Delayed Reaction Diffusion Research Articles

Synchronization of reaction–diffusion neural networks with random time-varying delay via intermittent boundary control

Under boundary measurement (BM), this paper solves the synchronization problem via intermittent boundary proportional–derivative (PD) control of delayed reaction–diffusion neural networks (RDNNs), where the random time-varying delay (RTVD) belongs to two intervals and is considered by a probabilistic way to take the influence of uncertain factors. The intermittent boundary PD synchronization scheme, which considers the inevitable actuator limitations, is based on BM and only available at some specified boundary position. By using the inequality techniques and switching Lyapunov functional, the mean square exponential stability (MSES) of the synchronization error system (SES) containing the response and drive dynamics is ensured by some synchronization criteria, which are established via an intermittent boundary PD controller under BM and expressed as linear matrix inequalities. Lastly, the proposed intermittent boundary PD synchronization approach is verified by a numerical example.

Intermittent State Observer Design for Neural Networks With Reaction–Diffusion Terms Using Partial Measurements

This article develops a novel state observer for delayed reaction–diffusion neural networks by utilizing incomplete measurements. To reduce the transmission cost efficiently, the space domain is divided into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L$</tex-math> </inline-formula> parts and only partial information needs to be measured in every subdomain, such as a point in one-dimensional space, a line and a plane in two-and three-dimensional space, respectively. In addition, the time domain is divided: the measured output signals are transmitted intermittently. Then, new conditions that assure the asymptotic stability of observation error system are derived based on the Lyapunov direct method and several inequality techniques. Finally, the proposed approach’s effectiveness is demonstrated via three numerical examples.

Boundary output feedback stabilization of state delayed reaction–diffusion PDEs

This paper addresses the problem of output feedback stabilization of a class of 1-D reaction–diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition. The system output is selected as either a Dirichlet or Neumann boundary trace. The developed control design procedure does not aim to compensate, but rather to dominate the state delayed term. This is achieved by introducing an observer that only estimates a finite number of modes of the PDE while applying a suitable pole shifting-based procedure on a truncated model that captures the unstable part of the plant. We show that the reported output feedback control strategy always achieves, for an arbitrarily given value of the state delay, the exponential stabilization of the plant in H1-norm. A notable feature of the domination-based approach is that the tuning of the controller and observer gains, as well as the subsequent stability conditions, appears to be independent of the value of the state delay.

Adaptive event-triggered extended dissipative synchronization of delayed reaction–diffusion neural networks under deception attacks

Under spatially averaged measurements (SAMs) and deception attacks, this article mainly studies the problem of extended dissipativity output synchronization of delayed reaction–diffusion neural networks via an adaptive event-triggered sampled-data (AETSD) control strategy. Compared with the existing ETSD control methods with constant thresholds, our scheme can be adaptively adjusted according to the current sampling and latest transmitted signals and is realized based on limited sensors and actuators. Firstly, an AETSD control scheme is proposed to save the limited transmission channel. Secondly, some synchronization criteria under SAMs and deception attacks are established by utilizing Lyapunov–Krasovskii functional and inequality techniques. Then, by solving linear matrix inequalities (LMIs), we obtain the desired AETSD controller, which can satisfy the specified level of extended dissipativity behaviors. Lastly, one numerical example is given to demonstrate the validity of the proposed method.

Fixed-time periodic stabilization of discontinuous reaction–diffusion Cohen–Grossberg neural networks

This paper aims to study the fixed-time stabilization of a class of delayed discontinuous reaction–diffusion Cohen–Grossberg neural networks. Firstly, by providing some relaxed conditions containing indefinite functions and based on inequality techniques, a new fixed-time stability lemma is given, which can improve the traditional ones. Secondly, based on state-dependent switching laws, the periodic wave solution of the formulated networks is transformed into the periodic solution of ordinary differential system. By utilizing differential inclusions theory and coincidence theorem, the existence of periodic solutions is obtained. Thirdly, based on the new fixed-time stability lemma, the periodic solutions are stabilized at zero in a fixed-time, which is a new topic on reaction–diffusion networks. Moreover, the established criteria are all delay-dependent, which are less conservative than the previous delay-independent ones for ensuring the stabilization of delayed reaction–diffusion networks. Finally, two examples give numerical explanations of the proposed results and highlight the influence of delays.

Spatiotemporal dynamics optimization of a delayed reaction–diffusion mussel–algae model based on PD control strategy

The dynamic control over time evolution of ordinary differential systems has developed rapidly in recent years, but how to control the spatiotemporal evolution dynamics of partial differential systems is still an open question. Turing pattern is a major spatiotemporal evolution behavior in mussel-algal ecosystems and its control can pull the ecosystem back from the borderline of collapse and make it more stable. However, there has been relatively little research on the optimal control of Turing pattern in the mussel-algal system. In this paper, we first attempt to propose a proportional-derivative (PD) control strategy for the reaction–diffusion mussel–algae model under the influence of time delays. Turing instability driven by diffusion occurs in the diffusive mussel–algae model without control. The introduction of PD control can effectively suppress the occurrence of Turing instability and keep the ecological model stable in both time and space. When the time delay exceeds a critical value, the diffusive mussel–algae model will lose stability and undergo a Hopf bifurcation. PD controller can make Hopf bifurcation occur in advance or in hysteresis by changing the threshold point of Hopf bifurcation. The intrinsic mechanism of Hopf bifurcation is analyzed by means of the formal theory and the central manifold theorem. The numerical simulations demonstrate the efficiency and feasibility of the PD control scheme.

