery model where the logistic carrying capacities of both predator and prey fish populations are variable based on the shared biotic resource. Distinct from most fishery models, we incorporate the independent harvesting strategies on the fisheries with different harvesting efforts by assuming them to have different economic values. In our model, the prey fish population is assumed to be infected directly by an anthropogenic toxicant from the environment while the predator is infected indirectly when they feed on prey. We investigate the local stability of trivial equilibria with Routh–Hurwitz criterion and the global stability of non-trivial equilibrium by constructing a Lyapunov function. Bifurcation and numerical analyses are presented to show distinctive result that a Hopf bifurcation occurs with respect to harvesting parameter instead of resource enrichment parameter as in the literature. Bionomic equilibrium is explored with several possibilities and restrictions. The nontrivial bionomic equilibrium is found to depend critically on the resource density. Finally, the optimal harvesting policy of the three dimensional model is derived by utilizing Pontryagin Maximum Principle. From this study, harvesting parameter is a crucial factor for the stability of the intraguild system. The objective is to obtain the optimal sustainable yield that provides maximum monetary profit while conserving the marine ecosystem.

This paper focuses on providing a novel high-order algorithm for the numerical solutions of fractional order Volterra integro-differential equations using Atangana–Baleanu approach by employing the reproducing kernel approximation. For this purpose, we investigate couples of Hilbert spaces and kernel functions, as well as, the regularity properties of Atangana–Baleanu derivative, and utilize that the representation theorem of its solution. To remove the singularity in the kernel function, using new Atangana–Baleanu approach the main operator posses smoothing solution with a better regularity properties and the reproducing kernel algorithm is designed for the required equation. The convergence properties of the proposed algorithm are also studied which proves that the new strategy exhibits a high-order of convergence with decreasing error bound. Some numerical examples of single and system formulation illustrate the performance of the approach. Summary and some notes are also provided in the case of conclusion and highlight.

In this research paper, we plan to study the relation among concentration of greenhouse gases, human population and environmental tax. For this purpose, a deterministic nonlinear mathematical model is proposed and analyzed in regard to the boundedness and persistence of its solutions, equilibria and their stabilities. Whenever, a policy is implemented, an automated delay comes into existence because outcomes of application of any policy always take some time to become visible. To see the effects of delay, analyses for the stability and direction of Hopf bifurcation for delay system are done. Critical value of the delay parameter is calculated theoretically and numerically, and then verified graphically. For verifications and descriptions of analytical findings, numerical simulations are performed. Graphical comparisons for different values of various parameters provides us some realistic and interesting results. Overall, model analysis shows that the increasing level of greenhouse gases can be controlled by applying environmental tax and imposing some restrictions on corruption.

The paper provides a theoretical analysis of some local bifurcations in the fractional Chen system. Contrary to the integer-order case, basic bifurcation properties of the fractional Chen system are shown to be qualitatively different from those described previously for the fractional Lorenz system. Further, the fractional Hopf bifurcation in the Chen system is expressed rigorously with respect to general entry parameters. Based on these observations, some particularities of the fractional dynamics of the Chen system are documented and its chaotic behavior for low derivative orders is discussed.

Fractional-order systems are utilized widely to model material or physical systems with the properties of memory, non-locality and history-dependent, thus massive studies have been drawn on their dynamics during the past decade. In this paper, we are motivated to study stochastic dynamics in a fractional-order system subjected to Gaussian noise. The statistical complexity measure (SCM), an approach of information theory, has been well defined and calculated in the fractional-order Langevin model based on its Shannon entropy, by which the stochastic resonance (SR) response has been detected and measured. To compare, we further derive the equivalent system by means of the minimum mean square error rule to estimate the signal-to-noise ratio (SNR). It has been found that the SCM and the SNR depict analogous profile and exhibit peaks almost at the same level of noise. Varying the signal frequency, the SCM even quantifies Multiple-SR that SNR does not measure in the fractional-order Langevin model. Thus, the SCM may be utilized as an alternative candidate to measure SR response in bistable fractional-order systems.

