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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.

This paper develops the analytical conditions for the existence of periodic solutions of a second-order,time-delayed, discontinuous dynamical system. A sample model consists of two linear delayed sub-systems with a switching boundary. The defined G-functions for the delayed, discontinuous systems are introduced, and sufficient and necessary conditions for a flow crossing, sliding and grazing along the switch boundary are developed for such a delayed, discontinuous system. Furthermore, nine (9) regular basic mappings in phase plane and thirty-three (33) delay-related mappings for the second-order, time-delayed, discontinuous systems are classified. Constraint equations are predicted analytically for two periodic orbits with initial functions provided posteriorly. Finally, three numerical examples are illustrated to verify the existence of generalized slowly oscillating periodic orbits without and with sliding portions. This paper improves and extends motion switchability conditions at the boundary in discontinuous dynamical systems without delay.

A three factor variance model introduced by Gatheral in 2008, called the double mean reverting (DMR) model, is well-known to reflect the empirical dynamics of the variance and prices of options on both SPX and VIX consistently with the market. One drawback of the DMR model is that calibration may not be easy as no closed form solution for European options exists, not like the Heston model. In this paper, we still use the double mean reverting nature to extend the Heston model and study the pricing of variance swaps given by simple returns in discrete sampling times. The constant mean level of Heston’s stochastic volatility is extended to a slowly varying process which is specified in two different ways in terms of the Ornstein-Uhlenbeck (OU) and Cox-Ingersoll-Ross (CIR) processes. So, two types of double mean reversion are considered and the corresponding models are called the double mean reverting Heston-OU model and the double mean reverting Heston-CIR models. We solve Riccati type nonlinear equations and derive closed form exact solutions or closed form approximations of the fair strike prices of the variance swaps depending on the correlation structure of the three factors. We verify the accuracy of our analytic solutions by comparing with values computed by Monte Carlo simulation. The impact of the double mean reverting formulation on the fair strike prices of the variance swaps are also scrutinized in the paper.

In this paper, we study the effect of stochastic noise on the virus infection model with nonlytic immune response. Firstly, the mathematical model describing the virus infection with nonlytic immune response is presented. The basic reproduction number is derived and the stability of disease-free state E0 and disease state E1 are analysed. Then the threshold conditions for extinction and persistence of the virus are derived by the rigorous theoretical proofs. It is found that when the noise is large enough, the virus will die out without constraint. When the noise is small, the virus will become extinct under the condition R 0 * 1 and persistence under R 0 * * > 1 . Besides, the upper bound and lower bound for persistence have been given. At last, some numerical simulations are carried out to support our results. The conclusion of this paper could help provide the theoretical basis for the further study of the virus infection.

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.

Stability in distribution, implying the existence of the invariant probability measure, is an important measure of stochastic hybrid system. However, the effect of Levy jumps on the stability in distribution is still unclear. In this paper, we consider a n-species competitive Lotka–Volterra model with Levy jumps under regime-switching. First, we prove the existence of the global positive solution, obtain the upper and lower boundedness. Then, asymptotic stability in distribution as the main result of our paper is derived under some sufficient conditions. Finally, numerical simulations are carried out to support our theoretical results and a brief discussion is given.

In this paper, we discuss the conditions of existence and uniqueness of solutions to a certain class of ordinary differential equations involving Atangana–Baleanu fractional derivative. Benefiting from the Gronwall inequality in the frame of Riemann–Liouville fractional integral, we establish a Gronwall inequality in the frame of Atangana–Baleanu fractional integral. Then, we study the stability of such equations in the sense of Ulam.

In this paper, we present a numerical technique for solving fractional Sturm–Liouville problems with variable coefficients subject to mixed boundary conditions. The proposed algorithm is a spectral Galerkin method based on fractional-order Legendre functions. Tedious manipulation of the series appearing in the implementation of the method have been carried out to obtain a system of algebraic equations for the coefficients. Our findings demonstrate the possibility of having no eigenvalues, finite number of eigenvalues or infinite number of eigenvalues depending on the fractional order. The convergence and effectiveness of the present algorithm are demonstrated through several numerical examples.

In this paper, we give analytical solutions of a fractional-time wave equation with memory effect and frictional memory kernel of Mittag–Leffler type via the Atangana–Baleanu fractional order derivative. The method of separation of variables and the Laplace transform has been used to obtain the exact solutions for the fractional order wave equations. Additionally, we present analytical solutions considering the Caputo–Fabrizio fractional derivative with exponential kernel. We showed that the solutions obtained via Caputo–Fabrizio fractional order derivative were a particular case of the solutions obtained with the new fractional derivative based in the Mittag–Leffler law.

