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International Journal of Dynamical Systems and Differential Equations

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 International Journal of Dynamical Systems and Differential Equations (29 papers in press)  Regular Issues  Dynamical behaviors of an impulsive food-chain system with Hassell-Varley functional response and mutual interference   by Si Zhou, Yuanfu Shao, Qin Liu, Zhen Wang Abstract: An impulsively controlled food-chain system with Hassell-Varley 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 prey-free and top predator-free 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; Hassell-Varley functional response; mutual interference. Alpha-stability of fractional-order Hopfield neural networks   by Changjin Xu, Peiluan Li Abstract: This paper deals with a class of fractional-order Hopfield neural networks. Applying the contraction mapping principle and the inequality technique, some very verifiable criteria on the alpha-stability of fractional-order 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; \\alpha-stability. 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 di erential 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 Leary-Schauder alternative and Bohnenblust-Karlin 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 periodic-like and soliton-like solutions of the generalized Fisher equation with time-variable coefficients   by Mohamed Abdel Latif, Entsar El-Shazly, Ahmed Elsaid, Hamed Nour Abstract: In this article, we perform the group classification of the generalized Fisher equation with time-variable coefficients. Some new periodic-like and soliton-like 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 right-hand sides and having an anti-saddle at the origin. Using our geometric approach and the Wintner-Perko 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; Wintner-Perko termination principle. Unique solutions for new fractional differential equations with p-Laplacian and infinite-point 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 p-Laplacian and infinite-point 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: Riemann-Liouville fractional derivative; p-Laplacian; infinite-point boundary value problem; $varphi-(h,e)-$concave operators. Behavior of Two-Dimensional 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-(2k-1)}}{varepsilon + delta x_{n-(2k-1)} y_{n-(k-1)}}, quad rn y_{n+1}=frac{y_{n-(2k-1)}}{rho + sigma y_{n-(2k-1)} x_{n-(k-1)}},rn ]rnwhere $varepsilon, delta, rho, sigma in {-1,1}$ and $kin mathbb{N}$ with real initial conditions $(x_n)_{n=-(2k-1)}^0$ and $(y_n)_{n=-(2k-1)}^0$ rnsuch that $varepsilon + delta x_{m-(2k-1)} y_{m-(k-1)} neq 0$ and $rho + sigma y_{m-(2k-1)} x_{m-(k-1)} neq0$ 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 one-dimensional 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 Riemann-Liouville fractional derivative. The applicability of the method has been illustrated through time fractional Burgers-Korteweg-de Vries with time dependent variable coefficients, time fractional dissipative Zabolotskaya-Khokhlov 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; Riemann-Liouville fractional derivative; Erd$acute{e}$lyi-Kober fractional operators. Leader following speed synchronization in multiple DC motor system using a hybrid controller   by Suhaib Masroor, Chen Peng, Syed Muhammad Fazal-ul-Karim Abstract: In this paper, we explore an innovative approach to design a Chopper fed DC motor coupled as a Multi-agent 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 i-e 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 i-e 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 Two-Dimensional 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 Cantor-type 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; saddle-node and homoclinic bifurcations. 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. Sum operator methods for the existence and uniqueness of solution to infinite-point boundary value problems for fractional differential equations   by Yupin Wang, Shurong Sun Abstract: In this paper, we study infinite-point boundary value problems for a class of higher-order nonlinear factional differential equations involving the Riemann-Liouville 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 Navier-Stokes 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 well-known 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 fourth-order differential equations without coercivity   by Adel Daouas Abstract: In this paper, we study the existence of homoclinic solutions for a class of fourth-order 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: fourth-order differential equations; homoclinic solutions; Mountain Pass Theorem.DOI: 10.1504/IJDSDE.2018.10009558  