Stability Switching Curves Research Articles

Hopf bifurcation control of memristor-based fractional delayed tri-diagonal bidirectional associative memory neural networks under various controllers

In this paper, authors present a memristor-based fractional order system of tri-diagonal bidirectional associative memory neural networks (TdBAMNNs) incorporating leakage and communication delays. The existence and uniqueness theorem is established for the memristor-based fractional-order TdBAMNNs system. Analysis of Hopf bifurcation anti-control is conducted using various feedback controllers by exploring single-parameter and double-parameter approaches. A study of the stability switching curves is undertaken to determine the direction of stability transitions in the system. The theoretical results and graphical illustrations are validated through numerical simulations to exhibit their feasibility. Comparative analysis of the system’s dynamical behavior under different controllers is performed. This paper comprehensively explores the proposed memristor-based system, proving its theoretical underpinnings, and analyzing stability through bifurcation control strategies and comparative assessments under different control scenarios.

Hopf bifurcation analysis of a two‐delayed diffusive predator–prey model with spatial memory of prey

In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results.

Cholera disease dynamics with vaccination control using delay differential equation

The COVID-19 pandemic came with many setbacks, be it to a country’s economy or the global missions of organizations like WHO, UNICEF or GTFCC. One of the setbacks is the rise in cholera cases in developing countries due to the lack of cholera vaccination. This model suggested a solution by introducing another public intervention, such as adding Chlorine to water bodies and vaccination. A novel delay differential model of fractional order was recommended, with two different delays, one representing the latent period of the disease and the other being the delay in adding a disinfectant to the aquatic environment. This model also takes into account the population that will receive a vaccination. This study utilized sensitivity analysis of reproduction number to analytically prove the effectiveness of control measures in preventing the spread of the disease. This analysis provided the mathematical evidence for adding disinfectants in water bodies and inoculating susceptible individuals. The stability of the equilibrium points has been discussed. The existence of stability switching curves is determined. Numerical simulation showed the effect of delay, resulting in fluctuations in some compartments. It also depicted the impact of the order of derivative on the oscillations.

Stability and Hopf Bifurcation Analysis of A Fractional-Order BAM Neural Network with Two Delays Under Hybrid Control

In this paper, considering that fractional-order calculus can more accurately describe memory and genetic properties, we introduce fractional integral operators into neural networks and discuss the stability and Hopf bifurcation of a fractional-order bidirectional associate memory (BAM) neural network with two delays. In addition, the hybrid controller is proposed to achieve Hopf bifurcation control of the system. By taking two time delays as the bifurcation parameters and analyzing of the corresponding characteristic equation, stability switching curves of the controllable system for two delays are obtained. The direction of the characteristic root crossing the imaginary axis in stability switching curves is determined. Sufficient criteria are sequentially given to judge the local stability and the existence of Hopf bifurcation of a fractional-order BAM neural network system. The numerical simulation results show that the hybrid controller can effectively control Hopf bifurcation of a fractional-order BAM neural network system with two delays.

On a Special Two-Person Dynamic Game

The asymptotical properties of a special dynamic two-person game are examined under best-response dynamics in both discrete and continuos time scales. The direction of strategy changes by the players depend on the best responses to the strategies of the competitors and on their own strategies. Conditions are given first for the local asymptotical stability of the equilibrium if instantaneous data are available to the players concerning all current strategies. Next, it is assumed that only delayed information is available about one or more strategies. In the discrete case, the presence of delays has an effect on only the order of the governing difference equations. Under continuous scales, several possibilities are considered: each player has a delay in the strategy of its competitor; player 1 has identical delays in both strategies; the players have identical delays in their own strategies; player 1 has different delays in both strategies; and the players have different delays in their own strategies. In all cases, it is assumed that the equilibrium is asymptotically stable without delays, and we examine how delays can make the equilibrium unstable. For small delays, the stability is preserved. In the cases of one-delay models, the critical value of the delay is determined when stability changes to instability. In the cases of two and three delays, the stability-switching curves are determined in the two-dimensional space of the delays, where stability becomes lost if the delay pair crosses this curve. The methodology is different for the one-, two-, and three-delay cases outlined in this paper.

