Showing papers in "Nonlinear Analysis-real World Applications in 2012"
TL;DR: In this paper, a nonlinear Langevin equation involving two fractional orders α ∈ ( 0, 1 ] and β ∈( 1, 2 ] with three-point boundary conditions was studied and the contraction mapping principle was applied to prove the existence of solutions.
Abstract: This paper studies a nonlinear Langevin equation involving two fractional orders α ∈ ( 0 , 1 ] and β ∈ ( 1 , 2 ] with three-point boundary conditions. The contraction mapping principle and Krasnoselskii’s fixed point theorem are applied to prove the existence of solutions for the problem. The existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results. Some illustrative examples are also discussed.
214 citations
TL;DR: In this article, the authors explored the dynamics of an SIR epidemic model to understand how limited medical resources and their supply efficiency affect the transmission of infectious diseases and revealed that with varying amount of medical resources, the target model admits both backward bifurcation and Hopf bifurbation.
Abstract: The dynamics of an SIR epidemic model is explored in this paper in order to understand how the limited medical resources and their supply efficiency affect the transmission of infectious diseases. The study reveals that, with varying amount of medical resources and their supply efficiency, the target model admits both backward bifurcation and Hopf bifurcation. Sufficient criteria are established for the existence of backward bifurcation, the existence, the stability and the direction of Hopf bifurcation. The mechanism of backward bifurcation and its implication for the control of the infectious disease are also explored. Numerical simulations are presented to support and complement the theoretical findings.
172 citations
TL;DR: In this article, a modified Lotka-Volterra interaction term is used as the functional response of the predator to the prey, which is proportional to the square root of the prey population, and the solution behavior near the origin is more subtle and interesting than standard models and makes sense ecologically.
Abstract: A predator–prey model is considered in which a modified Lotka–Volterra interaction term is used as the functional response of the predator to the prey. The interaction term is proportional to the square root of the prey population, which appropriately models systems in which the prey exhibits strong herd structure implying that the predator generally interacts with the prey along the outer corridor of the herd. Because of the square root term, the solution behavior near the origin is more subtle and interesting than standard models and makes sense ecologically.
166 citations
TL;DR: The threshold for determining whether the virus dies out completely is given, and the existence of equilibria is studied, and it is found that, depending on the anti-virus ability, a backward bifurcation or a Hopf bIfurcation may occur.
Abstract: In this paper, we propose a novel computer virus propagation model and study its dynamic behaviors; to our knowledge, this is the first time the effect of anti-virus ability has been taken into account in this way. In this context, we give the threshold for determining whether the virus dies out completely. Then, we study the existence of equilibria, and analyze their local and global asymptotic stability. Next, we find that, depending on the anti-virus ability, a backward bifurcation or a Hopf bifurcation may occur. Finally, we show that under appropriate conditions, bistable states may be around. Numerical results illustrate some typical phenomena that may occur in the virus propagation over computer network.
159 citations
TL;DR: In this article, the stability of delayed recurrent neural networks with impulse control and Markovian jump parameters is studied, where the jumping parameters are modeled as a continuous-time, discrete-state Markov process.
Abstract: This paper is concerned with the stability of delayed recurrent neural networks with impulse control and Markovian jump parameters. The jumping parameters are modeled as a continuous-time, discrete-state Markov process. By applying the Lyapunov stability theory, Dynkin’s formula and linear matrix inequality technique, some new delay-dependent conditions are derived to guarantee the exponential stability of the equilibrium point. Moreover, three numerical examples and their simulations are given to show the less conservatism and effectiveness of the obtained results. In particular, the traditional assumptions on the differentiability of the time varying delays and the boundedness of their derivatives are removed since the time varying delays considered in this paper may not be differentiable, even not continuous.
150 citations
TL;DR: In this paper, it was shown that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system, in contrast with integer-order derivatives.
Abstract: Using the Mellin transform approach, it is shown that, in contrast with integer-order derivatives, the fractional-order derivative of a periodic function cannot be a function with the same period. The three most widely used definitions of fractional-order derivatives are taken into account, namely, the Caputo, Riemann–Liouville and Grunwald–Letnikov definitions. As a consequence, the non-existence of exact periodic solutions in a wide class of fractional-order dynamical systems is obtained. As an application, it is emphasized that the limit cycle observed in numerical simulations of a simple fractional-order neural network cannot be an exact periodic solution of the system.