Dynamics of a discrete Leslie-Gower predator-prey model with feedback controls   by Changjin Xu, Peiluan Li Abstract: In this paper, we propose and deal with a discrete Leslie-Gower predator-prey 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; Leslie–Gower predator–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 Grünwald-Letnikov-type backward difference operator with the equivalent Riemann-Liouville-type 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; N-transform; state space representation; state transition matrix; system response.DOI: 10.1504/IJDSDE.2018.10009562  Special Issue on: International Meeting on Applied Mathematics in Errachidia 2016 New Trends on Dynamical Systems and Differential Equations On a non-scalar damping model in micromagnetism   by C. Ayouch, El-H. Essoufi, M. Tilioua Abstract: We consider a mathematical model describing magnetisation dynamics with non-scalar damping. The model consists on a generalised Landau-Lifshitz-Gilbert equation with a general damping tensor. We apply Faedo-Galerkin/penalty method to show the existence of global weak solutions in one-dimensional case. Keywords: Faedo-Galerkin method; ferromagnetic materials; global existence; LLG equation.DOI: 10.1504/IJDSDE.2018.10009155  The best strategy for local mesh refinement with the PCD method   by Ahmed Tahiri Abstract: We propose in this contribution numerical tests for local mesh refinement with the PCD method. We investigate two rates of local mesh refinement: the ratio 2 and the ratio 3. First we introduce a local mesh refinement by subdividing the elements of the zone to be refined by the ratio 2 in each direction, i. e. each coarse element is subdivided by 4 fine elements. Second, we subdivide the elements of the zone to be refined by the ratio 3 in each direction, i. e. each coarse element is subdivided by 9 fine elements. Here we investigate some numerical experiments to show that the optimal local refinement rate is 2. This is in agreement with the regularity assumptions required for the proposed discretisation. Keywords: boundary value problem; local mesh refinement; multilevel local refinement; optimal rate for local mesh refinement; PCD method; slave nodes.DOI: 10.1504/IJDSDE.2018.10009156  Periodic solutions to singular damped delay differential equations with impulses   by Fatima Dib, Naima Daoudi-Merzagui Abstract: In this paper, we discuss the existence of periodic solutions for nonautonomous second-order damped delay differential equations with singular non-linearities, in presence of impulsive effects. Simple sufficient conditions are provided that enable us to obtain positive periodic solutions. Our approach is based on a variational method. Some recent results in the literature are extended. Keywords: damped; delay differential equation; impulses; mountain-pass theorem; periodic solution; singular non-linearity.DOI: 10.1504/IJDSDE.2018.10009157  Positive solutions to a non-homogeneous elliptic system of fourth order   by Mohamed Talbi, El. M. Hssini, M. Massar, N. Tsouli Abstract: In this work, the existence of positive solutions to a non-linear elliptic system of fourth order with positive parameters is studied. We establish an existence of an interval of parameters such the system has at least two positive solutions. The behaviour of energy corresponding to these positive solutions, with respect to the real parameters is proved. Keywords: Ekeland's variational principle; Palais-Smale condition; p-biharmonic operator.DOI: 10.1504/IJDSDE.2018.10009158  Weighted eigenvalue problems involving a fourth-order elliptic equation with variable exponent   by Abdesslem Ayoujil Abstract: In this paper, we consider weighted fourth order nonhomogeneous elliptic problem with variable exponent. Under appropriate conditions, interval of a numerical parameter sufciently small is derived for which the existence results are obtained. The proof relies on simple variational arguments based on Ekeland's variational principle. Keywords: p(x)-biharmonic operator; eigenvalue; Ekeland's variational principle; generalised Sobolev spaces; mountain pass theorem; weak solution.DOI: 10.1504/IJDSDE.2018.10009159  Cryptanalysis of Farash et al.'s SIP authentication protocol   by Mourade Azrour, Yousef Farhaoui, Mohammed Ouanan Abstract: Session Initiation Protocol (SIP) is the most popular signalling protocol used in order to establish, maintain and terminate a multimedia sessions between different participants. Nowadays, the security of SIP is becoming more and more important. Authentication is the most important security service required for SIP. To provide secure communication, many SIP authentication schemes have been proposed. Very recently, Farash et al. proposed a new SIP authentication protocol based on elliptic curve cryptography. They