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### International Journal of Dynamical Systems and Differential Equations

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 International Journal of Dynamical Systems and Differential Equations (61 papers in press) Regular Issues Results on approximate controllability of second-order non-autonomous integrodifferential inclusions via resolvent operators   by M. Tamil Selvan, R. Murugesu Abstract: In this work, we establish a set of sufficient conditions for the approximate controllability for a class of non-autonomous second-order integrodifferential inclusions in Banach spaces. We establish our main results with the help of resolvent operators and Bohnenblust-Karlin's fi xed point theorem. Then we extend our study to second-order neutral systems with nonlocal conditions. An example is given to illustrate the main result. Keywords: Approximate controllability; Integrodifferential inclusions; Resolvent operators; Evolution equations; Nonlocal conditions. An Additive Separation of Variables 3D Solution to a Dynamical BVP for Neutron Cancer Therapy   by Nassar Haidar Abstract: We study the boundary-value problem (BVP) for irradiation of a rightrnparallelepipedal cancerous region in a (B/Gd) neutron cancer therapy (NCT)rncomposite-region setup by three mutually orthogonal, time-modulated, one-speed neutron beams. The technique of composite region coupling by a neutron source at a common boundary of different regions, that has been introduced in [1], is demonstrated to allow for an additive separation of variables (ASOV) regional neutron-density 3D wave solution to the posing four-regional boundary value problem (BVP). The beams, which may have different pulse shapes, have different modulation frequencies and variable relative time delays. Keywords: Accelerator Based Modulated Neutron Sources; Four-Regional Boundary-ValuernProblems; Additive Separation of Variables; One-Speed Neutron Diffusion; Neutron-Densityrn3D Wave; Dynamical NCT; Laplace Transforms; Three Mutually Orthogonal Neutron Beams. Lie group analysis for heat transfer in flow of second grade fluid   by Tarik Amtout, Houda Biyadi, Mustapha Er-Riani, Mustapha El Jarroudi Abstract: In this paper, the Lie symmetry analysis is performed for the heat transfer flow of a second grade fluid between two parallel heated plates. The symmetries for the coupled equations are given. The exact solutions and similarity reductions generated from the symmetry transformations are provided. Furthermore, translational symmetries were utilized to find a family of travelling wave solutions of the governing nonlinear problem. Keywords: Lie group analysis; Second grade fluid; Heat transfer flow; Similarity reduction; Travelling wave solutions. Lyapunov-type inequalities on fractional q-difference Schrodinger equation with the Woods-Saxon potential   by Kuikui Ma, Zhenlai Han Abstract: In this paper, the integer order Schr\"{o}dingerrnequation with the Woods-Saxon potential is extended to thernfractional $q$-difference field. We establish the Lyapunov-typerninequalities for nonlinear fractional $q$-difference equations, tornthe best of our knowledge, which is the first work dealing withrnLyapunov-type inequalities for nonlinear fractional $q$-differencernequations. Results in this paper even are new in integer order case.rnMoreover, we further investigate the two-point boundary valuernproblem of nonlinear fractional $q$-difference Schr\"{o}dingerrnequation with the Woods-Saxon potential. By applying thernLeray-Schauder degree theory, we get a sufficient condition of thernexistence of solutions that is relatively easy to verify comparedrnwith the result of existing literature. By utilizing thernLeggett-Williams fixed point theorem, an inequality is added to thernexistence condition of solutions of such problem studied in thernexisting literature, and we get the multiplicity of solutions ofrnthis problem. As applications, two examples are presented tornillustrate our main results. Keywords: Fractional $q$-difference equations; Lyapunov inequality; Boundary value problem. Global Dynamics of a Cancer Stem Cell Treatment Model   by Kristen Abernathy, Zachary Abernathy, Robert Dougherty-Bliss, Caleb Mayer, Heidi Whiteside Abstract: We provide global stability arguments for a cancer treatment model with chemotherapy and radiotherapy that accounts for the cancer stem cell hypothesis. Employing the method of localization of compact invariant sets, we resolve the global dynamics of the no-treatment, constant radiation, and combination chemotherapy and radiotherapy cases. In our analysis of the combination treatment model, we show that the presence of a chemotherapy agent lowers the required radiation strength for a globally asymptotically stable cure state. Keywords: cancer stem cells; global stability; cancer treatment; localization of compact invariant sets. Population Dynamic Caused by War Involvement via Fractional Derivative on Time Scales   by Mehdi Nategh, Dumitru Baleanu, Abdolali Neamaty, Bahram Agheli Abstract: In this work, a fractional derivative on time scales is discussed. Then by suggesting a new structure on the real line, we extend the objectivity of this derivative. A population dynamic problem caused by a confrontation or invasion is mentioned together with a model which led us to a non-homogeneous second order fractional PDE on time scales. Keywords: Time scales; Fractional dynamics; Population dynamic problem; War involvement. Front transition in higher order diffusion equations with a general reaction nonlinearity   by Samir Shamseldeen Abstract: In this paper, we investigate the wave front solutions of a class of higher order reaction-diffusion equations with a general reaction nonlinearity. Linear stability analysis with a modulated traveling wave perturbation is used to prove the existence of wave front solutions. We proved that the studied equation supports both monotonic translating front and patterned front solutions. Also, a minimal front speed and the condition for a transition between these front types (monotonic and patterned) are determined. Two numerical examples are discussed (the extended Fisher-Kolmogorov equation with two different reaction nonlinearities) to support the obtained results. Keywords: reaction-diffusion equations; traveling waves; Minimal front speed; pulled fronts. Role