140 citations
TL;DR: In this article, the dynamics of multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission were investigated and the basic reproduction number R 0 was derived, and it was shown that the global dynamics are completely determined by the values of R 0.
Abstract: The dynamics of multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission are investigated. We derive the basic reproduction number R 0 and establish that the global dynamics are completely determined by the values of R 0 : if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable; if R 0 > 1 , then there exists a unique endemic equilibrium which is globally asymptotically stable. Our results contain those for single-group SEIR models with distributed and infinite delays. In the proof of global stability of the endemic equilibrium, we exploit a graph-theoretical approach to the method of Lyapunov functionals. The biological significance of the results is also discussed.
138 citations
TL;DR: A modified SIS model with an infective vector on complex networks is proposed and analyzed, which incorporates some infectious diseases that are not only transmitted by a vector, but also spread by direct contacts between human beings.
Abstract: In this paper, a modified SIS model with an infective vector on complex networks is proposed and analyzed, which incorporates some infectious diseases that are not only transmitted by a vector, but also spread by direct contacts between human beings. We treat direct human contacts as a social network and assume spatially homogeneous mixing between vector and human populations. By mathematical analysis, we obtain the basic reproduction number R 0 and study the effects of various immunization schemes. For the network model, we prove that if R 0 1 , the disease-free equilibrium is globally asymptotically stable, otherwise there exists an unique endemic equilibrium such that it is globally attractive. Our theoretical results are confirmed by numerical simulations and suggest a promising way for the control of infectious diseases.
134 citations
TL;DR: In this paper, a reliable method for constructing a directed weighted complex network (DWCN) from a time series is proposed, which can be used to detect unstable periodic orbits of different periods.
Abstract: We propose a reliable method for constructing a directed weighted complex network (DWCN) from a time series Through investigating the DWCN for various time series, we find that time series with different dynamics exhibit distinct topological properties We indicate this topological distinction results from the hierarchy of unstable periodic orbits embedded in the chaotic attractor Furthermore, we associate different aspects of dynamics with the topological indices of the DWCN, and illustrate how the DWCN can be exploited to detect unstable periodic orbits of different periods Examples using time series from classical chaotic systems are provided to demonstrate the effectiveness of our approach
130 citations
TL;DR: A novel modified generalized projective synchronization (MGPS) is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix.
Abstract: This paper presents a new fractional-order hyperchaotic system. The chaotic behaviors of this system in phase portraits are analyzed by the fractional calculus theory and computer simulations. Numerical results have revealed that hyperchaos does exist in the new fractional-order four-dimensional system with order less than 4 and the lowest order to have hyperchaos in this system is 3.664. The existence of two positive Lyapunov exponents further verifies our results. Furthermore, a novel modified generalized projective synchronization (MGPS) for the fractional-order chaotic systems is proposed based on the stability theory of the fractional-order system, where the states of the drive and response systems are asymptotically synchronized up to a desired scaling matrix. The unpredictability of the scaling factors in projective synchronization can additionally enhance the security of communication. Thus MGPS of the new fractional-order hyperchaotic system is applied to secure communication. Computer simulations are done to verify the proposed methods and the numerical results show that the obtained theoretic results are feasible and efficient.
127 citations
TL;DR: In this article, an analysis for magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid inducing by an accelerating plate is presented, where the no-slip assumption between the wall and the fluid is no longer valid.
Abstract: This paper presents an analysis for magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid inducing by an accelerating plate. Where the no-slip assumption between the wall and the fluid is no longer valid. The fractional calculus approach is introduced to establish the constitutive relationship of a viscoelastic fluid. Closed form solutions for velocity and shear stress are obtained in terms of Fox H-function by using the discrete Laplace transform of the sequential fractional derivatives. The solutions for no-slip condition and no magnetic field can be derived as the special cases. Furthermore, the effects of various parameters on the corresponding flow and shear stress characteristics are analyzed and discussed in detail.
TL;DR: The global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo are investigated, and it is shown that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations.