proved that their scheme is secured against different attacks. However, in this paper we show that Farash et al.'s protocol suffers from Denning-Sacco attacks and denial of service attacks. Moreover, we propose our solution to solve the problem. The security analysis shows that our proposed solution is more secure and can resist to various attacks. Keywords: attack; authentication protocol; denial of service; Denning-Sacco; DoS; elliptic curve cryptography; security; session initiation protocol; SIP.DOI: 10.1504/IJDSDE.2018.10009162  Vector extrapolation method for non-overlapping Schwarz iterations   by Nabila Nagid, Hassan Belhadj, Mohamed Ridouan Amattouch Abstract: In this paper, we propose a vector extrapolation method for accelerating the non-overlapping Schwarz iterations in the case of the non-linear reaction diffusion equations. The acceleration occurs at two levels: the acceleration of sequences produced by the fixed point algorithm, and the acceleration of sequences generated by Schwarz method. Specifically, the acceleration occurs in the internal Dirichlet boundary conditions. In order to illustrate the interest of the proposed algorithm, we have performed different test-cases of analytical examples, all results show the efficiency of the proposed approach in terms of CPU-Time, number of iterations and the rate of convergence. Keywords: domain decomposition method; extrapolation method; finite element method; fixed point method; non-linear reaction diffusion problems; non-overlapping Schwarz algorithm; vector ε-algorithm.DOI: 10.1504/IJDSDE.2018.10009166  Nonlinear Steklov eigenvalue problem with variable exponents and without Ambrosetti-Rabinowitz condition   by Abdellah Zerouali, Belhadj Karim, Omar Chakrone Abstract: In this paper, we study a nonlinear Steklov eigenvalue problem involving the p(x)-Laplacian on a bounded domain. We introduce a new variational technic that allows us to investigate this problem without need of the Ambrosetti-Rabinowitz condition on the nonlinearity. Keywords: critical point; p(x)-Laplacian; Steklov problem; variable exponent Lebesgue-Sobolev spaces.DOI: 10.1504/IJDSDE.2018.10009167  Content frequency and shape features based on CBIR: application to colour images   by Chawki Youness, El Asnaoui Khalid, Ouanan Mohammed, Aksasse Brahim Abstract: Due to the diversity of the image content, we propose in this study a new technique for content-based image retrieval to characterise the image. In this scenario, all images are characterised by their frequency content and their shape information. Indeed, using the high-resolution spectral analysis methods, especially the 2-D estimation of signal parameters via rotational invariance techniques, we extract from the image its content frequency and with the angular radial transform its shape information to construct a new vector descriptor. The experimental results applied to the Coil_100 database show the effectiveness and the robustness of our approach compared with the method based on the shame information only, and the precision average can be reached as 74.49%. Keywords: ART; CBIR; 2-D ESPRIT; frequency content; high resolution; shape feature; spectral analysis.DOI: 10.1504/IJDSDE.2018.10009168  Numerical analysis of some time-fractional partial differential equations for noise removal   by Moulay Rchid Sidi Ammi, Ismail Jamiai Abstract: In this paper, we consider the numerical resolution of two time-fractional partial differential equations for processing image denoising, which are obtained from some classical partial differential equations for processing image denoising, edge preservation and compression by replacing the first-order time derivative with a fractional derivative of order α, with 0 < α < 1. Discrete schemes of these models are introduced based on the finite differences method. Then, stability and error estimates are derived on the approximate solutions. Numerical experiments are presented to show the robustness of these new time-fractional models to obtain better results in image denoising and restoration. Keywords: finite differences method; fractional calculus; image processing.DOI: 10.1504/IJDSDE.2018.10009169  Regional boundary controllability of semi-linear parabolic systems with state constraints   by Touria Karite, Ali Boutoulout Abstract: This work focuses on the controllability of semi-linear parabolic systems with state constraints. Sub-differential techniques are used to compute the control u that steers the system (S) from the initial state y0 to a final one between two prescribed functions, only on a boundary subregion Γ of the system evolution domain Ω. Keywords: boundary; constrained controllability; distributed systems; optimal control; parabolic systems; regional controllability; sub-differential.DOI: 10.1504/IJDSDE.2018.10009170