In this paper, dual-complex k-Fibonacci numbers are defined. Also, some algebraic properties of dual-complex k-Fibonacci numbers which are connected with dual-complex numbers and k-Fibonacci numbers are investigated. Furthermore, the Honsberger identity, the d’Ocagne’s identity, Binet’s formula, Cassini’s identity, Catalan’s identity for these numbers are given.

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.

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.

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 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 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.

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.

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 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 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.

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.

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).

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”.

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.

In this paper, we study the bifurcation and stability of a ratio-dependent predator-prey model with nonconstant predator harvesting rate. The analysis is carried out both analytically and numerically. We determine stability and dynamical behaviours of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Fold, Hopf, Cusp, and Bogdanov–Takens bifurcations. We use the numerical software MATCONT, to compute curves of equilibria and to compute several bifurcation curves. We especially approximate a family of limit cycles emanating from a Hopf point. Our results generalize and improve some known results and show that the model has more rich dynamics than the ratio-dependent predator-prey model without harvesting rate.

A stochastic three species food chain model with general L e ´ vy jumps is investigated in the paper. The criterion on the global stability in the mean with probability one for each species is established. The results show that the dynamics of the model can be significantly changed by L e ´ vy jumps, and strongly indicate that the general jump N(dt, du) is more complicated than compensated jump N ˜ ( dt , du ) .

In this paper, bicomplex Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex Fibonacci quaternions which are connected with bicomplex numbers and Fibonacci numbers are investigated. Furthermore, Binet’s formula, Cassini’s identity, Catalan’s identity for these quaternions and real representation of these quaternions are given.

Recently a new fractional differentiation was introduced to get rid of the singularity in the Riemann-Liouville and Caputo fractional derivative. The new fractional derivative has then generate a new class of ordinary differential equations. These class of fractional ordinary differential equations cannot be solved using conventional Adams–Bashforth numerical scheme, thus, in this paper a new three-step fractional Adams–Bashforth scheme with the Caputo–Fabrizio derivative is formulated for the solution linear and nonlinear fractional differential equations. Stability analysis result shows that the proposed scheme is conditionally stable. Applicability and suitability of the scheme is justified when applied to solve some novel chaotic systems with fractional order α ∈ (0, 1).

In this paper the chaotic behavior of second-order, discontinuous systems with a pseudo-equilibrium point on a discontinuity surface is analyzed. The discontinuous system is piecewise linear and approximated to a non-smooth continuous system. The discontinuous term is represented by a sign function that is replaced by a saturation function with high slope. Some of the conditions that determine the chaotic behavior of the approximate system are formally established. Besides, the convergence of its chaotic solutions to those of the discontinuous system is shown. Several bifurcation diagrams of both systems show the similarity of their dynamical behavior in a wide parameter range, and particularly for a parameter region determined from the application of the Melnikov technique to non-smooth systems, where a chaotic behavior can be displayed.

The vaccination, latent and relapse period are three important factors affecting the whole disease development. In this paper, we propose an SVEIR epidemic model with continuous age-dependent vaccination, latency and relapse, at the same time, the nonlinear incidence rate is also considered. Uniform persistence of the model is proved by reformulating it as the so called Volterra integral equations. The basic reproduction number R 0 , which completely determines the global dynamics of the model, is derived. By using Lyapunov functionals, the global stability of the equilibria is obtained. Namely, the disease-free equilibrium is globally asymptotically stable if R 0 1 , while if R 0 > 1 the endemic equilibrium is globally asymptotically stable. Finally, two numerical examples support our main analytical results.

In this paper, an age structured SIRS epidemic model with age of recovery is studied which allows the removed individuals to become susceptible again when they lose the protection property as the time goes. The age structured model is reformulated as an abstract non-densely defined Cauchy problem and the expression of the basic reproduction number, R 0 , is obtained. If R 0 1 , the model only has the disease-free steady state and it is global stability. If R 0 > 1 , besides the disease-free steady state the model also has an endemic steady state. By analyzing the associated characteristic equation, the existence of a local Hopf bifurcation is proved under certain conditions. Also, we considered the global continuation of the local Hopf bifurcation. Finally, some numerical simulations are carried out to illustrate the theoretical results.