Stability and bifurcation analysis of Alzheimer's disease model with diffusion and three delays.

A reaction-diffusion Alzheimer's disease model with three delays, which describes the interaction of β-amyloid deposition, pathologic tau, and neurodegeneration biomarkers, is investigated. The existence of delays promotes the model to display rich dynamics. Specifically, the conditions for stability of equilibrium and periodic oscillation behaviors generated by Hopf bifurcations can be deduced when delay σ (σ=σ1+σ2) or σ3 is selected as a bifurcation parameter. In addition, when delay σ and σ3 are selected as bifurcation parameters, the stability switching curves and the stable region are obtained by using an algebraic method, and the conditions for the existence of Hopf bifurcations can also be derived. The effects of time delays, diffusion, and treatment on biomarkers are discussed via numerical simulations. Furthermore, sensitivity analysis at multiple time points is drawn, indicating that different targeted therapies should be taken at different stages of development, which has certain guiding significance for the treatment of Alzheimer's disease.

Stability and Hopf bifurcation analysis of a fractional-order ring-hub structure neural network with delays under parameters delay feedback control.

In this paper, a fractional-order two delays neural network with ring-hub structure is investigated. Firstly, the stability and the existence of Hopf bifurcation of proposed system are obtained by taking the sum of two delays as the bifurcation parameter. Furthermore, a parameters delay feedback controller is introduced to control successfully Hopf bifurcation. The novelty of this paper is that the characteristic equation corresponding to system has two time delays and the parameters depend on one of them. Selecting two time delays as the bifurcation parameters simultaneously, stability switching curves in $ (\tau_{1}, \tau_{2}) $ plane and crossing direction are obtained. Sufficient criteria for the stability and the existence of Hopf bifurcation of controlled system are given. Ultimately, numerical simulation shows that parameters delay feedback controller can effectively control Hopf bifurcation of system.

The chaotic monopolist revisited with bounded rationality and delay dynamics

Two types of boundedly rational monopolists are studied when the marginal revenue is not necessarily negative sloping. A limited monopolist (ℓ-monopolist) knows only the price and output values in two previous periods. A knowledgeable monopolist (k-monopolist) knows the analytic expression of the marginal profit function and forms sales expectations based on past market information. The k-monopolist adjusts its output levels according to the usual gradient process, while the ℓ-monopolist approximates the marginal profit with a two-point finite difference formula. Stability conditions and complex dynamics are studied in discrete and continuous time scales. In the discrete model, the two monopolists exhibit different dynamics after the stability loss. The ℓ-monopolist has a cascade of a Neimark-Sacker bifurcation. In contrast, the k-monopolist has two different routes, a period-doubling and a Neimark-Sacker bifurcation, depending on how to form sales expectations. In the continuous case, the discrete models are transformed into continuous models via Euler transformation. The stability switching curves are analytically constructed and the directions of stability switching are characterized by computing the stability index for each point of the curves. It is numerically demonstrated that two monopolists exhibit similar dynamics and the k-monopolist approximates the ℓ-monopolist under a special circumstance.

Stability Switching Curves and Hopf Bifurcation of a Fractional Predator–Prey System with Two Nonidentical Delays

In this paper, we propose and analyze a three-dimensional fractional predator–prey system with two nonidentical delays. By choosing two delays as the bifurcation parameter, we first calculate the stability switching curves in the delay plane. By judging the direction of the characteristic root across the imaginary axis in stability switching curves, we obtain that the stability of the system changes when two delays cross the stability switching curves, and then, the system appears to have bifurcating periodic solutions near the positive equilibrium, which implies that the trajectory of the system is the axial symmetry. Secondly, we obtain the conditions for the existence of Hopf bifurcation. Finally, we give one example to verify the correctness of the theoretical analysis. In particular, the geometric stability switch criteria are applied to the stability analysis of the fractional differential predator–prey system with two delays for the first time.