In this paper, we propose and study a Logistic model with feedback control of fractional order that is extended directly from a system of ordinary differential equations. Uniform asymptotic stability of this model is established based on an appropriate Lyapunov function and an important consequence of this result; we present a simple proof for the global stability of the original system of ordinary differential equations. Besides, to numerically solve and simulate the proposed fractional model, unconditionally positive nonstandard finite difference schemes are constructed and analyzed. Finally, the numerical simulations obtained by the constructed nonstandard finite difference schemes are compared with the Grunwald–Letnikov scheme to reveal that the proposed scheme is convenient for solving the proposed model and confirm the validity of the established results.

With the development of nonlinear science, it is found that the fractional differential equation can more accurately describe the variation of natural phenomena.Therefore, the study of fractional differential equations and their boundary value problems is of great significance for solving nonlinear problems.The Dirichlet function, as an abstract mathematical model, has many unique properties in calculus.It points out special circumstances when describing many mathematical concepts, It can also be used to construct counterexamples in calculus and to clarify many fuzzy concepts to deepen the understanding of mathematical concepts.Therefore, this paper mainly studies the necessary and sufficient conditions for the controllability of fractional linear differential systems in abstract space.The finite difference decomposition method of fractional calculus in the abstract space for the Dirichlet function equation is also studied.The existence of solutions for boundary value problems of fractional differential equations with Dirichlet boundary value condition in the abstract space is discussed by using the critical point theory.

In this paper, we studied a diffusive predator-prey model with competitive interference and Crowley-Martin type functional response. The conditions for local stability of coexisting equilibrium are given by analyzing the eigenvalue spectrum. By using delay as bifurcation parameter, conditions for occurrence of Hopf bifurcation are also given. The property of bifurcating period solutions is investigated by calculating the normal form. Some numerical simulations are performed to support our theoretical result. Our conclusions show that diffusion and delay are two factors that should be considered in establishing the predator-prey model, since they can induce spatially bifurcating period solutions.

In this paper, a delayed predator-prey model with the presence of a social behavior for the prey population has been investigated. A new functional response is obtained. We studied the effect of the herd shape for the prey population on the prey and predator equilibrium densities. The analysis of the system has been also established where the boundedness, stability, Hopf bifurcation, stability of Hopf bifurcation are obtained.

In this paper, we present an upwind finite volume method to solve the convection-diffusion equations with Dirichlet boundary on rectangular mesh. By utilizing the technique of element-by-element analysis, the stability of the method has been proved and the H1-norm error estimate is presented. Furthermore, we provide the proofs of the maximum principle and L∞-norm error estimate. Finally, some numerical experiments are provided to confirm our theoretical results.

In this paper we construct the higher order Heisenberg supermagnet models in (2+1)-dimensions by introducing different auxiliary matrix variables. We investigate their structure and integrability with respect to two constraints. By virtue of the gauge transformation, we derive their gauge equivalent counterparts which are the super and fermionic higher order nonlinear Schrodinger equations in (2+1)-dimensions. Furthermore, we obtain their corresponding Backlund transformations.

The dynamics of the language competition model is considered in this paper. The classical system is converted to non-integer order case by replacing the second-order partial derivative with the Riesz fractional derivative. A well-known numerical approximation methods based on the Fourier spectral algorithm in space and the third-order exponential time-differencing scheme are formulated to numerically simulate the three component fractional-in-space reaction-diffusion system in one and high dimensions for different values of α. Numerical results indicate α ∈ (1, 1.5] as the key control parameter that can influence the coexistence of various speakers over a period of time.

In this paper, a new kind of alternating direction implicit (ADI) Crank-Nicolson-type orthogonal spline collocation (OSC) method is formulated for the two-dimensional fractional evolution equation with a weakly singular kernel arising in the theory of linear viscoelasticity. The novel OSC method is used for the spatial discretization, and ADI Crank-Nicolson-type method combined with the second order fractional quadrature rule are considered for the temporal component. The stability of proposed scheme is rigourously established, and nearly optimal order error estimate is also derived. Numerical experiments are conducted to support the predicted convergence rates and also exhibit expected super-convergence phenomena.