of harvesting in controlling chaos and disease propagation in predator-prey system with disease in prey   by Prodip Roy, Krishna Pada Das, Partha Karmakar, Seema Sarkar (Mondal) Abstract: The paper explores an eco-epidemiological model with harvesting in the species and the disease is circulating in the prey population. The curiosity of this paper is to consider the role of harvesting on chaotic dynamics. We perform the local and global stability analysis of the equilibrium points and the Hopf bifurcation analysis around the interior equilibrium point. Further we pay attention to the direction of Hopf Bifurcation. Our numerical simulations reveal that the three species eco-epidemiological system shows chaos in low level of infection. It is observed that when force of infection increases chaos becomes stable. We conclude that chaotic dynamics can be controlled by the harvesting parameter as well as the force of infection. We apply basic tools of non-linear dynamics such as Poincare section and maximum Lyapunov exponent to identify chaotic behavior of the system. Keywords: Disease in prey; stable; permanence; impermanence; chaos; Lyapunov exponent. Disappearance of limit cycle oscillations in a predator-prey model: role of mortality due to predation of infected prey   by Krishna Pada Das, Subhabrata Ghosh, Somnath Maiti Abstract: This present work establishes an eco-epidemiological model byrnthree element, namely a susceptible prey,an infected prey and preda-rntor.In this situation the prey population shows its migratory nature.rnIt is observed in real life situation that a predator dies due to preda-rntion of an infected prey or they can even survive after consumptionrnof an infected one. We make a general consideration on the growth ofrnthe predator population by consuming the infected prey population.rnWe found that if the predation of the infected prey has a negativerneffect on the growth of the predator population, then the predatorsrnmay be washed away from the system but it will survive if the intakernof the infected prey has a positive effect which boost up its growth.rnWe found specific conditions under the influence of predational effectsrnfor the stability of the system around the coexistence of steady state.rnHopf-bifurcation and persistence conditions are also worked out. We observe that simultaneously with an increase in the degree of infectionrnthe system loses its stability and indicate limit cycle oscillation. Wernobserve that the oscillatory coexistence of the species disappears andrnthe system reaches its steady state position regulated by the conver-rnsion efficiency of the predator. Finally we perform extensive numericalrnsimulations to establish our analytical approach. Keywords: Disease in prey; Malthus growth; non positive restricted parameter;rnstability; Hopf-bifurcation; permanence. First order fifth degree Fuchs differential equation with fixed critical points   by Khaled M'hamed-messaoud, Toufik Laadj, Arezki Kessi Abstract: In this paper, we are interested in studying the nonlinear differential equations of order one and fifth degree, whose general integral is uniform. We will give sufficient conditions, for the considered equations to be with fixed critical points. Keywords: Fuchs differential equations; fixed critical points; painlevé property.DOI: 10.1504/IJDSDE.2019.10013708  On the global existence of solutions to a class of semilinear fractional order system with delay   by Mengrui Xu, Shurong Sun Abstract: In this paper, we supplement the global existence of solutions to a class of nonlinear fractional order system with time delay by using the Leray-Schauder fixed point theorem, and improve conditions to make the estimate value of the above solution hold. An example as an application of our results is also given. Keywords: Fractional differential system; Initial value problem; Delay; Existence. Permanence and extinction for a stochastic two-species competitive system   by Yizhong Liu Abstract: This paper deals with a stochastic two-species competitive system. Some very verifiable criteria on the uniformly weakly persistent in the mean almost surely (a.s.) and extinction for each species are obtained. Some examples are given to illustrate our main theoretical findings. Our results are new and complement previously known results. Keywords: Stochastic Competitive model; Permanence;rnExtinction; Time delay. Existence and multiplicity of solutions for the fractional Schr   by Peiluan Li Abstract: In the present paper, we investigate the solutions of a fractional Schr Keywords: Fractional Schrodinger equations; Steep potential well; Infinitely many solutions. An Antiplane Electro-Elastic Contact Problem: Case Regularized Friction Law   by Mohamed Dalah Abstract: In this paper, we study an antiplane electro-elastic contact problem with regularized friction law. We suppose that the material is assumed to be electro-elastic and the foundation is assumed to be electrically conductive. In the first step, we write the mathematical model and its well-posedness. In the second step, we try to give the variational formulation and main results. Finally, we prove that the weak solution of the antiplane electro-elastic contact problem with regularized friction law is converge to the weak solution of problem P in [3] when the parameter of regularization $rho$ is very small. Keywords: Regularized friction law; Antiplane; Friction law; Variational formulation; Variational inequality; Weak solution. Intrinsic growth rates influence on the net economic rents of fishermen   by Meriem BENTOUNSI, Imane AGMOUR, Naceur ACHTAICH, Youssef EL FOUTAYENI Abstract: In this paper, a mathematical model consisting of Sardina pilchardus, Xiphias gladius and Bluefin tuna is considered. The differential system is based on the Lotka-Volterra scheme, considering a logistic growth for each population. Biological and bioeconomic equilibria of the system are obtained, and criteria for local stability and instability of the system are derived. This work is an attempt to study the effect of intrinsic growth coefficients changes on the net economic rents of several fishermen exploiting the three marine populations in the Atlantic coast of Morocco based on the parameters given by INRH (Institut National de Recherche Halieutique). To achieve this objective, we have to study the bioeconomic model. Numerical