Theory Of Functional Differential Equations Research Articles

Asymptotic collective behaviors of a delayed multi‐particle system with multiple processes

It is a significant issue to explore the collective behaviors of the delayed complex systems. This paper studies asymptotic collective behaviors of a delayed multi‐particle system with multiple processes while particles interact directionally and locally. Up to now, the studies on dynamic behaviors of multi‐particle systems do not involve limited processing capacity, that is, the information queuing phenomenon is ignored. To this aim, we introduce multiple processes to formulate the evolution of a delayed multi‐particle system. To optimize the delay parameters, we choose the greedy principle to set the processing order. The main idea is to decompose the velocity into each process and use functional differential equation theory and matrix theory to constrain the velocity variation rate while the processing order is fixed. As the new results, in the non‐critical and general situation, when the time delay is less than or equal to the threshold value and the eigenvalue 1 of the average matrix is semi‐simple, we achieve the criterion of flocking and multi‐cluster with exponential convergence rate involving multiple processes. Furthermore, we get the lower bound estimate of the processing capacity.

Novel Results on Persistence and Attractivity of Delayed Nicholson's Blowflies System with Patch Structure

This paper is concerned with the dynamic characteristics of a class of Nicholson's blowflies system with patch structure and multiple pairs of distinct time-varying delays. We aim to find the influence of the distinct time-varying delays in the same reproductive function on its asymptotic behavior. First, we derive the global existence, positiveness and uniform persistence of solutions for the addressed system. Then, by employing the theory of functional differential equations, the fluctuation lemma and the technique of differential inequalities, we build up some new delay-dependent criteria for the global attractivity of the positive equilibrium point vector, which does not possess the same components. In addition, we exam the effectiveness and feasibility of the theoretical achievements by some numerical simulations.

The effects of fear and delay on a predator-prey model with Crowley-Martin functional response and stage structure for predator

<abstract><p>Taking into account the delayed fear induced by predators on the birth rate of prey, the counter-predation sensitiveness of prey, and the direct consumption by predators with stage structure and interference impacts, we proposed a prey-predator model with fear, Crowley-Martin functional response, stage structure and time delays. By use of the functional differential equation theory and Sotomayor's bifurcation theorem, we established some criteria of the local asymptotical stability and bifurcations of the system equilibrium points. Numerically, we validated the theoretical findings and explored the effects of fear, counter-predation sensitivity, direct predation rate and the transversion rate of the immature predator. We found that the functional response as well as the stage structure of predators affected the system stability. The fear and anti-predation sensitivity have positive and negative impacts to the system stability. Low fear level and high anti-predation sensitivity are beneficial to the system stability and the survival of prey. Meanwhile, low anti-predation sensitivity can make the system jump from one equilibrium point to another or make it oscillate between stability and instability frequently, leading to such phenomena as the bubble, or bistability. The fear and mature delays can make the system change from unstable to stable and cause chaos if they are too large. Finally, some ecological suggestions were given to overcome the negative effect induced by fear on the system stability.</p></abstract>

Dynamics of Bacterial white spot disease spreads in Litopenaeus Vannamei with time-varying delay.

In this paper, we mainly consider a eco-epidemiological predator-prey system where delay is time-varying to study the transmission dynamics of Bacterial white spot disease in Litopenaeus Vannamei, which will contribute to the sustainable development of shrimp. First, the permanence and the positiveness of solutions are given. Then, the conditions for the local asymptotic stability of the equilibriums are established. Next, the global asymptotic stability for the system around the positive equilibrium is gained by applying the functional differential equation theory and constructing a proper Lyapunov function. Last, some numerical examples verify the validity and feasibility of previous theoretical results.

Two-parameter bifurcations analysis of a delayed high-temperature superconducting maglev model with guidance force.

A modified high-temperature superconducting maglev model is studied in this paper, mainly considering the influence of time delay on the dynamic properties of the system. For the original model without time delay, there are periodic equilibrium points. We investigate its stability and Hopf bifurcation and study the bifurcation properties by using the center manifold theorem and the normal form theory. For the delayed model, we mainly study the co-dimension two bifurcations (Bautin and Hopf-Hopf bifurcations) of the system. Specifically, we prove the existence of Bautin bifurcation and calculate the normal form of Hopf-Hopf bifurcation through the bifurcation theory of functional differential equations. Finally, we numerically simulate the abundant dynamic phenomena of the system. The two-parameter bifurcation diagram in the delayed model is given directly. Based on this, some nontrivial phenomena of the system, such as periodic coexistence and multistability, are well presented. Compared with the original ordinary differential equation system, the introduction of time delay makes the system appear chaotic behavior, and with the increase in delay, the variation law between displacement and velocity becomes more complex, which provides further insights into the dynamics of the high-temperature superconducting maglev model.