Bifurcation Analysis of a Diffusive Virus Infection and Immune Response Model with Two Delays

A reaction–diffusion system with two delays, which describes viral infection spreading in the lymphoid tissue, is investigated. The delays promote very complex dynamics of the model. We get the Hopf bifurcation curves and the stability region for the coexisting steady state on a two-parameter plane by finding the stability switching curves and the subsets of stable region. When two delays cross the boundary of the stable region, the system will undergo stability switches. It is shown that two types of bistability are possible: the coexistence of the stable virus-free steady state and the stable coexisting steady state; the coexistence of the stable virus-free steady state and a stable periodic solution. Numerical simulations suggest that delays can also induce tristability including a steady state and two stable periodic solutions.

Receptance-Based Computation of Stability Crossing Curves for Single-Input-Multiple-Output Second-Order Linear Systems with Two Time-Delays

For a complete stability analysis of multi-dimensional controlled systems modeled in the framework of second-order linear differential equations with two time-delays, the determination of stability crossing curves (or stability switching curves) within the domain of the delays is significantly important. This paper presents a simple receptance-based approach to solve this problem for a single-input-multiple-output controlled system using its second-order model. The proposed approach is based on a reduced characteristic function of the controlled system. This characteristic function is directly related to the receptance of the uncontrolled system and has a peculiar form that is well-suited for an effective method of calculation of these curves. Moreover, this method can find the direction in which the characteristic roots cross the imaginary axis as the delays deviate from a stability crossing curve. An example case study with two independent and constant delays is given to demonstrate the effectiveness of the proposed approach.

Bifurcation analysis in delayed Nicholson blowflies equation with delayed harvest

In this paper, we investigate a delayed Nicholson equation with delay harvesting term which was proposed in open problems and conjectures formulated by Berezansky et al. (Appl. Math. Model. 34: 1405–1417, 2010). The stability switching curves by taking two delays as parameters are obtained via the method introduced by An et al.(J. Differ. Equ. 266: 7073–7100, 2019). The existence of Hopf singularity on a two-parameter plane is determined by the varying direction of two parameters. Furthermore, the normal form near the Hopf singularity is derived via applying the center manifolds theory and normal forms method of FDEs. Finally, some numerical simulations are carried out to illustrate the theoretical conclusions.

Mathematical modeling of intervention and low medical resource availability with delays: Applications to COVID-19 outbreaks in Spain and Italy.

Infectious diseases have been one of the major causes of human mortality, and mathematical models have been playing significant roles in understanding the spread mechanism and controlling contagious diseases. In this paper, we propose a delayed SEIR epidemic model with intervention strategies and recovery under the low availability of resources. Non-delayed and delayed models both possess two equilibria: the disease-free equilibrium and the endemic equilibrium. When the basic reproduction number $ R_0 = 1 $, the non-delayed system undergoes a transcritical bifurcation. For the delayed system, we incorporate two important time delays: $ \tau_1 $ represents the latent period of the intervention strategies, and $ \tau_2 $ represents the period for curing the infected individuals. Time delays change the system dynamics via Hopf-bifurcation and oscillations. The direction and stability of delay induced Hopf-bifurcation are established using normal form theory and center manifold theorem. Furthermore, we rigorously prove that local Hopf bifurcation implies global Hopf bifurcation. Stability switching curves and crossing directions are analyzed on the two delay parameter plane, which allows both delays varying simultaneously. Numerical results demonstrate that by increasing the intervention strength, the infection level decays; by increasing the limitation of treatment, the infection level increases. Our quantitative observations can be useful for exploring the relative importance of intervention and medical resources. As a timing application, we parameterize the model for COVID-19 in Spain and Italy. With strict intervention policies, the infection numbers would have been greatly reduced in the early phase of COVID-19 in Spain and Italy. We also show that reducing the time delays in intervention and recovery would have decreased the total number of cases in the early phase of COVID-19 in Spain and Italy. Our work highlights the necessity to consider the time delays in intervention and recovery in an epidemic model.

