International Journal of Dynamical Systems and Differential Equations (24 papers in press)
Regular Issues
 Dynamical behaviors of an impulsive foodchain system with HassellVarley functional response and mutual interference
by Si Zhou, Yuanfu Shao, Qin Liu, Zhen Wang Abstract: An impulsively controlled foodchain system with HassellVarley functional response and mutual interference is established in this article. By applying theories and methods of ecology and ordinary differential equation, the dynamical complexity of this system is investigated. We give conditions of the extinction of prey and top predator and show that this system is uniformly bounded. By use of Floquet theory of impulsive equation and small amplitude perturbation skills, we consider the local stability and global stability of the preyfree and top predatorfree periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent via impulsive comparison theorem. Finally, numerical simulations are given to substantiate our theoretical results and to illustrate various dynamical behaviors of this system. Keywords: stability; permanence; impulsive; HassellVarley functional response; mutual interference.
 Alphastability of fractionalorder Hopfield neural networks
by Changjin Xu, Peiluan Li Abstract: This paper deals with a class of fractionalorder Hopfield neural networks. Applying the contraction mapping principle and the inequality technique, some very verifiable criteria on the alphastability of fractionalorder Hopfield neural networks are obtained. Finally, an example is given to illustrate our main theoretical findings. Our results are new and complement previously known results. Keywords: Hopfield neural networks; Fractional order; \\alphastability.
 EXISTENCE AND UNIQUENESS OF SOLUTIONS TO LYAPUNOV MATRIX STOCHASTIC DIFFERENTIAL EQUATIONS
by Deekshitulu GVSR, Sastry M.V.S.S.B.B. K. Abstract: In this paper, we establish the existence and uniqueness of solutions to Lyapunov matrix stochastic dierential equations by the method of successive approximations. The continuous dependence of the solutions on parameters and initial conditions are also discussed. An example is presented to illustrate the established results. Keywords: Existence; Uniqueness; Lyapunov matrix; Stochastic differential equations; Successive Approximations.
 Solvability of coupled systems of hybrid fractional differential equations and inclusions
by Nana Jin, Shurong Sun Abstract: In this paper, we investigate the boundary value problems of coupled systems of hybrid fractional\r\n differential equations and inclusions with coupled boundary conditions involving Caputo fractional\r\n derivative. By means of LearySchauder alternative and BohnenblustKarlin fixed point theorem,\r\n some results concerning the existence of solutions are obtained.\r\n At the same time, we also give the relationship between the solutions and\r\n upper and lower solutions. Finally, examples are presented to illustrate our main results. Keywords: upper and lower solutions; coupled system;\r\n hybrid fractional differential equations and inclusions; existence of solutions.
 Group classification and some new periodiclike and solitonlike solutions of the generalized Fisher equation with timevariable coefficients
by Mohamed Abdel Latif, Entsar ElShazly, Ahmed Elsaid, Hamed Nour Abstract: In this article, we perform the group classification of the generalized
Fisher equation with timevariable coefficients. Some new periodiclike and
solitonlike solutions for some specific forms of the arbitrary functions are
obtained. The power series solutions are obtained for some cases when the exact
solutions are difficult to be obtained. Also, the convergence of these solutions is
investigated.
Keywords: Lie symmetries; Group classification; Generalized Fisher equation.
 Global bifurcation analysis of the Kukles cubic system
by Valery Gaiko Abstract: In this paper, we carry out the global bifurcation analysis of the Kukles system representing a planar polynomial dynamical system with arbitrary linear and cubic righthand sides and having an antisaddle at the origin. Using our geometric approach and the WintnerPerko termination principle, we solve the problem on the maximum number and distribution of limit cycles in this system. Keywords: planar polynomial dynamical system; Kukles cubic system; field rotation parameter; bifurcation; limit cycle; WintnerPerko termination principle.
 Unique solutions for new fractional differential equations with pLaplacian and infinitepoint boundary conditions
by Li Wang, Chengbo Zhai Abstract: In this paper, we study the uniqueness and existence of solutions for a new fractional differential equation with pLaplacian and infinitepoint boundary conditions. The main method is a new fixed point theorem of $varphi(h,e)$concave operators. An example is given to illstrute the main result. Keywords: RiemannLiouville fractional derivative; pLaplacian; infinitepoint boundary value problem; $varphi(h,e)$concave operators.
