Theory Of Functional Differential Equations Research Articles

Symmetric periodic solutions of delay-coupled optoelectronic oscillators

Delay-coupled optoelectronic oscillators are considered. These structures are based on mutually coupled oscillators which oscillate at the same frequency. By taking the time delay as a bifurcation parameter, the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay are investigated, and then stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Using the symmetric functional differential equation theories combined with the representation theory of Lie groups, the multiple Hopf bifurcations of the equilibrium are demonstrated. In particular, we find that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling. The existence of multiple branches of bifurcating periodic solutions and their spatio-temporal patterns are obtained. Some numerical simulations are used to illustrate the effectiveness of the obtained results.

Globally Asymptotic Stability Analysis for Genetic Regulatory Networks with Mixed Delays: An M-Matrix-Based Approach

This paper deals with the problem of globally asymptotic stability for nonnegative equilibrium points of genetic regulatory networks (GRNs) with mixed delays (i.e., time-varying discrete delays and constant distributed delays). Up to now, all existing stability criteria for equilibrium points of the kind of considered GRNs are in the form of the linear matrix inequalities (LMIs). In this paper, the Brouwer's fixed point theorem is employed to obtain sufficient conditions such that the kind of GRNs under consideration here has at least one nonnegative equilibrium point. Then, by using the nonsingular M-matrix theory and the functional differential equation theory, M-matrix-based sufficient conditions are proposed to guarantee that the kind of GRNs under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. The M-matrix-based sufficient conditions derived here are to check whether a constant matrix is a nonsingular M-matrix, which can be easily verified, as there are many equivalent statements on the nonsingular M-matrices. So, in terms of computational complexity, the M-matrix-based stability criteria established in this paper are superior to the LMI-based ones in literature. To illustrate the effectiveness of the approach proposed in this paper, several numerical examples and their simulations are given.

EPIDEMIC SPREADING AND GLOBAL STABILITY OF A NEW SIS MODEL WITH DELAY ON HETEROGENEOUS NETWORKS

In this paper, we study a susceptible-infected-susceptible (SIS) model with time delay on complex heterogeneous networks. Here, the delay describes the incubation period in the vector population. We calculate the epidemic threshold by using a Lyapunov functional and some analytical methods, and find that adding delay increases the epidemic threshold. Then, we prove the global stability of disease-free and endemic equilibria by using the theory of functional differential equations. Furthermore, we show numerically that the epidemic threshold of the new model may change along with other factors, such as the infectivity function, the heterogeneity of the network, and the degrees of nodes. Finally, we find numerically that the delay can affect the convergence speed at which the disease reaches equilibria.

M-matrix-based globally asymptotic stability criteria for genetic regulatory networks with time-varying discrete and unbounded distributed delays

The problem of globally asymptotic stability for nonnegative equilibrium points of genetic regulatory networks (GRNs) with time-varying discrete delays and unbounded distributed delays is considered. So far, there are very few results concerning the problem; and in which the nonnegativity of equilibrium points is neglected. In this paper, the existence of nonnegative equilibrium points is firstly presented. Then, by using the nonsingular M-matrix theory and the functional differential equation theory, M-matrix-based sufficient conditions are proposed to guarantee that the kind of GRNs under consideration here has a unique nonnegative equilibrium point which is globally asymptotically stable. The M-matrix-based stability criteria derived here can be easily verified, since they are to check whether a constant matrix is a nonsingular M-matrix. Several numerical examples are offered to illustrate the effectiveness of the approach proposed in this paper.

On the Lyapunov theory for functional differential equations of fractional order

On the Lyapunov theory for functional differential equations of fractional order

Local determinacy of prices in an overlapping generations model with continuous trading

We characterize the determinacy properties of the intertemporal equilibrium for a continuous-time, pure-exchange, overlapping generations economy with logarithmic preferences. Using recent advances in the theory of functional differential equations, we show that the equilibrium is locally unique and that prices converge to a balanced growth path and are determined.

Numerical Solution of the Inventory Balance Delay Differential Equation

Inventory represents an essential part of current assets, which are typically characterized by their transience. This paper aims to outline a numerical solution of the inventory balance equation supplemented by an order-up-to replenishment policy for a case in which the problem is described by a differential equation with delayed argument. The results are demonstrated on a specific example and the behaviour of the model is presented using a computer simulation. The results are graphically shown in the Maple system. The solution makes use of the theory of functional differential equations, especially the part dealing with differential equations with delayed arguments.

