Showing papers in "Nonlinear Analysis-real World Applications in 2011"

A class of fractional evolution equations and optimal controls

Abstract: This paper concerns the existence of mild solutions for semilinear fractional evolution equations and optimal controls in the α -norm. A suitable α -mild solution of the semilinear fractional evolution equations is introduced. The existence and uniqueness of α -mild solutions are proved by means of fractional calculus, singular version Gronwall inequality and Leray–Schauder fixed point theorem. The existence of optimal pairs of system governed by fractional evolution equations is also presented. Finally, an example is given for demonstration.

Existence of nontrivial solutions and high energy solutions for Schrödinger–Kirchhoff-type equations in RN

Abstract: In the present paper, the following Schrodinger–Kirchhoff-type problem: (1.1) − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in R N is studied and four new existence results for nontrivial solutions and a sequence of high energy solutions for problem (1.1) are obtained by using a symmetric Mountain Pass Theorem.

Magnetic field effects on free convection flow of a nanofluid past a vertical semi-infinite flat plate

Abstract: In this paper, we discuss similarity reductions for problems of magnetic field effects on free convection flow of a nanofluid past a semi-infinite vertical flat plate. The application of a one-parameter group reduces the number of independent variables by 1, and consequently the governing partial differential equation with the auxiliary conditions to an ordinary differential equation with the appropriate corresponding conditions. The differential equations obtained are solved numerically and the effects of the parameters governing the problem are discussed. Different kinds of nanoparticles were tested.

Existence and controllability results for fractional semilinear differential inclusions

Abstract: In this paper, we prove the existence and controllability results for fractional semilinear differential inclusions involving the Caputo derivative in Banach spaces. The results are obtained by using fractional calculation, operator semigroups and Bohnenblust–Karlin’s fixed point theorem. At last, an example is given to illustrate the theory.

Modeling herd behavior in population systems

Abstract: In this paper, we show that under suitable simple assumptions the classical two populations system may exhibit unexpected behaviors. Considering a more elaborated social model, in which the individuals of one population gather together in herds, while the other one shows a more individualistic behavior, we model the fact that interactions among the two occur mainly through the perimeter of the herd. We account for all types of populations’ interactions, symbiosis, competition and the predator–prey interactions. There is a situation in which competitive exclusion does not hold: the socialized herd behavior prevents the competing individualistic population from becoming extinct. For the predator–prey case, sustained limit cycles are possible, the existence of Hopf bifurcations representing a distinctive feature of this model compared with other classical predator–prey models. The system’s behavior is fully captured by just one suitably introduced new threshold parameter, defined in terms of the original model parameters.

Bifurcation and chaotic behavior of a discrete-time predator–prey system ☆

Abstract: The dynamics of a discrete-time predator–prey system is investigated in the closed first quadrant R + 2 . It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of R + 2 by using a center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as orbits of period 7, 14, 21, 63, 70, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, quasi-periodic orbits and chaotic sets. These results show far richer dynamics of the discrete model compared with the continuous model. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

Synchronization of delayed complex dynamical networks with impulsive and stochastic effects

Abstract: In this paper, the globally exponential synchronization of delayed complex dynamical networks with impulsive and stochastic perturbations is studied The concept named “average impulsive interval” with “elasticity number” of impulsive sequence is introduced to get a less conservative synchronization criterion By comparing with existing results, in which maximum or minimum of impulsive intervals are used to derive the synchronization criterion, the proposed synchronization criterion increases (or decreases) the impulse distances, which leads to the reduction of the control cost (or enhance the robustness of anti-interference) as the most important characteristic of impulsive synchronization techniques It is discovered in our criterion that “elasticity number” has influence on synchronization of delayed complex dynamical networks but has no influence on that of non-delayed complex dynamical networks Numerical simulations including a small-world network coupled with delayed Chua’s circuit are given to show the effectiveness and less conservativeness of the theoretical results

Function projective synchronization for fractional-order chaotic systems

Abstract: This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between the fractional-order Lorenz systems with different orders, and achieve synchronization between the fractional-order Lorenz system and fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.

Solutions of incompressible hydrodynamic flow of liquid crystals

Abstract: In this paper, we investigate the incompressible hydrodynamic flow of the nematic liquid crystals in dimension N ( N = 2 or 3). We obtain the local existence and uniqueness of the solution if the initial density ρ 0 ≥ 0 . Particularly, if ρ 0 has a positive bound from below, and N = 2 , we get the global existence and uniqueness of the solution with small initial data.