In this paper, the invariance properties of time fractional coupled Drinfeld–Sokolov–Satsuma–Hirota (DSSH) equations have been investigated using the Lie group analysis method. In this regard a systematic research to derive Lie point symmetries of time fractional coupled DSSH equations is performed. Using the Lie group analysis method, the vector fields and the symmetry reduction of the time fractional coupled DSSH equations are obtained. It is shown that, the time fractional coupled DSSH equations can be reduced to the fractional coupled ordinary differential equations by using fractional Erdelyi–Kober differential operator with Riemann–Liouville derivative. Finally using new conservation theorem with formal Lagrangian, the new conserved vectors are well constructed with a detailed derivation, which present the conservation analysis for time fractional coupled Drinfeld–Sokolov–Satsuma–Hirota (DSSH) equations.

In this paper, two numerical techniques are introduced to study numerically the general fractional advertising model. This system describes the flux of the consumers from unaware individuals group to aware or purchased group. The first technique is an asymptotically stable difference scheme, which was structured depending on the nonstandard finite difference method. This scheme preserves the properties of the solutions of the model problem as the positivity and the boundedness. The second technique is the Jacobi–Gauss–Lobatto spectral collocation method which is exponentially accurate. By means of this approach, such problem is reduced to solve a system of nonlinear algebraic equations and are greatly simplified the problem. Numerical comparisons to test the behavior of the used techniques are run out. We conclude from the computational work that: the Jacobi–Gauss–Lobatto spectral collocation method is more accurate whereas the nonstandard finite difference method requires less computational time.

Let M be a closed connected smooth Riemannian manifold with dimM ≥ 2 and let f: M → M be a diffeomorphism. In the paper, we show that C1 generically, if a diffeomorphism f does not present a homoclinic tangency then it is weak Lebesgue measure expansive and, as an example, we find a partially hyperbolic diffeomorphism which is not weak measure expansive. Moreover, for a surface, if a diffeomorphism f has a homoclinic tangency then there is a diffeomorphism g C1 close to f such that g is not weak measure expansive.

This paper is concerned with existence, uniqueness and asymptotic behavior of traveling wave fronts for a generalized Fisher equation with nonlocal delay. The existence of traveling wave fronts is established by linear chain trick and geometric singular perturbation theory. The strategy is to reformulate the problem as the existence of a heteroclinic connection in R 4 . The problem is then tackled by using Fenichel’s invariant manifold theory. The asymptotic behavior and uniqueness of traveling wave fronts are also obtained by using standard asymptotic theory and sliding method.

In this paper, we establish sufficient conditions for the existence and uniqueness of the solution in functional weighted Sobolev space for a class of initial-boundary value problems with integral condition for a class of nonlinear partial fractional reaction-diffusion (RD) equations. The results are established by using a priori estimate in Bouziani fractional spaces and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the nonlinear problem.

In this paper, we consider the existence of periodic solutions for discrete time periodic time-varying coupled systems on networks (DPTCSN). Some novel sufficient conditions are obtained to guarantee the existence of periodic solutions for DPTCSN, which have a close relation to the topology property of the corresponding network. Our approach is based on the continuation theorem of coincidence degree theory, generalized Kirchhoff’s matrix tree theorem in graph theory, Lyapunov method and some new analysis techniques. The approach is applied to the existence of periodic solutions for discrete time Cohen–Grossberg Neural Networks (CGNN). Finally, an example and numerical simulations are provided to illustrate the effectiveness of our theoretical results.

In this paper, we study the Chebyshev’s property of the three-dimensional vector space E = J 0 , J 1 , J 2 > , where J i ( h ) = ∫ H = h x i d x y and H ( x , y ) = 1 2 y 2 + V ( x ) is a hyperelliptic Hamiltonian of degree 7. Our main result asserts that in two specific cases, namely (a) V ′ ( x ) = x 3 ( x − 4 7 ) ( x − 1 ) 2 and (b) V ′ ( x ) = x ( x − 2 7 ) ( x − 1 ) 4 , E is an extended complete Chebyshev system. To this end, we use the criterion and the tools developed by Grau–Manosas–Villadelprat in Trans. Amer. Math. Soc. in 2011.

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \xi\sigma (u_t(x)) \dot F(t,x), \end{equation*} where $\xi$ is a positive parameter and $\sigma$ is a globally Lipschitz continuous function. The stochastic forcing term $\dot F(t,x)$ is white in time but possibly colored in space. The operator $\mathcal{L}$ is a non-local operator. We study the behaviour of the solution with respect to the parameter $\xi$, extending the results in \cite{FoonNual} and \cite{Bin}, Comment: 14 pages; updated versionof the original paper titled "Large time behaviour for the non-linear fractional stochastic heat equation"