 Behavior of TwoDimensional Competitive System of Nonlinear Difference Equations of Higher Order
by Jerico Bacani, Julius Fergy Rabago Abstract: We generalize the result of Mansour et. al (2012) cite{mansour} and study other related systems that deal with the dynamics of a competitive population model described by a system of nonlinear difference equations. rnMore precisely, we consider the systemrn [rn x_{n+1}=frac{x_{n(2k1)}}{varepsilon + delta x_{n(2k1)} y_{n(k1)}}, quad rn y_{n+1}=frac{y_{n(2k1)}}{rho + sigma y_{n(2k1)} x_{n(k1)}},rn ]rnwhere $varepsilon, delta, rho, sigma in {1,1}$ and $kin mathbb{N}$ with real initial conditions $(x_n)_{n=(2k1)}^0$ and $(y_n)_{n=(2k1)}^0$ rnsuch that $varepsilon + delta x_{m(2k1)} y_{m(k1)} neq 0$ and $rho + sigma y_{m(2k1)} x_{m(k1)} neq0$ for all possible values of $m$ and $k$rnand study the form and behavior of its solutions for all values of $varepsilon, delta, rho$, and $sigma$ in ${1,1}$. rnThis work also generalizes several other results on system of nonlinear difference equations (see cite{algham}, cite{elsayed5}, cite{ibrahim5}, cite{kurbanli} and cite{touafek1}).rnFurthermore, the onedimensional case of the given system provides a generalization of a series of paper of E. M. Elsayed on nonlinear diffierence equations (see cite{elsayed1}, cite{elsayed2} and cite{elsayed6}) Keywords: discrete dynamical system; nonlinear difference equation; form of solutions; convergence; periodicity; competitive system.
 Lie Symmetry Analysis and Conservation Laws of Certain Time Fractional Partial Differential Equations
by Ramajayam Sahadevan, P. Prakash Abstract: A method is presented to derive the Lie point symmetries of time fractional partial differential equations in the sense of RiemannLiouville fractional derivative. The applicability of the method has been illustrated through time fractional BurgersKortewegde Vries with time dependent variable coefficients, time fractional dissipative ZabolotskayaKhokhlov equation, time fractional generalized Benjamin equation and time fractional diffusion equation with variable coefficients. Using the obtained Lie point symmetries, it is shown that each of the above mentioned time fractional partial differential equations can be transformed into a ordinary differential equations of fractional order. Exact solutions of the above mentioned time fractional equations are derived wherever possible. It is also explained how conservation laws can be derived to time fractional partial differential equations. Keywords: Time fractional partial differential equations; Lie group formalism; conservation laws; RiemannLiouville fractional derivative; Erd$acute{e}$lyiKober fractional operators.
 Leader following speed synchronization in multiple DC motor system using a hybrid controller
by Suhaib Masroor, Chen Peng, Syed Muhammad FazalulKarim Abstract: In this paper, we explore an innovative approach to design a Chopper fed DC motor coupled as a Multiagent System (MAS), predominantly leader following MAS, to achieve consensus on speed regulated by a Hybrid controller. The hybrid controller incorporates pole placement, tracking and regulation (RST) controller accompanied by adaptive model reference adaptive control (MRAC), after that incorporating MIT rule in the design analysis to endorse system stability. Leader following algorithm is fused with the system model to make the speed of following agents equivalent to that of leader. In the proposed method, every motor with its chopper circuit is treated as sole agent ie i_th agent. In this paper, we assume that communication among leader and follower is fixed moreover, we also consider two possible scenarios of communication ie in the presence of delay and without delay. For model simulation, MATLAB is used and the obtained results endorse effectiveness of the proposed design. Keywords: Leader following MAS; Consensus; DC Chopper; DC motor; Hybrid Control.
 On Some Attractors of a TwoDimensional Quadratic Map
by Mohamed Réda Ferchichi, Abla Yousfi Abstract: In this paper, we study the appearance, evolution and neighborhood of two attractors of a dynamical system defined by a quadratic polynomial map T:R^2→R^2. The first is a Cantortype attractor located on an invariant straight line. Thus, it suffices to study the restriction of the map T to this invariant line. The second is a closed curves cycle of period 2. We show, by a numerical approach, that when a parameter of the system varies, the evolution of the orbits in the region close to this second attractor is dependent on the evolution of the stable and unstable sets (homoclinic tangency) of a saddle cycle of period 2 located in this region. Keywords: Discrete dynamical systems; attractors; Cantor sets; invariant curves; saddlenode and homoclinic bifurcations.
 Results on approximate controllability of secondorder nonautonomous 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 nonautonomous secondorder integrodifferential inclusions in Banach spaces. We establish our main results with the help of resolvent operators and BohnenblustKarlin's fixed point theorem. Then we extend our study to secondorder 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 boundaryvalue problem (BVP) for irradiation of a rightrnparallelepipedal cancerous region in a (B/Gd) neutron cancer therapy (NCT)rncompositeregion setup by three mutually orthogonal, timemodulated, onespeed 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 neutrondensity 3D wave solution to the posing fourregional 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; FourRegional BoundaryValuernProblems; Additive Separation of Variables; OneSpeed Neutron Diffusion; NeutronDensityrn3D 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 ErRiani, 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.