Global attractivity of Nicholson’s blowflies system with patch structure and multiple pairs of distinct time-varying delays

This paper is intended to investigate a class of Nicholson’s blowflies system with patch structure and multiple pairs of distinct time-varying delays, we are interested in finding the influence of the distinct time-varying delays in the same reproductive function on its asymptotic behavior. By using the theory of functional differential equations, the fluctuation lemma, and the technique of differential inequalities, some new delay-dependent criteria on the global attractivity of the positive equilibrium point are established. In addition, the effectiveness and feasibility of the theoretical achievements are illustrated by some numerical simulations.

Bifurcations and multistability in a virotherapy model with two time delays

In this paper, we establish a delayed virotherapy model including infected tumor cells, uninfected tumor cells and free virus. In this model, both infected and uninfected tumor cells have special growth patterns, and there are at most two positive equilibria. We mainly analyze the stability and Hopf bifurcation of the model under different time delays. For the model without delay, we study the Hopf and Bogdanov–Takens bifurcations. For the delayed model, by center manifold theorem and normal form theory of functional differential equation, we study the direction of Hopf bifurcation and stability of the bifurcated periodic solution. Moreover, we prove the existence of Zero-Hopf bifurcation. Finally, some numerical simulations show the results of our theoretical calculations, and the dynamic behaviors near Zero-Hopf and Bogdanov–Takens point of the system are also observed in the simulations, such as bistability, periodic coexistence and chaotic behavior.

Bifurcation analysis of a food chain chemostat model with Michaelis-Menten functional response and double delays

<abstract><p>In this paper, we study a food chain chemostat model with Michaelis-Menten function response and double delays. Applying the stability theory of functional differential equations, we discuss the conditions for the stability of three equilibria, respectively. Furthermore, we analyze the sufficient conditions for the Hopf bifurcation of the system at the positive equilibrium. Finally, we present some numerical examples to verify the correctness of the theoretical analysis and give some valuable conclusions and further discussions at the end of the paper.</p></abstract>

Co-dimension two bifurcations analysis of a delayed tumor model with Allee effect

The mathematical model has become an important means to study tumor treatment and has developed with the discovery of medical phenomena. In this paper, we establish a delayed tumor model, in which the Allee effect is considered. Different from the previous similar tumor models, this model is mainly studied from the point of view of stability and co-dimension two bifurcations, and some nontrivial phenomena and conclusions are obtained. By calculation, there are at most two positive equilibria in the system, and their stability is investigated. Based on these, we find that the system undergoes Bautin bifurcation, zero-Hopf bifurcation, and Hopf–Hopf bifurcation with time delay and tumor growth rate as bifurcation parameters. The interesting thing is that there is a Zero-Hopf bifurcation, which is not common in tumor models, making abundant dynamic phenomena appear in the system. By using the bifurcation theory of functional differential equations, we calculate the normal form of these Co-dimension two bifurcations. Finally, with the aid of MATLAB package DDE-BIFTOOL, some numerical simulations have been performed to support our theoretical results. In particular, we obtain the bifurcation diagram of the system in the two parameter plane and divide its regions according to the bifurcation curves. Meanwhile, the phenomena of multistability and periodic coexistence of some regions can be also demonstrated. Combined with the simulation results, we can know that when the tumor growth rate and the delay of immune cell apoptosis are small, the tumor may tend to be stable, and vice versa.

The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the fundamental theory of functional differential equations with infinite delay and use the associated stability concept and known theorems regarding Lyapunov functionals including a generalized invariance principle. The formulation of Lyapunov functionals in the case of fractional damping is derived from a mechanical interpretation of the fractional derivative in infinite state representation. The method is applied on a single degree-of-freedom oscillator first, and the developed Lyapunov functionals are subsequently generalized for the finite-dimensional case. This opens the way to a stability analysis of nonlinear (controlled) systems with fractional damping. An important result of the paper is the solution of a tracking control problem with fractional and nonlinear damping. For this problem, the classical concepts of convergence and incremental stability are generalized to systems with fractional-order derivatives of state variables. The application of the related method is illustrated on a fractionally damped two degree-of-freedom oscillator with regularized Coulomb friction and non-collocated control.

Impacts of Multiple Time Delays on a Gene Regulatory Network Mediated by Small Noncoding RNA

In this paper, the impacts of multiple time delays on a gene regulatory network mediated by small noncoding RNA is studied. By analyzing the associated characteristic equation of the corresponding linearized system, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Furthermore, the explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are given by the center manifold theorem and the normal form theory for functional differential equations. Finally, some numerical simulations are demonstrated for supporting the theoretical results.