Abstract: Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4+ T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8+ cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R 0 and R 1 , basic reproduction numbers for viral infection and for CTL response, respectively. If R 0 ≤ 1 , the infection-free equilibrium P 0 is globally asymptotically stable, and the HTLV-I viruses are cleared. If R 1 ≤ 1 R 0 , the asymptomatic-carrier equilibrium P 1 is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R 1 > 1 , a unique HAM/TSP equilibrium P 2 exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.
TL;DR: It is proved that the virus is cleared and the disease dies out if the basic reproduction number R 0 ≤ 1 while the virus persists in the host and the infection becomes endemic if R 0 > 1 .
Abstract: The rate of infection in many virus dynamics models is assumed to be bilinear in the virus and uninfected target cells. In this paper, the dynamical behavior of a virus dynamics model with general incidence rate and cure rate is studied. Global dynamics of the model is established. We prove that the virus is cleared and the disease dies out if the basic reproduction number R 0 ≤ 1 while the virus persists in the host and the infection becomes endemic if R 0 > 1 .
TL;DR: In this article, the dynamical behaviors of a class of discrete-time SIRS epidemic models are discussed and conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained.
Abstract: In this paper, the dynamical behaviors of a class of discrete-time SIRS epidemic models are discussed. The conditions for the existence and local stability of the disease-free equilibrium and endemic equilibrium are obtained. The numerical simulations not only illustrate the validity of our results, but also exhibit more complex dynamical behaviors, such as flip bifurcation, Hopf bifurcation and chaos phenomenon. These results reveal far richer dynamical behaviors of the discrete epidemic model compared with the continuous epidemic models.
TL;DR: In this article, an SEIQV model with saturated incidence rate is considered and the basic reproduction number R 0 is found, where R 0 ≤ 1, the disease-free equilibrium is globally asymptotically stable.
Abstract: In this article, an SEIQV epidemic model with saturated incidence rate is considered. The basic reproduction number R 0 is found. If R 0 ≤ 1 , the disease-free equilibrium is globally asymptotically stable; if R 0 > 1 , endemic equilibrium is globally asymptotically stable and the disease is persistent. Numerical simulations are carried out to illustrate the feasibility of the obtained results, especially the effect of vaccination to eliminate the disease.
TL;DR: In this paper, an active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems, and sufficient conditions for phase synchronization of the fractional models of Lorenz, Lu and Rossler systems are derived.
Abstract: The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lu and Rossler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lu and Rossler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.
TL;DR: In this paper, the authors established two new regularity criteria for the 3D incompressible MHD equations involving partial components of the velocity and magnetic fields, and proved that the local strong solution (u, b) remains smooth on [ 0, T ].
Abstract: In this paper, we establish two new regularity criteria for the 3D incompressible MHD equations involving partial components of the velocity and magnetic fields. It is proved that if u 3 , b ∈ L α ( 0 , T ; L γ ( R 3 ) ) , 2 α + 3 γ ≤ 3 4 + 1 2 γ , γ > 10 3 or u 3 , b ∈ L α 1 ( 0 , T ; L γ 1 ( R 3 ) ) , with 2 α 1 + 3 γ 1 ≤ 1 , 3 γ 1 ≤ ∞ , ∂ 3 u 1 , ∂ 3 u 2 ∈ L α 2 ( 0 , T ; L γ 2 ( R 3 ) ) , with 2 α 2 + 3 γ 2 ≤ 2 , 3 2 γ 2 ≤ ∞ , then the local strong solution ( u , b ) remains smooth on [ 0 , T ] .
TL;DR: In this article, the optimal control applied to a vector borne disease with direct transmission in host population is presented, where three control functions are used: vector reduction, personal protection and blood screening, respectively.
Abstract: The paper presents the optimal control applied to a vector borne disease with direct transmission in host population. First, we show the existence of the control problem and then use both analytical and numerical techniques to investigate that there are cost effective control efforts for prevention of direct and indirect transmission of disease. In order to do this three control functions are used, one for vector-reduction strategies and the other two for personal (human) protection and blood screening, respectively. We completely characterize the optimal control and compute the numerical solution of the optimality system using an iterative method.
TL;DR: In this article, a new exponential passivity criterion was proposed via the full use of the information of neuron activation functions and the involved time-varying delays, which has less conservativeness and less number of decision variables than the existing ones.