Global weak solutions in a chemotaxis system with large singular sensitivity

Abstract: We study nonnegative radially symmetric solutions of the chemotaxis system (⋆){ut=Δu−χ∇⋅(uv∇v),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, in a ball Ω⊂Rn, n≥2, with parameter χ>0 and radially symmetric initial data u0∈C0(Ω) and v0∈W1,∞(Ω) satisfying u0≥0 and v0>0 in Ω. A generalized solution concept is introduced for the Neumann problem associated with (⋆), and within this concept global-in-time solutions are shown to exist regardless of the size of χ>0. This extends previous results which assert global existence of some weak solutions in the case χ

Adaptive fuzzy tracking control for the chaotic permanent magnet synchronous motor drive system via backstepping

Abstract: An adaptive fuzzy control method is developed to suppress chaos in the permanent magnet synchronous motor drive system via backstepping technology. Fuzzy logic systems are used to approximate unknown nonlinearities and an adaptive backstepping technique is employed to construct controllers. Compared with the conventional backstepping, the designed fuzzy controllers’ structure is very simple. The simulation results indicate that the proposed control scheme can suppress the chaos of PMSM drive systems and track the reference signal successfully even under the parameter uncertainties.

Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria

Abstract: This paper presents a new 3-D autonomous chaotic system, which is topologically non-equivalent to the original Lorenz and all Lorenz-like systems. Of particular interest is that the chaotic system can generate double-scroll chaotic attractors in a very wide parameter domain with only two stable equilibria. The existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear. Finally, the complicated dynamics are studied by virtue of theoretical analysis, numerical simulation and Lyapunov exponents spectrum. The obtained results clearly show that the chaotic system deserves further detailed investigation.

Lag synchronization of complex networks via pinning control

Abstract: This paper mainly investigates the lag synchronization of complex networks via pinning control. Without assuming the symmetry and irreducibility of the coupling matrix, sufficient conditions of lag synchronization are obtained by adding controllers to a part of nodes. Particularly, the following two questions are solved: (1) How many controllers are needed to pin a coupled complex network to a homogeneous solution? (2) How should we distribute these controllers? Finally, a simple example is provided to demonstrate the effectiveness of the theory.

Stability and bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response

Abstract: The paper studies the dynamical behaviors of a discrete predator–prey system with nonmonotonic functional response. The local stability of equilibria of the model is obtained. The model undergoes flip bifurcation and Hopf bifurcation by using the center manifold theorem and the bifurcation theory. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the model, such as the period-doubling bifurcation in periods 2, 4 and 8, and quasi-periodic orbits and chaotic sets. The most interesting aspect is choosing the same parameters and the initial value of the model; then we vary the parameter K , and obtain series bifurcations, such as flip bifurcation and Hopf bifurcation.

Stability and Hopf bifurcation in a diffusive predator–prey system with delay effect

Abstract: This paper is concerned with a delayed predator–prey system with diffusion effect. First, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the distribution of the eigenvalues. Next the direction and the stability of Hopf bifurcation are determined by the normal form theory and the center manifold reduction for partial functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Global existence for a haptotaxis model of cancer invasion with tissue remodeling

Abstract: This paper considers a 3×3 haptotaxis model of cancer invasion with tissue remodeling. The model consists of a parabolic haptotaxis partial differential equation (PDE) describing the evolution of tumor cell density, an ordinary differential equation modeling the proteolysis and the remodeling of the extracellular matrix (ECM), and a parabolic PDE governing the evolution of the matrix degrading enzyme (MDE) concentration. In addition to random diffusion, tumor cells are biased towards higher ECM density, which is referred to as haptotaxis. Under a restrictive assumption on the coefficients, the global existence and uniqueness of classical solutions to the model in two dimensions, along with the boundedness of solutions in two and three dimensions, is proved by establishing some delicate a priori estimates.

Anti-periodic solution for delayed cellular neural networks with impulsive effects

Abstract: In this paper, we discuss anti-periodic solution for delayed cellular neural networks with impulsive effects. By means of contraction mapping principle and Krasnoselski’s fixed point theorem, we obtain the existence of anti-periodic solution for neural networks. By establishing a new impulsive differential inequality, using Lyapunov functions and inequality techniques, some new results for exponential stability of anti-periodic solution are obtained. Meanwhile, an example and numerical simulations are given to show that impulses may change the exponentially stable behavior of anti-periodic solution.

Global stability of an epidemic model with latent stage and vaccination

Abstract: For an epidemic model with latent stage and vaccination for the newborns and susceptibles, we establish that the global dynamics are completely determined by the basic reproduction number R 0 . More specifically, we prove that, if R 0 ≤ 1 , then the disease-free equilibrium is globally asymptotically stable, that is, the disease dies out eventually; if R 0 > 1 , then there exists a unique endemic equilibrium and it is globally asymptotically stable in the interior of the feasible region, that is, the disease persists in the population. In this paper, by the proof of global stability, we propose an approach for determining the Lyapunov function and proving the negative definiteness or semidefiniteness of its derivative. Our proof shows that, for a given Lyapunov function, its derivative should be arranged in different forms for the different values of parameters to prove the negative definiteness or semidefiniteness of its derivative.

Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters

Abstract: This paper presents the adaptive anti-synchronization of a class of chaotic complex nonlinear systems described by a united mathematical expression with fully uncertain parameters. Based on Lyapunov stability theory, an adaptive control scheme and adaptive laws of parameters are developed to anti-synchronize two chaotic complex systems. The anti-synchronization of two identical complex Lorenz systems and two different complex Chen and Lu systems are taken as two examples to verify the feasibility and effectiveness of the presented scheme.

On electromagnetic wave propagation in fractional space

Abstract: The wave equation has very important role in many areas of physics. It has a fundamental meaning in classical as well as quantum field theory. With this view, one is strongly motivated to discuss solutions of the wave equation in all possible situations. The wave equation in fractional space can effectively describe the wave propagation phenomenon in fractal media. In this chapter, exact solutions of different forms of wave equation in \(D\)-dimensional fractional space are provided, which describe the phenomenon of electromagnetic wave propagation in fractional space.

Global dynamics of Nicholson-type delay systems with applications

Abstract: Models of marine protected areas and B-cell chronic lymphocytic leukemia dynamics that belong to the Nicholson-type delay differential systems are proposed. To study the global stability of the Nicholson-type models we construct an exponentially stable linear system such that its solution is a solution of the nonlinear model. Explicit conditions of the existence of positive global solutions, lower and upper estimations of solutions, and the existence and uniqueness of a positive equilibrium were obtained. New results, obtained for the global stability and instability of equilibria solutions, extend known results for the scalar Nicholson models. The conditions for the stability test are quite practical, and the methods developed are applicable to the modeling of a broad spectrum of biological processes. To illustrate our finding, we study the dynamics of the fish populations in Marine Protected Areas.

Spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge

Abstract: In this paper, we investigate the spatiotemporal dynamics of a two-dimensional predator–prey model, which is based on a modified version of the Leslie–Gower scheme incorporating a prey refuge. We establish a Lyapunov function to prove the global stability of the equilibria with diffusion and determine the Turing space in the spatial domain. Furthermore, we perform a series of numerical simulations and find that the model dynamics exhibits complex Turing pattern replication: stripes, cold/hot spots-stripes coexistence and cold/hot spots patterns. The results indicate that the effect of the prey refuge for pattern formation is tremendous. This may enrich the dynamics of the effect of refuge on the predator-prey systems.

Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design

Abstract: In this paper, the adaptive backstepping design is proposed for the full state hybrid projective synchronization between two different chaotic systems with fully unknown parameters. Based on the design, the synchronization of two uncertain chaotic systems is realized only by using one controller, and the unknown parameters are identified through the corresponding parameter update laws. The uncertain Genesio–Tesi chaotic system and Lorenz system are chosen as examples for detailed description of the method. Finally, some numerical simulations are given to illustrate the effectiveness of the proposed method.

Hyperchaotic secure communication via generalized function projective synchronization

Abstract: This paper presents two different hyperchaotic secure communication schemes by using generalized function projective synchronization (GFPS), where the drive and response systems could be synchronized up to a desired scaling function matrix. The unpredictability of the scaling functions can additionally enhance the security of communication. First, a hyperchaotic secure communication scheme applying GFPS of the uncertain Chen hyperchaotic system is proposed. The transmitted information signal is modulated into the parameter of the Chen hyperchaotic system in the transmitter and it is assumed that the parameter of the receiver system is unknown. Based on the Lyapunov stability theory and the adaptive control technique, the controllers are designed to make two identical Chen hyperchaotic systems with unknown parameter asymptotically synchronized; thus, the uncertain parameter of the receiver system is identified. The information signal can be recovered accurately by the estimated parameter. Secondly, another secure communication scheme by the coupled GFPS of the Chen hyperchaotic system is introduced. The information signal transmitted can be extracted exactly through simple operation in the receiver. The corresponding theoretical proofs and numerical simulations demonstrate the validity and feasibility of the proposed hyperchaotic secure communication schemes.

Stabilization and tracking control for a class of nonlinear systems

Abstract: This paper discusses stabilization and tracking control using linear matrix inequalities for a class of systems with Lipschitz nonlinearities. A nonlinear state feedback stabilization control is proposed for systems containing a more general case of Lipschitz nonlinearity. The main objective of the present study is to provide, for multi-input multi-output nonlinear systems, a tracking control approach based on nonlinear state feedback, which guarantees global asymptotic output and state tracking with zero tracking error in the steady state. Further, the tracking control is formulated for optimal disturbance rejection, using L 2 gain reduction based performance criteria. The proposed methodologies are illustrated herein using two simulation examples of chaotic and unstable dynamical systems.