simulations are given to support the theoretical results. Keywords: Bioeconomic model; intrinsic growth coefficients; Stability analysis; Net economic rent; Generalized Nash equilibrium problem. Some Oscillation Theorems for Generalized Lienard Type System   by Tohid Kasbi, Vahid Roomi, Aliasghar Jodayree Akbarfam Abstract: In this work a generalized Li Keywords: Oscillation; Liénard system; Differential Equations. AXI-SYMMETRIC THERMOELASTIC STRESS ANALYSIS OF A THIN CIRCULAR PLATE DUE TO HEAT GENERATION   by Kishor Gaikwad Abstract: The aim of this work is to determine the temperature, displacement function, thermal stresses and thermal deflection of a thin circular plate defined as 0  r  a, 0  z  h under an unsteady temperature field due to internal heat generation within it. Initially, the plate is kept at an arbitrary temperature F(r, z). For times t > 0, heat is generated within the thin circular plate at a rate of g(r, z, t) W.m−3. The governing heat conduction equation has been solved by generalized finite Fourier transform and finite Hankel transform technique. The results are obtained in a series form in terms of Bessels functions. The results for temperature, displacement function, thermal stresses and thermal deflection have been computed numerically and are illustrated graphically. Keywords: Inverse thermoelastic problem; Thermal deflection; Circular plate; Heat generation; Thermal stresses; Axi-symmetric. MULTIPLE PERIODIC SOLUTIONS FOR SECOND ORDER IMPULSIVE DELAY DIFFERENTIAL EQUATIONS   by Fatima Dib, Naima Daoudi-Merzagui Abstract: In this paper we discuss the existence of periodic solutions for an impulsive nonautonomous second order delay differential equation. We obtain multiple periodic solutions by a variational approach. Keywords: Second order delay differential equation; periodic solution; impulses; mountain pass theorem. ERGODIC SHADOWING OF NON-AUTONOMOUS DISCRETE-TIME DYNAMICAL SYSTEMS   by Hadi Parham, Fateme Helen Ghane, Esmaeel Rezaali Abstract: The present work is concerned with the ergodic shadowing property of non-autonomous discrete-time dynamical systems. The concept of the ergodic shadowing was introduced by Fakhari and Ghane (2010) for a continuous onto map. Here, we will generalize the notion of ergodic shadowing to non-autonomous discrete-time dynamical systems. We show that every uniformly equicontinuous non-autonomous discrete-time system with the ordinary shadowing and topologically mixing properties enjoys the ergodic shadowing property. Moreover, we define some kind of specification property and investigate its relation to the ergodic shadowing property. Finally, we prove that every uniformly expanding non-autonomous discrete-time system satisfies the ergodic shadowing property. Keywords: non-autonomous dynamical systems; ergodic shadowing; pseudo orbital pecification. Global exponential convergence of CNNs with multi-proportional delays and leakage delays   by Changjin Xu Abstract: In this article, we deal with cellular neural networks (CNNs) with multi-proportional delays and leakage delays. Applying the differential inequality techniques, a new sufficient criterion which ensures the global exponential convergence of CNNs with multi-proportional delays and leakage delays is obtained. Numerical simulations are given to illustrate the obtained results. The theoretical predictions of this article are novel and complement some earlier publications. Keywords: Cellular neural networks; Leakage delay; Exponential convergence;rnProportional delay. Multivariate vector sampling expansion in shift-invariant subspaces   by Qingyue Zhang Abstract: Sampling theorems on a shift-invariant subspace are having a significant impact, since they avoid most of the problems associated with classical Shannon's theory. Vector sampling theorems on a shift-invariant subspace which are motivated by applications in multi-channel deconvolution and multi-source separation are active field of study. In this paper, we consider vector sampling theorems on a multivariate vector shift-invariant subspace. We give a multivariate vector sampling expansion on a multivariate vector shift-invariant subspace. Some equivalence conditions for the multivariate vector sampling expansion to hold are given. We also give several examples to illustrate the main result. Keywords: sampling theorems; vector sampling theorems; shift-invariant subspaces; super Hilbert space; frames. On invariant analysis, group classification and conservation laws of two component Novikov equation}   by Manjit Singh Abstract: The two-component Novikov equation is investigated for group classification and non-trivial local conservation laws. In addition to Lie group analysis, the existing classification of 4-dimensional Lie algebra is used to improve the classifications of Lie algebra of Novikov equations. Apart from this, the direct method is used in the construction of conservation laws using multipliers. Keywords: Lie symmetries; Group classifications; Conservation laws. A Convergence Computational Scheme for System of Integral Equation Using Finite Element Method   by Hamed Zeidabadi, Mohammad Heidari Abstract: In this paper, a computational scheme for extracting approximate solutions of system of integral equations is proposed. For this purpose, by considering the variational form of the problem and using finite element method, the system of integral equations are reduced to a system of algebraic equations, that are solved by an efficient algorithm. Also, the existence and uniqueness of the system of integral equations are illustrated and the convergence of the approximate solution to the exact solution is investigated. Finally, the effectiveness of the proposed method is discussed by comparing with the results of the given approaches in Babolian et al. [1] and Jafarian et al. [2]. Keywords: System of Volterra Integral Equation; Finite Element Method; Existence; Uniqueness. Approximate Controllability of Hilfer Fractional Sobolev Type Integrodifferential Inclusions with Nonlocal Conditions   by JinRong Wang Abstract: In this paper, we investigate approximate controllability of Hilfer fractional Sobolev type differential inclusions with nonlocal conditions. The main techniques rely on the fixed point theorem combined with the semigroup theory, fractional calculus, and multivalued analysis. An interesting example is provided