Exponentially fitted numerical method for solving singularly perturbed delay reaction-diffusion problem with nonlocal boundary condition

ObjectivesIn this article, a singularly perturbed delay reaction-diffusion problem with nonlocal boundary conditions is considered. The exponential fitting factor is introduced to treat the solutions inside the boundary layer which occur due to perturbation parameter. The considered problem has interior layer at s = 1 and strong boundary layers at s = 0 and s= 2. We proposed an exponentially fitted finite difference method to solve the considered problem. The nonlocal boundary condition is treated using Composite Simpson’s frac{1}{3} rule.ResultThe stability and uniform convergence analysis of the proposed approach are established. The error estimation of the developed method is shown to be second-order uniform convergent. Two test examples were carried out to validate the applicability of the developed numerical method. The numerical results reflect the theoretical estimations.

Analysis and simulation of a delayed HIV model with reaction–diffusion and sliding control

Selection of an appropriate therapy threshold to restrain the virus load is still challenging on structured treatment interruptions (STIs) for HIV. In this paper, we ponder that how the sliding control and multistability to regulate the treatment effect through comprehensive dynamics of a virus-immune model. Firstly, based on piecewise therapy, we propose a delayed reaction–diffusion virus-immune model under the homogeneous Neumann boundary condition. Secondly, the existence and stabilities of five kinds of equilibria as well as the direction and stability of spatial Hopf bifurcation at regular equilibrium are investigated. Thirdly, the sliding domain and the boundary node bifurcations are addressed by theoretical analysis. Finally, we appraise the effects of therapy threshold, sliding domain and multistability on HIV therapy by simulations, and further seek out the appropriate therapy threshold for infected patients with given physiological parameters and present the corresponding principles. Our explorations will provide evidence for HIV and other disease therapies.

Intermittent Boundary Control for Synchronization of Fractional Delay Neural Networks With Diffusion Terms

This article studies the synchronization of new coupled fractional delayed reaction–diffusion neural networks with reaction terms satisfying the global Lipschitz condition via time-continuous and time-discontinuous boundary controllers. The realization of neural networks inevitably involves diffusion phenomena and time delays, and all the neurons of neural networks are interrelated. Considering these aspects, this study focuses on coupled fractional neural networks with time-delay and diffusion terms. A state-dependent boundary control (BC) is designed for when the state information is available, and a criterion is presented to ensure the synchronization of the considered systems. Considering the advantages of a time-discontinuous controller, an intermittent BC and a criterion of synchronization are given. When the state information cannot be fully obtained, a boundary-output-based observer is provided for estimating the states. Then, an observer-based intermittent boundary controller is given to ensure the synchronization. From the given criteria, the effects of time delay and the control time length on synchronization are analyzed. This research involves two key challenges: 1) consideration of the BC and intermittent control parameters in the system performance analysis and 2) clarification of the influence of system parameters on synchronization. These challenges are addressed using Poincaré’s inequality, the fractional Razumikhin-type theorem, and several properties of the Mittag-Leffler function are used to deal with the above difficulties. Examples show that our results are valid.

Symmetry-breaking bifurcations in a delayed reaction–diffusion equation

Symmetry-breaking bifurcations in a delayed reaction–diffusion equation

Stability and Bifurcation of a Delayed Reaction–Diffusion Model with Robin Boundary Condition in Heterogeneous Environment

In this paper, we investigate a reaction–diffusion model with delay and Robin boundary condition in heterogeneous environment. The existence, multiplicity and stability of spatially nonhomogeneous steady-state solutions and periodic solutions are studied by employing the Lyapunov–Schmidt reduction method. Moreover, the Hopf bifurcation direction is derived. It is observed that Robin boundary condition plays a crucial role in the Hopf bifurcation. More precisely, when the boundary effect is stronger than the interaction of the species within the region, there is no Hopf bifurcation no matter how the time delay [Formula: see text] changes. Finally, we illustrate our general theoretical results by an application to the Nicholson’s blowflies model.

Hopf bifurcation in a spatial heterogeneous and nonlocal delayed reaction–diffusion equation

A spatial heterogeneous and nonlocal delayed reaction–diffusion equation is investigated. The existence of the nontrivial steady-state solution is shown by steady-state bifurcation at the trivial solution. The form of the positive spatially nonhomogeneous steady state is given under certain conditions. Based on the form, its stability and Hopf bifurcation are analyzed so that the periodic and the quasi-periodic solutions near the spatially nonhomogeneous steady state are demonstrated.