 Lyapunovtype inequalities on fractional qdifference Schrodinger equation with the WoodsSaxon potential
by Kuikui Ma, Zhenlai Han Abstract: In this paper, the integer order Schr\"{o}dingerrnequation with the WoodsSaxon potential is extended to thernfractional $q$difference field. We establish the Lyapunovtyperninequalities for nonlinear fractional $q$difference equations, tornthe best of our knowledge, which is the first work dealing withrnLyapunovtype inequalities for nonlinear fractional $q$differencernequations. Results in this paper even are new in integer order case.rnMoreover, we further investigate the twopoint boundary valuernproblem of nonlinear fractional $q$difference Schr\"{o}dingerrnequation with the WoodsSaxon potential. By applying thernLeraySchauder 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 thernLeggettWilliams 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 DoughertyBliss, 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 notreatment, 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 nonhomogeneous 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 reactiondiffusion 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 FisherKolmogorov equation with two different reaction nonlinearities) to support the obtained results. Keywords: reactiondiffusion equations; traveling waves; Minimal front speed; pulled fronts.
 Sum operator methods for the existence and uniqueness of solution to infinitepoint boundary value problems for fractional differential equations
by Yupin Wang, Shurong Sun Abstract: In this paper, we study infinitepoint boundary value problems for a class of higherorder nonlinear factional differential equations involving the RiemannLiouville derivative. By using sum operator methods, the existence and uniqueness of solution to this kind of problems is obtained and iterative sequence of the positive solution is structured. Two examples are provided for our new results. Keywords: existence and uniqueness; fractional differential equation; Krasnoselskii's fixed point theorem; positive solution. DOI: 10.1504/IJDSDE.2018.10009553
 Dynamical behaviour of miscibles fluids in porous media
by A. Assala, N. Djedaidi, F.Z. Nouri Abstract: In this paper, we are interested in studying the dynamics of miscible fluids in porous media. The model describing this issue is a system of equations, coupling the standard NavierStokes equations with gravity g as external force and a convective diffusion equation for a dilute concentration in the carrier fluid. By assuming that the fluids are incompressible, we first derive a new system of equations, by taking into account additional terms, due to the concentration inhomogeneties and an interfacial tension between the fluids. Then we propose a numerical approach to solve our system in order to illustrate the effectiveness of the dynamics during the fluid miscibility process. At this stage, we start by showing a stability result for our numerical scheme and then present numerical results. Keywords: finite elements; fluid dynamics; porous media; stability. DOI: 10.1504/IJDSDE.2018.10009554
 Weak solutions for a class of generalised image restoration models
by Shuaijie Li, Peng Li Abstract: In this paper, based on some wellknown restoration models, first, we propose a general form of image restoration functional model under some conditions, and get a class of related nonlinear parabolic partial differential equations. Second, we have established the existence and uniqueness of weak solutions for these equations. This work has supplied theoretical basis for image restoration models, and has great significance in image restoration field. Keywords: anisotropic diffusion; approximate solution; existence; image restoration; isotropic diffusion; nonlinear diffusion; parabolic PDE; ROF model; total variation; uniqueness; weak solution. DOI: 10.1504/IJDSDE.2018.10009557
 A note on homoclinic solutions for a class of semilinear fourthorder differential equations without coercivity
by Adel Daouas Abstract: In this paper, we study the existence of homoclinic solutions for a class of fourthorder nonautonomous differential equations. Indeed, without coercive condition on the coefficient of the linear term and under suitable assumptions on the growth of the linearity, we establish the existence and the multiplicity of homoclinic solutions by using the Mountain Pass Theorem. Some recent results in the literature are generalized. Particularly, the open problem proposed by Zhang and Yuan (2015) is solved. Keywords: fourthorder differential equations; homoclinic solutions; Mountain Pass Theorem. DOI: 10.1504/IJDSDE.2018.10009558
 Dynamics of a discrete LeslieGower predatorprey model with feedback controls
by Changjin Xu, Peiluan Li Abstract: In this paper, we propose and deal with a discrete LeslieGower predatorprey model with feedback controls. By using the difference inequality theory, some sufficient conditions to ensure the permanence of the system are derived. The paper ends with brief conclusions. Our results are new and complete previously known results. Keywords: discrete; feedback control; Leslieâ€“Gower predatorâ€“prey model; permanence. DOI: 10.1504/IJDSDE.2018.10009561
 Discrete state space systems of fractional order
by Jagan Mohan Jonnalagadda Abstract: The present article devotes to the study of linear time invariant discrete fractional order state space systems using a novel approach. First, we replace the conventional GrünwaldLetnikovtype backward difference operator with the equivalent RiemannLiouvilletype nabla difference operator and obtain the system response using variation of constants and discrete Laplace transform methods. Next, we present three different algorithms to construct state transition matrix of the system. Finally, we provide an example to illustrate the applicability of established result. Keywords: backward (nabla) difference; fractional order; matrix exponential function; Ntransform; state space representation; state transition matrix; system response. DOI: 10.1504/IJDSDE.2018.10009562