Исследование одной периодической задачи методами теории функционально-дифференциальных уравнений

Исследование одной периодической задачи методами теории функционально-дифференциальных уравнений

Partial stability of some guidance dynamic systems with delayed line-of-sight angular rate

In this paper, for a class of the guidance dynamic system proposed by Moon, Kim, and Kim (Journal of Guidance, Control and Dynamics, vol. 24, no. 4, pp. 659–664, 2001) and Shafiei and Binazadeh (ISA Transactions, vol. 51, pp. 298–303, 2012), a new nonlinear missile guidance law with delayed line-of-sight (LOS) angular rate is introduced, and then partial stability is studied by using the method of Lyapunov functionals and stability theory of functional differential equations. The time delay in the LOS angular rate comes from the missile guidance dynamic system in which a low-cost sensor is used to obtain LOS rate information by image processing techniques. Furthermore, for a class of classical proportional navigation control system with delayed line-of-sight rate studied by Dhananjay, Lum, and Xu (IEEE Transactions on Control Systems Technology, vol. 21, pp. 247–253, 2013), a new stability region is obtained by using the method of Lyapunov-Razumikhin.

Delay-dependent asymptotic stability of mobile ad-hoc networks: A descriptor system approach

In order to analyze the capacity stability of the time-varying-propagation and delay-dependent of mobile ad-hoc networks (MANETs), in this paper, a novel approach is proposed to explore the capacity asymptotic stability for the delay-dependent of MANETs based on non-cooperative game theory, where the delay-dependent conditions are explicitly taken into consideration. This approach is based on the Lyapunov—Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) technique. A corresponding Lyapunov—Krasovskii functional is introduced for the stability analysis of this system with use of the descriptor and “neutral-type” model transformation without producing any additional dynamics. The delay-dependent stability criteria are derived for this system. Conditions are given in terms of linear matrix inequalities, and for the first time referred to neutral systems with the time-varying propagation and delay-dependent stability for capacity analysis of MANETs. The proposed criteria are less conservative since they are based on an equivalent model transformation. Furthermore, we also provide an effective and efficient iterative algorithm to solve the constrained stability control model. Simulation experiments have verified the effectiveness and efficiency of our algorithm.

On discreteness of spectrum and positivity of the Green’s function for a second order functional-differential operator on semiaxis

We study conditions of discreteness of spectrum of the operator defined by , . The operator has two singularities at the ends of the interval . The second question is positivity of solutions of the equation under boundary conditions , . The used abstract scheme is close to the well-known MS Birman’s method in the spectral theory of self-adjoint operators. Conditions for discreteness of spectrum and positivity of the Green’s operator are obtained. The result relates to the MS Birman’s result on the necessary and sufficient condition for discreteness of spectrum of a polar-differential operation. The results may be interesting for researchers in qualitative theory of functional-differential equations and spectral theory of self-adjoint operators. MSC: Primary 34K08; 34K10; secondary 34K12.

Analysis of bifurcation in a symmetric system of m coupled oscillators with delay

We consider a system of m coupled oscillators with k coupling parts with delay. Using the symmetric functional differential equation theories, we demonstrate the multiple Hopf bifurcation of the equilibrium at the origin. The existence of multiple branches of bifurcating periodic solution is obtained. Then we study the case of coupling BVP oscillators. Global existence of periodic solutions is established.

Stability and estimate of solution to uncertain neutral delay systems

The coefficients and delays in models describing various processes are usually obtained as a results of measurements and can be obtained only approximately. We deal with the question of how to estimate the influence of ‘mistakes’ in coefficients and delays on solutions’ behavior of the delay differential neutral system x i ′ (t)− q i (t) x i ′ (t− θ i (t))+ ∑ j = 1 n ( p i j (t)−Δ p i j (t)) x j (t− τ i j (t)−Δ τ i j (t))= f i (t), i=1,…,n, t∈[0,∞). This topic is known in the literature as uncertain systems or systems with interval defined coefficients. The goal of this paper is to obtain stability of uncertain systems and to estimate the difference between solutions of a ‘real’ system with uncertain coefficients and/or delays and corresponding ‘model’ system. We develop the so-called Azbelev W-transform, which is a sort of the right regularization allowing researchers to reduce analysis of boundary value problems to study of systems of functional equations in the space of measurable essentially bounded functions. In corresponding cases estimates of norms of auxiliary linear operators (obtained as a result of W-transform) lead researchers to conclusions about existence, uniqueness, positivity and stability of solutions of given boundary value problems. This method works efficiently in the case when a ‘model’ used in W-transform is ‘close’ to a given ‘real’ system. In this paper we choose, as the ‘models’, systems for which we know estimates of the resolvent Cauchy operators. We demonstrate that systems with positive Cauchy matrices present a class of convenient ‘models’. We use the W-transform and other methods of the general theory of functional differential equations. Positivity of the Cauchy operators is studied and then used in the analysis of stability and estimates of solutions.Results: We propose results about exponential stability of the given system and obtain estimates of difference between the solution of this uncertain system and the ‘model’ system x i ′ (t)− q i (t) x i ′ (t− θ i (t))+ ∑ j = 1 n p i j (t) x j (t− τ i j (t))= f i (t), i=1,…,n, t∈[0,∞). New tests of stability and in the future of existence and uniqueness of boundary value problems for neutral delay systems can be obtained on the basis of this technique.