to illustrate the obtained results. Keywords: Approximate controllability; Hilfer fractional derivative; Sobolev type differential inclusions; Semigroup theory. ON ERGODICITY OF MARKOVIAN MOSTLY EXPANDING SEMI-GROUP ACTIONS   by Azam Ehsani, Fateme Helen Ghane, Marzie Zaj Abstract: We consider finitely generated semigroup actions on a compact manifold and discuss their ergodic properties. We introduce Markovian mostly expanding semigroups and show that each $C^{1+\alpha}$ Markovian mostly expanding semigroup action is ergodic (with respect to the Lebesgue measure) whenever it is strongly transitive. Moreover, it is proved that each Markovian mostly expanding semigroup is non uniformly expanding. Our approach provides a large class of non-uniformly expanding semigroups. Keywords: finitely generated semigroup action; backward minimality; strong transitivity; ergodicity of semigroup actions; non-uniformly expanding property. Harvesting in tri-trophic food chain stabilizes the chaotic dynamics-conclusion drawn from Hastings and Powell model   by Binayak Nath, Krishna Pada Das Abstract: The paper explores a tri-trophic food chain model with harvesting in the species. The curiosity of this paper is to observe chaotic dynamics and its control. We perform the local stability analysis of the equilibrium points. The Hopf bifurcation analysis and global stability around the interior equilibrium point are also performed. Our numerical simulations reveal that the three species food chain model induces chaos from period-doubling, limit cycle and stable focus for increasing values of half saturation constant. We conclude that chaotic dynamics can be controlled by the harvesting parameter. We apply basic tools of non-linear dynamics such as Poincare section and Lyapunov exponent to identify chaotic behavior of the system. Keywords: Harvesting; Chaos; stable focus; Poincare map; Lyapunov exponent; persistence; Hopf bifurcation. Rough center in a 3-dimensional Lotka-Volterra system   by Yusen Wu Abstract: This paper identifies rough center for a Lotka-Volterra system, a 3-dimensional quadratic polynomial differential system with four parameters $h,n,lambda,mu$. The known work shows the appearance of four limit cycles, but the center condition is not determined. In this paper, we verify the existence of at least four limit cycles in the positive equilibrium due to Hopf bifurcations by computing normal forms. Furthermore, applying algorithms of computational commutative algebra we find Darboux polynomial and give a center manifold in closed form globally, showing that the positive equilibrium of center-focus is actually a rough center on a center manifold. Keywords: Rough center; 3-dimensional Lotka-Volterra system; Normal form theory. Lyapunov-Type Inequalities for m-point fractional boundary value problem   by Rabiaa Aouafi, Nacer Adjeroud Abstract: In this work, we establish some Lyapunov-type inequalities for the fractional boundary value problem of higher order under multi-point boundary conditions. In order to illustrate the results, we provide numerical example and we use this inequality to obtain an interval where linear combinations of certain Mettag-Leffler functions have no real zeros. Keywords: Fractional differential equation; Caputo's fractional derivative; Lyapunov inequalities; multi-point; Mittag-Leffler function. Existence and uniqueness of (w,c)-periodic solutions of semilinear evolution equations   by Makrina Agaoglou, Michal Feckan, Angeliki Panagiotidou Abstract: In this work we study the existence and uniqueness of (o; c)-periodicrnsolutions for semilinear evolution equations in complex Banach spaces. Keywords: (w,c)-periodic solutions; semilinear evolution equations; nonresonance conditions. Diagnosis of time-delay fractional systems using observer-based methods   by Halima Atitallah, Asma Aribi, Mohamed Aoun Abstract: In this paper, two model-based methods are considered for the diagnosis of time-delay fractional systems. Time-delay fractional Luenberger observer without unknown input and time-delay fractional unknown input observer are developed and used for fault detection and isolation. A single-observer scheme is needed for fault detection and a bank of generalized (respectively dedicated) observers is required for fault isolation. A theoretical study investigating the convergence condition for each observer-based method in terms of matrix inequalities is presented. Residual sensitivities to faults and to disturbances are studied. Time-delay fractional unknown input observer parameters are computed to obtain structured residuals. This observer ensures unknown input decoupling from the state which results residual insensitive to unknown inputs. Two numerical examples to validate the efficiency of the proposed approaches for diagnosis of time-delay fractional systems are given. Example 1 illustrates the efficiency of the two observers in fault detection and the difference between them in unknown input influence. Example 2 illustrates the efficiency of the proposed approach for fault isolation. Keywords: Luenberger observer; unknown input observer; diagnosis; time-delay fractional system; fault detection; fault isolation. Existence of multiple positive solutions for nonlinear three-point Riemann-Liouville fractional differential equation   by Yunhong Li, Weihua Jiang Abstract: In this article, the existence of multiple positive solutions is considered for nonlinear three-point Riemann-Liouville fractional differential equation. We use the Avery-Peterson fixed point theorem to acquire the existence of multiple positive solutions for the boundary value problem. An example is also presented to illustrate the effectiveness of the main result. Keywords: Riemann-Liouville fractional derivative; positive solutions; Avery-Peterson fixed point theorem. Non-uniqueness of solution for initial value problem of impulsive Caputo-Katugampola fractional differential equations   by Xianmin Zhang Abstract: In this paper, the non-uniqueness of solution is mainly considered to the initial value problem (IVP) for the system of impulsive fractional differential equations (IFrDE) with Caputo-Katugampola derivative. The IVP for IFrDE with Caputo- Katugampola derivative is equivalent to the integral equations with an arbitrary constant, which means that the solution is non-unique. Finally, a numerical example is