Dynamics of a delayed Lotka–Volterra competition model with directed dispersal

We consider a diffusive Lotka–Volterra competition system with stage structure, where the intrinsic growth rates and the carrying capacities of the species are assumed spatially heterogeneous. Here, we also assume each of the competing populations chooses its dispersal strategy as the tendency to have a distribution proportional to a certain positive prescribed function. We give the effects of dispersal strategy, delay, the intrinsic growth rates and the competition parameters on the global dynamics of the delayed reaction diffusion model. Our result shows that competitive exclusion occurs when one of the diffusion strategies is proportional to the carrying capacity, while the other is not; while both populations can coexist if the competition favors the latter species. Finally, we point out that the method is also applied to the global dynamics of other kinds of delayed competition models.

Hopf bifurcation in a delayed reaction diffusion predator-prey model with weak Allee effect on prey and fear effect on predator

&lt;abstract&gt;&lt;p&gt;In this work, a Leslie-Gower model with a weak Allee effect on the prey and a fear effect on the predator is proposed. By using qualitative analyses, the local stability of the coexisting equilibrium and the existence of Turing instable are discussed. By analyzing the distribution of eigenvalues, the existence of a Hopf bifurcation is studied by using the gestation time delay as a bifurcation parameter. By utilizing the normal form method and the center manifold theorem, we calculate the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. We indicate that both the weak Allee effect on the prey and fear effect on the predator have an important impact on the dynamical behaviour of the new Leslie-Gower model. We also verify the obtained results by some numerical examples.&lt;/p&gt;&lt;/abstract&gt;

Asymptotic behavior of the principal eigenvalue and basic reproduction ratio for time-periodic reaction-diffusion systems with time delay

This paper is devoted to asymptotic behavior of the basic reproduction ratio for time-periodic and delay reaction-diffusion systems in the cases of small and large diffusion coefficients. First, we employ the generalized Kerin-Rutman theorem to establish the existence of the principal eigenvalue for periodic and cooperative reaction-diffusion systems with delay. Then, motivated by Kerin-Rutman theorem, we provide a method to estimate the upper and lower bounds of principal eigenvalues so as to obtain the asymptotic behavior of the principal eigenvalue. In view of the relationship between the basic reproduction ratio and the principal eigenvalue, we obtain the limiting profile of the basic reproduction ratio according to the asymptotic behavior of the principal eigenvalue.

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

Fixed-time boundary stabilisation for delay reaction–diffusion systems

This paper considers the problem of the boundary fixed-time stabilisation for delay reaction–diffusion systems (DRDSs). A new boundary controller is designed to achieve fixed time stability of DRDSs. Firstly, a boundary fixed-time controller is presented for DRDSs. Then based on the designed boundary controller, using Lyapunov functional method, fixed-time stability lemma and Poincaré's inequality, we obtain the criteria to guarantee the fixed-time stability of DRDSs. In addition, boundary fixed-time controller is proposed for uncertain DRDSs to reach robust fixed-time stability. Both the cases, the systems with or without uncertainties, are severally followed by a fixed time, which is an upper-bound of the settling time and irrelevant to the initial conditions. Finally, two simulation examples are given to illustrate the fixed-time stability of the controlled systems, which show the validity of our theoretical results.

Equivalence of MTS and CMR methods associated with the normal form of Hopf bifurcation for delayed reaction–diffusion equations

In this paper, we prove the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method for deriving the normal form of Hopf bifurcation for delayed reaction–diffusion equations. First, we generalize the MTS method associated with functional differential equations to delayed reaction–diffusion equation for solving the normal form of Hopf bifurcation. This provides a user-friendly approach of analyzing the stability of time-periodic solutions, particularly, for the engineering researchers who are not familiar with CMR method. Second, we outline the CMR method associated with the computation of normal form of Hopf bifurcation for delayed reaction–diffusion equations. Third, we prove that the two methods can derive equivalent up to the third order normal form of Hopf bifurcation. Finally, different types of examples are presented, such as reaction–diffusion equations with Neumann boundary condition and Dirichlet boundary conditions, single delay and two delays, spatially homogeneous and spatially inhomogeneous periodic solutions. We obtain the equivalent Hopf bifurcation normal forms results derived by using the CMR and MTS methods for above examples, respectively.

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

A uniformly convergent numerical method is presented for solving singularly perturbed time delay reaction-diffusion problems. Properties of the continuous solution are discussed. The Crank–Nicolson method is used for discretizing the temporal derivative, and an exponentially fitted tension spline method is applied for the spatial derivative. Using the comparison principle and solution bound, the stability of the method is analyzed. The proposed numerical method is second-order uniformly convergent. The theoretical analysis is supported by numerical test examples for various values of perturbation parameters and mesh size.

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

AbstractIn this article, we propose and analyze several numerical methods for the nonlinear delay reaction–diffusion system with smooth and nonsmooth solutions, by using Quasi‐Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2‐1σ schemes) in time. The optimal convergence results in the senses of L2‐norm and broken H1‐norm, and H1‐norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum‐of‐exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.