Solutions for a p(x)-Kirchhoff type equation with Neumann boundary data☆

Abstract: This paper is concerned with the existence and multiplicity of solutions to a class of p ( x ) -Kirchhoff type problem with Neumann boundary data of the following form { − M ( ∫ Ω 1 p ( x ) ( | ∇ u | p ( x ) + | u | p ( x ) ) d x ) ( div ( | ∇ u | p ( x ) − 2 ∇ u ) − | u | p ( x ) − 2 u ) = f ( x , u ) in Ω , ∂ u ∂ υ = 0 on ∂ Ω . By means of a direct variational approach and the theory of the variable exponent Sobolev spaces, under appropriate assumptions on f and M , we obtain a number of results on the existence and multiplicity of solutions for the problem. In particular, we also obtain some results which can be considered as extensions of the classical result named “combined effects of concave and convex nonlinearities”. Moreover, the positive solutions and the regularity of weak solutions of the problem are considered.

Stabilization of complex dynamical networks with noise disturbance under performance constraint

Abstract: Under given performance constraint, this paper studies the stabilization problem of general dynamical network subject to noise disturbance. The newly presented dynamical network model includes both intrinsic disturbance of single node and communication noise over the network connections, which appear typically in a network environment. Single controller is pinned into one of the nodes for the exponential stabilization of dynamical network, and the prescribed performance constraint is satisfied. The reason why only one controller is valid for stabilization of dynamical network is the full utilization of network’s local connections. One important feature of this paper is the introduction of the performance constraint concept into the stabilization of complex dynamical network with intrinsic and communication noises. The derived criteria are expressed in terms of linear matrix inequalities (LMIs), which are easy to be verified by resorting to recently developed algorithm. Numerical example is utilized to illustrate the effectiveness of the proposed results.

Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model

Abstract: For an age-structured SIR epidemic model, which is described by a system of partial differential equations, the global asymptotic stability of an endemic equilibrium in the situation where the basic reproduction number R 0 is greater than unity has been an open problem for decades. In the present paper, we construct a multigroup epidemic model regarded as a generalization of the model, and study the global asymptotic stability of each of its equilibria. By discretizing the multigroup model with respect to the age variable under some parameter assumptions, we first rewrite the PDE system into an ODE system, and then, applying the classical method of Lyapunov functions and a recently developed graph-theoretic approach with an original idea of maximum value functions, we prove that the global asymptotic stability of each equilibrium of the discretized system is completely determined by R 0 , namely, the disease-free equilibrium is globally asymptotically stable if R 0 ≤ 1 , while an endemic equilibrium exists uniquely and is globally asymptotically stable if R 0 > 1 . A numerical example illustrates that a numerical solution of R 0 for the discretized ODE system becomes closer to that for the original PDE system as the step size of the discretization decreases.

Individually-based Markov processes modeling nonlinear systems in mathematical biology

Abstract: In the present paper the general approach that allows one to construct the individually-based Markov processes describing various systems in mathematical biology (or in other applied sciences) is presented. The Markov processes are of a jump type and the starting point is the related linear equations. They describe at the micro–scale level the behavior of a large number N of interacting entities (particles, agents, cells, individuals, …). The large entity limit (“ N → ∞ ”) is studied and the intermediate level (the meso–scale level) is given in terms of nonlinear kinetic–type equations. Finally the corresponding systems of nonlinear ODEs (or PDEs) at the macroscopic level (in terms of concentrations of the interacting subpopulations) are obtained. Mathematical relationships between these three possible descriptions are presented and explicit error estimates are given. The general framework is applied to propose the microscopic and mesoscopic models that correspond to well known systems of nonlinear equations in biomathematics. The paper generalises the previous approach resulting in bilinear equations of the Boltzmann–type at the mesoscopic level.

Bifurcation and chaos analysis of spur gear pair with and without nonlinear suspension

Abstract: This study performs a systematic analysis of the dynamic behavior of a single degree-of-freedom spur gear system with and without nonlinear suspension. The dynamic orbits of the system are observed using bifurcation diagrams plotted using the dimensionless damping coefficient and the dimensionless rotational speed ratio as control parameters. The onset of chaotic motion is identified from the phase diagrams, power spectra, Poincare maps, Lyapunov exponents and fractal dimension of the gear system. The numerical results reveal that the system exhibits a diverse range of periodic, sub-harmonic and chaotic behaviors. The results presented in this study provide an understanding of the operating conditions under which undesirable dynamic motion takes place in a spur gear system and therefore serve as a useful source of reference for engineers in designing and controlling such systems.