provided to show the main result. Keywords: fractional differential equation; impulsive fractional differential equation; impulse; Caputo-Katugampola derivative. Time feedback control in a modified Sprott E model   by Yizhong Liu Abstract: This paper is concerned with chaos control for a modified Sprott E system.rn Applying time-delayed feedback control method, we establish some new conditions to control chaotic behavior of modified Sprott E system.rn With the aid of local stability analysis, we theoretically provern the occurrences of Hopf bifurcation.rn Computer simulations are implemented to support analytical results. Finally,rn a brief conclusion is included. Keywords: Modified Sprott E system; Chaos; Stability; Hopf bifurcation; Time-delayed feedback. Oscillation of one kind of second order neutral delay differential equations   by Hui Li, Yige Zhao, Shurong Sun Abstract: This paper is dedicated to discussing the oscillation of the second order neutral delay differential equations $$left(r(t)(z'(t))^alpharight)'+q(t)f(x^beta(sigma(t)))=0,$$ where $z(t)=x(t)+p(t)x(tau(t)).$ Sufficient conditions are provided by Riccati transformation comparing with related first order differential inequalities and differential equations. Results obtained in this paper have extended and improved conclusions contained in other literatures. Several illustrative examples are presented. Keywords: Oscillation; Neutral; Delay; Differential equation. Effects of Computer Networks' Viruses under the Influence of Removable Devices   by Ashraf Ahmad, Yousef Abu Hour, Mahmoud DarAssi Abstract: The Removable devices (RD) is one of the important factors that affects the virus spreading. We assumed that the infected RD could affect the nodes of $S$ and $E$ compartments at the rates, $\theta_1$ and $\theta_2,$ respectively. While the previous studies considered this effect on susceptible compartment only. {Moreover, we considered the effect of the rate of the nodes which are break down from network because of infected RD, $\mu_1$.} This model has no virus-free equilibrium and has a unique endemic equilibrium. The theorems of asymptotically autonomous systems and the generalized Poincare-Bendixson are used to show that the endemic equilibrium is globally asymptotically stable. Numerical methods are used to solve the obtained system of differential equations and the solutions are illustrated in several examples. The effects of $\xi, \, \epsilon,\, \theta_1$ and $\theta_2$ rates on the devices that moved from latent to recovered nodes are investigated. The behavior of the susceptible, exposed and infected nodes in the computer network have been analyzed. Keywords: Epidemiology; Modeling; Computer virus spread; Equilibrium; Infected Removable Devices. Coexistence of Three Limit Cycles for a Septic Polynomial Differential Systems   by Mohamed GRAZEM, Ahmed BENDJEDDOU, Rachid CHEURFA Abstract: The existence of limit cycles is interesting and very important in applications. It is a key to understand the dynamic of polynomial differential systems. The aim of this paper is to investigate a class of planar differential systems of degree seven. Under some suitable conditions, the existence of three limit cycles two of them are non-algebraic while the third is algebraic is proved. Furthermore, these limit cycles are explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our results. Keywords: Planar polynomial differential system; First integral; Periodic orbits; algebraic and non-algebraic limit cycle. Approximate controllability results for neutral stochastic integrodifferential equations of Sobolev type with unbounded delay via resolvent operators   by Nirmalkumar Rajendran, Murugesu Rangasamy Abstract: In this paper, we consider a class of approximate controllablility of stochastic integrodifferential equations of Sobolev type with unbounded delay via resolvent operators. Using the stochastic analysis technique and fixed point theorem, a set of sufficient conditions is obtained for the required result of approximate controllability of stochastic integrodifferential equations of Sobolev type with unbounded delay. Finally, an example is provided to illustrate the obtained result. Keywords: Approximate Controllability; Fixed point theorem; Stochastic differential equation; Mild solution. Simulation of large deflections of a flexible cantilever beam fabricated from functionally graded materials by the Adomian decomposition method   by Randolph Rach, Jun-Sheng Duan, Abdul-Majid Wazwaz Abstract: In this work, we use the Adomian decomposition method to study large deflections of a flexible cantilever beam fabricated from functionally graded materials with a sinusoidal nonlinearity. We convert the specified nonlinear boundary value problem with Dirichlet and Neumann boundary conditions, that governs the large deflections, to an equivalent nonlinear Fredholm-Volterra integral equation. We illustrate the obtained approximations by appropriate graphs and examine the resulting possible errors. Finally, we discuss the relationship of the deflection and the model parameters. Keywords: large deflection; boundary value problem; flexible cantilever beam; functionally graded material; Adomian decomposition method; Adomian polynomials. Global existence and blow up of positive initial energy solution of a quasilinear wave equation with nonlinear damping and source terms   by Paul Ogbiyele Abstract: In this paper, we consider a quasilinear wave equation having nonlinear damping and source terms $$u_{tt} - \Delta u_t - \sum^N_{i=1} \frac{\partial}{\partial x_i} \Bigl[\sigma_i(x , u_{x_i}) + \beta_i(x , u_{tx_i }) \Bigr] + f(x , u_t) = g(x , u)$$ and obtained global existence and blow up results under certain polynomial growth conditions on the nonlinear functions $\sigma_i,; \beta_i, (i= 1, 2, ..., N)$, $f$ and $g$. We obtain global existence result for positive initial energy solution using Galerkin approximation procedure and nonexistence(blow up) result using the technique introduced by Georgiev and Todorova cite{GT} with little modification for our problem. Keywords: Galerkin approximation procedure; Global solution; Blow up; Potential well. 4-dimensional zero-Hopf bifurcation for polynomial differentials systems with cubic homogeneous nonlinearities via averaging theory   by Amina Feddaoui, Jaume Llibre, Amar Makhlouf Abstract: The averaging theory of