Infinity dynamics and DDF control for a chaotic system with one stable equilibrium

Hidden attractors in chaotic dynamical systems can be found by exploring the basin of attraction which has no intersect with any equilibria. Controlling chaos in these systems are complicated, which needs developed methods. In this paper, a 3D jerk system with only one stable equilibrium and hidden attractor is analyzed in infinity by the help of the Poincare compactification in R3. Meanwhile, a distributed delayed feedback (DDF) control scheme for this system is proposed. By using the center manifold theory of functional differential equation (FDE), Hopf bifurcation for the DDF control system is analyzed and obtained. Results confirm the accuracy of the bifurcation analysis and the effectiveness of the proposed DDF control strategy.

К оценке значений линейных функционалов на решениях систем с последействием

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse

We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number $R_{0}$ for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when $R_{0} < 1$. Moreover, we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when $R_{0} > 1$.

Further Improvement on Delay-derivative-dependent Stochastic Stability Criteria for Markovian Jumping Neutral-type Interval Time-varying Delay Systems with Mixed Delays

This paper is concerned with the problem of delay-derivative-dependent stability analysis for Markovian jumping neutral-type interval time-varying delay systems with mixed delays. The first, based on the Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality approach, a delay-range-dependent condition for neutral-type Markovian jumping systems (NMJSs) with time-varying delays is obtained, which can guarantee global asymptotical stability of these NMJSs. Unlike the previous methods, the upper bound of the discrete delay and neutral delay derivative is taken into consideration even if this upper bound is larger than or equal to $$1(\dot{h}(t)<1,\dot{\tau}(t)<1)$$ . It is proved that the obtained results are less conservative than the existing ones. In our result, the time-varying delays are only assumed to be bounded. This has undoubtedly extended its application range. To better handle the problem on stability for neutral control systems, in which time-varying delay was involved, a stability criterion with less conservatism was put forward. Moreover, because our results in this paper are all based on the linear matrix inequality (LMI) approach, we can utilize Matlab’s LMI Control Toolbox to verify the global stability of correlation systems conveniently. As far as we are aware, this paper seems to be the first to discuss stability problems for neutral-type Markovian jumping systems with delay-derivative-dependence and delay-range-dependence.

Bifurcation analysis in a nonlinear electro-optical oscillator with delayed bandpass feedback

The dynamics of a nonlinear electro-optical oscillator with delayed bandpass feedback is investigated. By analyzing the distribution of the eigenvalues, the existence and stability of Hopf bifurcations are obtained. Particularly, the stability switches are found as the delay varies, where the time delay is regarded as bifurcation parameter. And then, by applying the normal form method and center manifold theory of functional differential equations, an algorithm for determining the sense of the Hopf bifurcations and stability of the bifurcating periodic solutions is derived. For illustrating the theoretical results, some numerical simulations are performed.

Dynamical stability in a delayed neural network with reaction–diffusion and coupling

In this paper, a delayed neural network with reaction–diffusion and coupling is considered. The network consists of two sub-networks each with two neurons. In the first instance, some parameter regions are identified by employing partial functional differential equation theory. Moreover, sufficient conditions of stationary bifurcation and Bogdanov–Takens bifurcation are also derived. Further, analytical results and illustrations are proved for the case where the unstable trivial equilibrium point becomes stable in the presence of reaction–diffusion terms with appropriate values. We emphasize that the non-trivial role of diffusions is enlarging the stability region in the system described by PDE, comparing with the corresponding system described by DDE. Finally, numerical simulations are carried out to verify the efficiency of the theoretical analysis and provide comparisons with some existing literature.

Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays

In this paper we study a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays. It is assumed that the process of proliferation is delayed compared with apoptosis. The delay represents the time taken for cells to undergo mitosis. By employing stability theory for functional differential equations, comparison principle and some meticulous mathematical analysis, we mainly study the asymptotic behavior of the solution, and prove that in the case $c$ (the ratio of the diffusion time scale to the tumor doubling time scale) is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to one of two dormant states as $t\to ∞$ . The results show that dynamical behavior of solutions of the model are similar to that of solutions for corresponding nonretarded problems under some conditions.

Qualitative properties of a p-Laplacian population model with delay

This paper is concerned with qualitative properties of the evolutionary p-Laplacian population model with delay. We first establish the existence of solutions of the model by using the method of parabolic regularization and energy estimate and give the uniqueness by a recursive process. Then, combining the upper and lower solution method and the oscillation theory of functional differential equations, we obtain the oscillation of all positive solutions about the positive equilibrium.

Stability of Traveling Wave Fronts for Nonlocal Diffusion Equation with Delayed Nonlocal Response

In this paper, we consider with the stability of traveling wave fronts for the nonlocal diffusion equation with delay and global response. We first establish the existence and comparison theorem of solutions for the nonlocal reaction-diffusion equation by appealing to the theory of abstract functional differential equation. Then we further show that the traveling wave fronts are asymptotical stability with phase shift. Our main technique is the super and subsolution method coupled with the comparison principle and squeezing method.