Abstract: The problem of delay-dependent exponential passivity analysis is investigated for neural networks with time-varying delays. By use of a linear matrix inequality (LMI) approach, a new exponential passivity criterion is proposed via the full use of the information of neuron activation functions and the involved time-varying delays. The obtained results have less conservativeness and less number of decision variables than the existing ones. A numerical example is given to demonstrate the effectiveness and the reduced conservatism of the derived results.
TL;DR: In this paper, the authors proposed a general method to achieve projective synchronization of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal.
Abstract: This paper investigates the projective synchronization (PS) of different fractional order chaotic systems while the derivative orders of the states in drive and response systems are unequal. Based on some essential properties on fractional calculus and the stability theorems of fractional-order systems, we propose a general method to achieve the PS in such cases. The fractional operators are introduced into the controller to transform the problem into synchronization problem between chaotic systems with identical orders, and the nonlinear feedback controller is proposed based on the concept of active control technique. The method is both theoretically rigorous and practically feasible. We present two examples that illustrate the effectiveness and applications of the method, which include the PS between two 3-D commensurate fractional-order chaotic systems and the PS between two 4-D fractional-order hyperchaotic systems with incommensurate and commensurate orders, respectively. Abundant numerical simulations are given which agree well with the analytical results. Our investigations show that PS can also be achieved between different chaotic systems with non-identical orders. We have further reviewed and compared some relevant methods on this topic reported in several recent papers. A discussion on the physical implementation of the proposed method is also presented in this paper.
TL;DR: In this article, the global dynamics of a delayed SIRS epidemic model for transmission of disease with a class of nonlinear incidence rates of the form βS(t) ∫h 0 f (τ )G(I(t τ ))dτ were studied.
Abstract: In this paper, we study the global dynamics of a delayed SIRS epidemic model for transmission of disease with a class of nonlinear incidence rates of the form βS(t) ∫h 0 f (τ )G(I(t τ ))dτ . Applying Lyapunov functional techniques in the recent paper (Y. Nakata, Y. Enatsu and Y. Muroya, On the global stability of an SIRS epidemic model with distributed delays, accepted), we establish sufficient conditions of the rate of immunity loss for the global asymptotic stability of an endemic equilibrium for the model. In particular, we offer a unified construction of Lyapunov functionals for both cases of R0 1 and R0 > 1, where R0 is the basic reproduction number.
TL;DR: In this paper, two kinds of solutions to random fuzzy differential equations (RFDEs) are generated by the usage of two different concepts of fuzzy derivative in the formulation of a differential problem.
Abstract: We present the studies on two kinds of solutions to random fuzzy differential equations (RFDEs). The different types of solutions to RFDEs are generated by the usage of two different concepts of fuzzy derivative in the formulation of a differential problem. Under generalized Lipschitz condition, the existence and uniqueness of both kinds of solutions to RFDEs are obtained. We show that solutions (of the same kind) are close to each other in the case when the data of the equation did not differ much. By an example, we present an application of each type of solutions in a population growth model which is subjected to two kinds of uncertainties: fuzziness and randomness.
TL;DR: In this article, the peristaltic flow of a Williamson fluid in asymmetric channels with permeable walls is investigated, and numerical results are obtained using the perturbation technique for the pumping and trapping phenomena, and these are used to bring out the qualitative features of the solutions.
Abstract: The peristaltic flow of a Williamson fluid in asymmetric channels with permeable walls is investigated. The channel asymmetry is produced by choosing a peristaltic wave train on the wall with different amplitudes and phases. The solutions for stream function, axial velocity and pressure gradient are obtained for small Weissenberg number, We, via a perturbation expansion about We, while an exact solution method is discussed for large values of We. The exact solutions become singular as We tends to zero; hence the separate perturbation solutions are essential. Also, numerical results are obtained using the perturbation technique for the pumping and trapping phenomena, and these are used to bring out the qualitative features of the solutions. It is noted that the size of the trapped bolus decreases and its symmetry disappears for large values of the permeability parameter. The effects of various wave forms (namely, sinusoidal, triangular, square and trapezoidal) on the fluid flow are discussed.
TL;DR: This work introduces the robust plug and play chaotic circuit designed to be easily realized using standard components in a rigorous, fast and inexpensive way and finds that experimental results display periodicity, bifurcations and chaos that match with high accuracy the corresponding theoretical values.