second order shows that for polynomial differential systems in R^4 with cubic homogeneous nonlinearities at least nine limit cycles can be born in a zero-Hopf bifurcation. Keywords: Hopf bifurcation; averaging theory; cubic polynomial differential systems;. Bifurcation behavior of a nonlinear innovation diffusion model with external influences   by Rakesh Kumar, Anuj Sharma, Kulbhushan Agnihotri Abstract: A nonlinear form of Bass model for innovation diffusion consisting of two differential equations governed by two variables viz. for adopters and non-adopters population densities is proposed to lay stress on the evaluation period. By investigating the corresponding characteristic equation, the local stability of a positive equilibrium and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. The critical value of evaluation period is determined beyond which small amplitude oscillations of the adopter and non-adopters population occur, and this critical value goes on decreasing with the increase in carrying capacity of the non-adopters population. Basic results are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions by using the normal form theory and center manifold theorem. Sensitivity analysis is performed for state variables at positive steady state on model parameters. It is observed that the cumulative density of external influences has a significant role in developing the maturity stage (final adoption stage) in the system. Numerical computations are executed to confirm the correctness of theoretical investigations. Keywords: Innovation diffusion model; Stability analysis; Sensitivity analysis; Hopf bifurcation; Center manifold theorem; Normal form theory. SEMIGROUP DYNAMICS FOR FLIGHT VECTORS   by Robert OBrien Abstract: A commutative semigroup of contractions S on a Hilbert space, H, has a natural order and net structure which deﬁnes stability, system dynamics, and α and ω limits for the ﬂight vectors H0. The space of pure ﬂight vectors (no weakly stable components) are spanned by the ω limits of weakly-wandering vectors which are weakly Poisson recurrent. The result H0 splits: H0 = Hm ⊕Hw, Hw the weakly stable subspace and Hm the weakly Poisson recurrent space. Hm = ⊕M(xτ,S) where M(xτ,S) is the closed subspace spanned by the weak limit points of xτ, {xτ} an orthonormal set of weakly-wandering vectors in Hm. Two examples illustrate the results. Keywords: Hilbert Space; Recurrence; Flight Vectors; Dynamical System; Weakly Wandering; Almost Weakly Stable; Unitary; Weak Stability; Contraction; Semigroup. Parameter Estimation for Chan-Karoli-Longstaff-Saunders Model Driven by Small Levy Noises from Discrete Observations   by Chao Wei Abstract: This paper is concerned with the parameter estimation problem for discrete observed Chan-Karoli-Longstaff-Saunders model driven by small L'{e}vy noises. The explicit formula of the least squares estimators are obtained and the estimation error is given. By using Cauchy-Schwarz inequality, Gronwall's inequality, Markov inequality and dominated convergence, the consistency of the least squares estimators are proved when a small dispersion coefficient $varepsilonrightarrow 0$ and $nrightarrowinfty$ simultaneously. The simulation is made to verify the effectiveness of the estimators. Keywords: Least squares estimator; L'{e}vy noises; discrete observations; consistency. Matrix Mittag-Leffler function and solution of multi-term fractional differential equations   by Junsheng Duan Abstract: In this paper, we first derive the solution of fractional differential equation system expressed in matrix Mittag-Leffler function by using the Adomian decomposition method. Then we consider the initial value problem (IVP) for multi-term fractional differential equation. By introducing new unknown functions, we rewrite the IVP for multi-term fractional differential equation into the IVP for a fractional differential equation system. Thus the solution can be given in terms of matrix Mittag-Leffler functions. We demonstrate the method using four numerical examples and the results are simulated using MATHEMATICA 8. Keywords: fractional calculus; fractional derivative; Mittag-Leffler function; fractional differential equation. Analysis of migration pattern of prey species with reserved zone   by JYOTIRMOY ROY, Shariful Alam Abstract: In this article a generalized prey-predator system has been analyzed, where the whole habitat is divided into two different zones, namely free zone and reserved zone. It is assumed that in the reserved zone only prey species can access and predation is strictly prohibited, whereas in the free zone both the species can cohabit and naturally predation is allowed. The migration rates of the prey species from reserved zone to unreserved zone and vice-versa both depends on predator's availability and accordingly suitable functions has been incorporated in the model system. The local and global stability analysis of the model system have been performed in a systematic manner and system persistence criterion has been established. The role of prey migration rate from reserved zone to unreserved zone has been investigated and it is found that Hopf bifurcation occurs when the prey migration rate from reserved zone to unreserved zone crosses a certain threshold value. It is also found that the prey migration rate has stabilizing effect on the dynamics of the system and has significant effect on the coexistence of all the species. Finally numerical simulation has been carried out to support our analytical findings. Keywords: Prey-predator model; Reserved zone; Stability and persistence; Hopf bifurcation; Limit cycle. Space time fractional Boussinesq equation with singular and non singular kernels   by Ritu Agarwal, Mahaveer Yadav, Ravi P. Agarwal Abstract: Recently, many authors have found analytical and numerical solutions of fractional Boussinesq equation by applying various fractional operators with singular kernels. Motivated by recently introduced fractional operators with non-singular kernels, in this paper a comparison of the solution of linearized fractional Boussinesq equation has been made for the fractional operators Caputo (with singular kernel) and Caputo-Fabrizio (with non-singular kernel). Linearized Boussinesq equation is derived by assuming that the average thickness of saturated layer of an aquifer is constant. Keywords: Linearized