In this paper, we present a nonlinear fractional order model in the Caputo sense to explore and simulate the TB dynamics. Using the TB confirmed notified cases from the year 2002 to 2017 in Khyber Pakhtunkhwa, Pakistan, we estimate the model parameters and demonstrate that the proposed fractional model provides a good fit to the real data. Initially, we compute the basic reproduction number and the model equilibria. Then, the existence and uniqueness of the model are shown via generalized mean value theorem. Further, we explore the local and global stability of the disease free equilibria in fractional environment. Finally, numerical results are obtained in order to validate the importance of the arbitrary order derivative and theoretical results. We conclude that the fractional epidemic model is more generalized than the classical model, which give more information about the disease dynamics and give a good agreement to the real data of TB infection.

The objective of this paper is to investigate the question of modifying a given generalized Bessel sequence to yield a generalized frame or a tight generalized frame by finite extension. Some necessary and sufficient conditions for the finite extensions of generalized Bessel sequences to generalized frames or tight generalized frames are provided, and every result is illustrated by the corresponding example.

In this paper, we consider the mathematical analysis and numerical simulation of time-fractional multicomponent systems. Here, the classical time derivatives in such systems are replace with the Atangana–Baleanu fractional derivative in the sense of Caputo. This derivative is found useful in the sense that it combines both the non-local and nonsingular kernels in its formulation. A two-step family of Adams–Bashforth method is derived for the approximation of the Atangana–Baleanu derivative. Numerical experiments presented for different instances of α, 0

Stability and Hopf bifurcation of a class of delayed complex-valued neural networks are investigated in this paper. First, using proper translations and coordinate transformations, we decompose the activation functions and connection weights into their real and imaginary parts, so as to construct an equivalent real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided through normal form theory and central manifold theorem. In the end, some numerical simulations are given to illustrate the correctness of the results.

In this paper, we numerically study a nonlinear time-fractional reaction-diffusion equation involving the Caputo and Atangana–Baleanu fractional derivatives of order α ∈ (0, 1). A novel algorithm known as the Laplace Adams–Bashforth method is formulated for the approximation of these derivatives. In the simulation framework, a tri-tropic food chain system is considered in which the classical time-derivatives are replaced with non-integer order derivatives. Mathematical analysis of the main system is examined for both stability and Hopf-bifurcations to occur. Numerical simulation results show the existence of chaotic behaviours and spatiotemporal oscillations as well as the emergence of some Turing patterns (such as, spots and stripes) in two-dimensional space.

In this paper, we sought and presented an 8-Dimensional deterministic mathematical COVID-19 dynamic model that accounted for the global stability analysis of the role of dual-bilinear treatment protocols of COVID-19 infection. The model, which is characterized by human-to-human transmission mode was investigated using dual non-pharmaceutical (face-masking and social distancing) and dual pharmaceutical (hydroxylchloroquine and azithromycin) as control functions following the interplay of susceptible population and varying infectious population. First, we investigated the model state-space and then established and computed the system reproduction number for both off-treatment ℜ 0 ( 1 ) = 10.94 and for onset-treatment ℜ 0 ( 2 ) = 3.224 . We considered the model for off-treatment and thereafter by incorporating the theory of LaSalle's invariant principle into the classical method of Lyapunov functions, we presented an approach for global stability analysis of COVID-19 dynamics. Numerical verification of system theoretical predictions was computed using in-built Runge-Kutta of order of precision 4 in a Mathcad surface. The set approach produces highly significant results in the main text. For example, while rapid population extinction was observed by the susceptible under off-treatment scenario in the first t f ≤ 18 days, the application of non-pharmaceuticals at early stage of infection proved very effective strategy in curtailing the spread of the virus. Moreso, the implementation of dual pharmacotherapies in conjunction with non-pharmaceuticals yields tremendous rejuvenation of susceptible population ( 0.5 ≤ S p ( t ) ≤ 3.143 c e l l s / m l 3 ) with maximal reduction in the rates of isolation, super spreaders and hospitalization of the infectives. Thus, experimental results of investigation affirm the suitability of proposed model for the control and treatment of the deadly disease provided individuals adheres to treatment protocols.

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.

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.

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.

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.

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.