Hopf bifurcation in a partial dependent predator–prey system with multiple delays

In this paper, a partial dependent predator–prey system with multiple delays is investigated. By choosing τ1, τ2 and τ3 as bifurcating parameters, we show that Hopf bifurcations occur. In addition, by using theory of functional differential equation and Hassard's method, explicit algorithms for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, numerical simulations are performed to support the analytical results, and the chaotic behaviors are observed.

Control of Hopf Bifurcation and Chaos in a Delayed Lotka-Volterra Predator-Prey System with Time-Delayed Feedbacks

A delayed Lotka-Volterra predator-prey system with time delayed feedback is studied by using the theory of functional differential equation and Hassard’s method. By choosing appropriate control parameter, we investigate the existence of Hopf bifurcation. An explicit algorithm is given to determine the directions and stabilities of the bifurcating periodic solutions. We find that these control laws can be applied to control Hopf bifurcation and chaotic attractor. Finally, some numerical simulations are given to illustrate the effectiveness of the results found.

Hopf bifurcation of traveling wave solutions of delayed Fisher-KPP equation

The dynamical behavior of the traveling wave front solutions of the delayed Fisher-KPP equation is studied by employing the theory of functional differential equation. It is found that, when the delay term τ comes through a threshold, one equilibrium point is always unstable and there exists a periodic solution near the other equilibrium point in the traveling wave equation. By using the theory of center manifold and normal form, the stability of the bifurcating periodic solution, which corresponds to the periodic traveling wave solution of the original equation, is obtained. Finally, some numerical simulations are also given to illustrate our results of theoretical analysis.

DYNAMIC PROPERTIES OF A SYMMETRICALLY CONSERVATIVE TWO-MASS SYSTEM

A symmetrically conservative two-mass system with delay is considered. Using the symmetric functional differential equation theories, multiple Hopf bifurcations of the equilibrium at the origin are demonstrated. The existence of multiple branches of bifurcating periodic solution and their spatiotemporal patterns are obtained. Some numerical examples and the corresponding numerical simulations are used to illustrate the effectiveness of the obtained results.

Delayed Feedback Control and Bifurcation Analysis of an Autonomy System

An autonomy system with time-delayed feedback is studied by using the theory of functional differential equation and Hassard’s method; the conditions on which zero equilibrium exists and Hopf bifurcation occurs are given, the qualities of the Hopf bifurcation are also studied. Finally, several numerical simulations are given; which indicate that when the delay passes through certain critical values, chaotic oscillation is converted into a stable state or a stable periodic orbit.

A new delayed projection neural network for solving quadratic programming problems subject to linear constraints

In this paper, a new projection neural network with two time delays is proposed for solving quadratic programming problems subject to linear constraints. And we give the existence of the continuous solution. By the theory of functional differential equation, the proposed neural network is proved to be globally exponentially stable under some conditions. Simulation results with some applications show the performance and characteristic of the proposed neural network.

Pinning synchronization of time-varying delay coupled complex networks with time-varying delayed dynamical nodes

This paper deals with the pinning synchronization of nonlinearly coupled complex networks with time-varying coupling delays and time-varying delays in the dynamical nodes. We control a part of the nodes of the complex networks by using adaptive feedback controllers and adjusting the time-varying coupling strengths. Based on the Lyapunov—Krasovskii stability theory for functional differential equations and a linear matrix inequality (LMI), some sufficient conditions for the synchronization are derived. A numerical simulation example is also provided to verify the correctness and the effectiveness of the proposed scheme.