Boussinesq equation; Caputo fractional derivative; Caputo Fabrizio fractional derivative; Fractional Laplacian Operator; Mittag Leffler function. Dynamics of a predator-prey model with discrete and distributed delay   by Bootan Rahman, Muhammad Yau, Yuliya Kyrychko, Konstantin Blyuss Abstract: This paper considers a predator-prey model with discrete time delay representing prey handling time and assumed equal to the predator maturation period, and a distributed time delay describing intra-species interactions. We show that due to the delayed logistic growth of the prey, it is impossible for the species to become extinct through predation. Conditions for existence and local stability of the co-existence equilibrium are derived in terms of system parameters. Using techniques of centre manifold reduction and the normal form theory, we establish the direction of Hopf bifurcation of the co-existence equilibrium, as well as the stability of the bifurcating period solution. Numerical bifurcation analysis and simulations are performed to illustrate regions of stability of the co-existence equilibrium, to investigate how the amplitude and the period of bifurcating periodic solutions depend on parameters, and to demonstrate different types of dynamics of the system. Keywords: Stability; discrete and distributed delay; predator-prey model; Hopf bifurcation; periodic solutions. On the Oscillation of Conformable Fractional Partial Delay Differential Systems   by George E. Chatzarakis, Muthusamy Deepa, Nagamanickam Nagajothi, Vadivel Sadhasivam Abstract: In this article, we investigate the oscillation of a conformable fractional three dimensional nonlinear partial delay differential system. We establish some new oscillation criteria of the solutions of the differential system by using the generalized Riccati transformation and the integral averaging method. The obtained results are illustrated by various examples. Keywords: Oscillation; Delay; Partial differential system; Conformable fractional derivative. Global dynamics analysis of a stochastic SIRS epidemic model with vertical transmission and different periods of immunity   by Driss KIOUACH, Yassine SABBAR Abstract: In this work, we analyze a stochastic SIRS (Susceptible-InfectedRecovered-Susceptible) epidemic model with vertical transmission and different periods of immunity. This model has a global positive solution. Firstly, we establish sufficient conditions for extinction and persistence in the mean of a disease. Then, we prove the global stability of the system under a suitable condition of perturbation intensity. In the case of the non-autonomous system, we show that there exists at least one positive periodic solution. Finally, some numerical examples are introduced to show the validity of our results. Keywords: Stochastic SIRS model; vertical transmission; global stability; extinction; persistence; periodic solution. EXISTENCE OF POSITIVE QUASI-HOMOCLINIC SOLUTIONS FOR DAMPED p-LAPLACIAN DIFFERENTIAL EQUATIONS   by Monia Boujlida Abstract: In this paper we prove the existence of nontrivial homoclinic so-rnlutions for the damped $p$-Laplacian differential equationrnrn$$('u''^{p-2}u')' + c('u''^{p-2}u')+ a(t)'u'^{p-2}u + f(t,u)=0 ; t in matbb{R};$$rnwhere $p geq 2$, c geq 0 is a constant and the functions $a$ and $f$ are continuous andrnnot necessarily periodic in $t$. Using the Mountain-Pass Theorem, we obtainrnthe existence of positive homoclinic solution in both cases sub-quadratic andrnsuper-quadratic. Keywords: Quasi-homoclinic solution; the (PS)-condition; Mountain Pass The-rnorem; damped $p$-Laplacian equation. Cheap controls for disturbances compensation in hyperbolic delayed systems.   by Salma Souhaile, Larbi Afifi Abstract: Thiswork applies to the remediability problem for a class of hyperbolic perturbed systems with constant or time-varying delays.With a convenient choice of input operator (control) and through the observation (output), we show how to remedy the effect of any disturbance f on the considered system. We give the main properties and characterizations of the concept according to the delay. Then, under the appropriate hypothesis, we prove howto find the optimal control ensures the compensation of a disturbance using the corresponding observation only. The usual case of actuators and sensors is examined. An application and numerical results for a one-dimensional wave equation with delay are also presented. Keywords: Hyperbolic systems; Disturbance; Control; Observation; Delay; Remediability. Oscillation of delay difference equations with finite non-monotone arguments   by Limei Feng, Zhenlai Han Abstract: In this paper, the oscillation of delay difference equations with finite non-monotone delayrn$$triangle x(t)+sum_{i=1}^mp_i(t)x(tau_i(t))=0, tin mathbb{N}$$rnis studied. Three criteria of these equations are obtained for oscillation. And examples are given to show the meanings of the theorems.rn Keywords: delay difference equation; non-monotone argument; oscillatory solution. Delay feedback strategy for a fractional-order chaotic financial system   by Changjin Xu Abstract: In this paper, we are concerned with a new fractional incommensurate order financial system\r\nwhich is a generalized version of the\r\nfinancial model investigated in earlier works. Designing a suitable time-delayed feedback controller, we have controlled the chaotic phenomenon of the\r\nfractional incommensurate order financial system. By analyzing the characteristic equation of the involved financial system and regarding the delay as the bifurcation\r\nparameter, we establish a set of new sufficient conditions to guarantee the stability and the existence of Hopf bifurcation\r\n for fractional incommensurate order financial system.\r\n The study reveals that the delay and the fractional order have an important influence on the stability and Hopf bifurcation of considered financial system.\r\n Computer simulations are presented to illustrate the correctness of the theoretical results.