Abstract: Long run growth of the US national economic system, for example, reveals a strong oscillatory behavior due to complex interactions of aggregates. However, modelizations of such dynamics often assume that instability is the outcome of linear and additive cycles determined by exogenous shocks. In this work, a modelization of endogenous nonlinear and inseparable cycles is retained to explain the highly complex business cycle phenomenon. Bouali’s system is built to this scope. Its numerical simulations exhibit a rich repertoire of nonlinear dynamical phenomena, but this paper introduces its electronic implementation. The robust plug and play chaotic circuit is designed to be easily realized using standard components in a rigorous, fast and inexpensive way. We find that experimental results display periodicity, bifurcations and chaos that match with high accuracy the corresponding theoretical values.
TL;DR: In this article, a continuous-time version of the Hegselmann-Krause opinion dynamics, which models bounded confidence by a discontinuous interaction, is studied, and results of existence, completeness and convergence to clusters of agents sharing a common opinion are provided.
Abstract: In this paper, we study a continuous-time version of the Hegselmann–Krause opinion dynamics, which models bounded confidence by a discontinuous interaction. Intending solutions in the sense of Krasovskii, we provide results of existence, completeness and convergence to clusters of agents sharing a common opinion. For a deeper understanding of the role of the mentioned discontinuity, we study a class of continuous approximating systems, and their convergence to the original one. Our results indicate that their qualitative behavior is similar, and we argue that discontinuity is not an essential feature in bounded confidence opinion dynamics.
TL;DR: In this article, the complex dynamics of a reaction diffusion S − I model incorporating demographic and epidemiological processes with zero-flux boundary conditions were investigated, and the global stability of the disease free equilibrium and the epidemic equilibrium was established.
Abstract: In this paper, we investigate the complex dynamics of a reaction–diffusion S − I model incorporating demographic and epidemiological processes with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the disease free equilibrium and the epidemic equilibrium was established. In addition, the conditions of Turing instability were obtained and the Turing space in the parameters space were given. Based on these results, we present the evolutionary processes that involves organism distribution and their interaction of spatially distributed population with local diffusion, and find that the model dynamics exhibits a diffusion-controlled formation growth to “holes, holes–stripes, stripes, spots–stripes and spots” pattern replication. Furthermore, we indicate that the diseases’ spread is getting smaller with R 0 increasing, and the increasing the diffusion of infectious will increase the speed of diseases spreading. Our results indicate that the diffusion has a great influence on the spread of the epidemic and extend well the finding of spatiotemporal dynamics in the epidemic model.
TL;DR: In this paper, the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus was analyzed using Krasnoselskii's fixed point theorem.
Abstract: We present some results for the global attractivity of solutions for fractional differential equations involving Riemann–Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results.
TL;DR: In this article, the authors study a second order nonlinear boundary value problem subject to some nonlocal boundary conditions, which models a thermostat, and determine the Green's function and establish its useful properties which enables them to establish the existence of multiple positive solutions and to prove nonexistence results.
Abstract: We study a second order nonlinear boundary value problem subject to some nonlocal boundary conditions, which models a thermostat. The sensors are linear functionals, one gives feedback from part of the interval to a controller at one endpoint, the other gives feedback to a controller at the other endpoint. We determine the Green’s function and establish its useful properties which enables us to establish the existence of multiple positive solutions and to prove nonexistence results. We calculate all relevant constants in some explicit examples.
TL;DR: In this article, a free boundary problem for a system of partial difierential equations, which arises in a model of tumor growth with a necrotic core, is considered.
Abstract: We consider a free boundary problem for a system of partial difierential equations, which arises in a model of tumor growth with a necrotic core. For any positive numbers ‰ < R, there exists a radially symmetric stationary solution with tumor boundary r = R and necrotic core boundary r = ‰. The system depends on a positive parameter „, which describes the tumor aggressiveness. There also exists a sequence of values „2 < „3 < ¢¢¢ for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric solution branch.
Abstract: In this paper, we study fractional Schrodinger equations with potential and optimal controls. The first novelty is a suitable concept on a mild solution for our problems. Existence, uniqueness, local stability and attractivity, and data continuous dependence of mild solutions are also presented respectively. The second novelty is an initial study on the optimal control problems for the controlled fractional Schrodinger equations with potential. Existence and uniqueness of optimal pairs for the standard Lagrange problem are obtained.