\r\n The theoretical findings of this paper are new and have important meanings in dealing with the economic and financial problems. Keywords: Chaos control; financial system; stability; Hopf bifurcation; fractional order; delay. Residual power series method for the time fractional Fornberg-Whitham equation   by Jianke Zhang, Luyang Yin Abstract: The purpose of this paper is to solve the time fractional Fornberg-Whitham equation by the residual power series method, where the fractional derivatives are in Caputo sense. According to the definition of generalized fractional power series, the solutions of the fractional differential equations are approximatively expanded and substituted into the differential equations. The coefficients to be determined in the approximate solutions are calculated according to the residual functions and the initial conditions, and the approximate analytical solutions of the equations can be obtained. Finally, the approximate analytical solutions are compared with the exact solutions. The results show that the residual power series method is convenient and effective for solving the time fractional Fornberg-Whitham equation. Keywords: Residual power series method; Time-fractional Fornberg-Whitham equation; Caputo derivative. Numerical Solution of Time-Delay Systems by Hermite Wavelet   by Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri, Sohrab Effati Abstract: This paper presents a direct numerical method based on Hermite wavelet to fi nd the solution of time-delay systems. The operational matrices of integration, differentiation, production, and delay are derived and utilized to reduce the time-delay dynamical system to a set of algebraic equations. Thus, the problem is simpli fied greatly. The method is easy to implement. The illustrative examples with time-invariant and time-varying coefficients demonstrate the validity of the method. Keywords: Time-delay system; Hermite wavelet; Operational matrix; Direct method. Solving Nonlinear Fredholm integral equations with PQWs in complex plane   by Majid Erfanian Abstract: In this article, we propose a numerical scheme to solve a kind of nonlinear Fredholm integral equations of the second kind in the complex plane. The periodic quasi-wavelets (PQWs) constructed on [0,2pi] are utilized as a basis of the iteration method. Using the Banach fixed point theorem, we obtain some results concerning the error analysis. Illustrative examples are included to demonstrate the validity and applicability of the technique. Keywords: Nonlinear Fredholm integral equation; Periodic quasi-wavelet; Complex plane; fixed point theorem; error analysis. A Discrete Viral Infection Model with Both Modes of Transmission and Distributed Delays   by E.L. Boukari Brahim, Hattaf Khalid, E.L. Ghordaf Jalila Abstract: The aim of this work is to propose and analyze a discrete virus dynamics model with distributed delays and both modes of transmission, one by virus-to-cell infection and the other by cell-to-cell transfer. In the proposed model, the first distributed delays describes the time needed for infected cells to produce new virions, and the second portrays the time necessary for the newly produced virions to become mature and infectious. In addition, the infection transmission process is modeled by general incidence functions for both modes. Furthermore, we prove that the proposed discrete model has the same dynamics as the corresponding continuous model, such as positivity, boundedness and global behaviors of solutions with no restriction on the time step size. Moreover, numerical simulations are given to illustrate and confirm our main analytical results. Keywords: Viral infection; distributed delay; difference equation; global stability. Single controller for synchronization of coupled neural networks with distributed time-varying delays   by Cheng-De Zheng, Fan Xie Abstract: This paper deals with global synchronization in arrays of delayed chaotic neural networks with nonlinear hybrid coupling. By constructing a new Lyapunov-Krasovskii functional, a novel synchronization criterion is presented in terms of matrix inequalities based on Chen's integral inequalities and reciprocal convex technique. These established conditions are heavily dependent on the bounds of both time-delay and its derivative. Through employing Matlab Toolbox and adjusting some matrix parameters in the derived results, the design and applications of the generalized networks can be realized. The effectiveness and applicability of the proposed methods is demonstrated by a numerical example with simulations. Keywords: synchronization; matrix inequality; hybrid coupled neural networks; reciprocal convex technique. Numerical approach for solving nonlinear stochastic It   by Rebiha Zeghdane Abstract: In this paper, we give a new method for solving stochastic nonlinear Volterra integral equations by using shifted Legendre operational matrix. It is discussed that how the stochastic differential equations (SDE) could numerically be solved as matrix problems. By using this new operational matrix of integration and the so-called collocation method, nonlinear Volterra integral equations is reduced to systems of algebraic equations with unknown Legendre coefficients. Finally, the high accuracy of approximated solutions are illustrated by several experiment. Keywords: Stochastic Volterra integral equation; Brownian motion; Approximate solution; Best approximation; Legendre polynomials; Collocation method. Interval oscillation criteria for damped second-order delay differential equation with nonlinearities given by Riemann-Stieltjes integral   by MUTHULAKSHMI V, MANJURAM R Abstract: The purpose of this paper is to investigate the oscillatory behavior of certain types of damped second-order forced delay differential equation with nonlinearities given by Riemann-Stieltjes integral. By using the Riccati transformation, some inequalitiess and integral averaging technique, interval oscillation criteria of both El-Sayed type and Kong type are established. Finally, two examples are presented to illustrate the theoretical results. Keywords: Interval criteria; Oscillation; Delay differential equation; Damping term; Riemann-Stieltjes integral. Eventually periodicity of solutions for some discrete max-type system of third order   by Huili Ma, Haixia Wang Abstract: This paper is concerned with the eventually periodicity of the following max-type difference equation systemrn$$x_{n+1}=maxleft{frac{A}{x_{n}y_{n-1}},x_{n-2}right},$$rn$$y_{n+1}=maxleft{frac{A}{y_{n}x_{n-1}},y_{n-2}right},$$rnwhere $nin N$, $Ain R$, and the initial values $x_{-2}, x_{-1}, x_{0}, y_{-2}, y_{-1}, y_{0}$ are arbitrary non-zero numbers. Keywords: Periodic solutions; Difference equations; Max-type system.