Double Hopf bifurcation of a diffusive predator–prey system with strong Allee effect and two delays

In this paper, we consider a diffusive predator–prey system with strong Allee effect and two delays. First, we explore the stability region of the positive constant steady state by calculating the stability switching curves. Then we derive the Hopf and double Hopf bifurcation theorem via the crossing directions of the stability switching curves. Moreover, we calculate the normal forms near the double Hopf singularities by taking two delays as parameters. We carry out some numerical simulations for illustrating the theoretical results. Both theoretical analysis and numerical simulation show that the system near double Hopf singularity has rich dynamics, including stable spatially homogeneous and inhomogeneous periodic solutions. Finally, we evaluate the influence of two parameters on the existence of double Hopf bifurcation.

Stability switching and its directions in cournot duopoly game with three delays

<p style='text-indent:20px;'>A three-delay duopoly is considered where the firms have identical implementation delays with different information delays. The equilibrium is locally asymptotically stable without delays however this stability is lost with increasing values of the delays. The stability properties of the equilibrium depend on the common implementation delay of the firms and on the sum of the two information delays. The stability switching curves are first analytically characterized and illustrated, and then the direction of the stability switching is determined at each point of the curves. The possibility of multiple pure imaginary eigenvalues is also discussed when the directions of the stability switches cannot be determined. Simulation examples illustrate the theoretical results.</p>

Delay Cournot Duopoly Game with Gradient Adjustment: Berezowski Transition from a Discrete Model to a Continuous Model

This paper investigates the asymptotical behavior of the equilibrium of linear classical duopolies by reconsidering the two-delay model with two different positive delays. In a two-dimensional analysis, the stability switching curves were first analytically determined. Numerical studies verified and illustrated the theoretical results. In the sensitivity analysis it was demonstrated that the inertia coefficient has a twofold effect: enlarges the stability region as well as simplifies the complicated dynamics with period-halving cascade. In contrary, the adjustment speed contracts the stability region and complicates simple dynamics with period-doubling bifurcation. In addition, for various values of τ1 and τ2, a wide variety of dynamics appears ranging from simple cycle via a Hopf bifurcation to chaotic oscillations.

Time delays and chaos in two competing species revisited

This paper reexamines the Lotka-Volterra competition model with two delays. The steady state is shown to be locally asymptotically stable without delay. If the two delays are identical, then the model becomes a one-delay system. The critical value of the delay is determined when stability might be lost. If the delays are different, then the stability switching curves are analytically defined and numerically verified. It is demonstrated that the unstable two-delay system may exhibit periodic behavior, multistability, quasi period-doubling cascade and even complicated dynamics depending on model parameters.

Delay Stability of n-Firm Cournot Oligopolies

The dynamic behavior of n-firm oligopolies is examined without product differentiation and with linear price and cost functions. Continuous time scales are assumed with best response dynamics, in which case the equilibrium is asymptotically stable without delays. The firms are assumed to face both implementation and information delays. If the delays are equal, then the model is a single delay case, and the equilibrium is oscillatory stable if the delay is small, at the threshold stability is lost by Hopf bifurcation with cyclic behavior, and for larger delays, the trajectories show expanding cycles. In the case of the non-equal delays, the stability switching curves are constructed and the directions of stability switches are determined. In the case of growth rate dynamics, the local behavior of the trajectories is similar to that of the best response dynamics. Simulation studies verify and illustrate the theoretical findings.

Stability switching curves in a Lotka–Volterra competition system with two delays

This study is devoted to consider the dynamic properties of a Lotka–Volterra competition model with two discrete delays. First, it reviews stability conditions of the no-delay model as a benchmark. Second, it examines the delay model and analytically establishes the stability switching curve on which stability is switched to instability and vice versa. Finally, numerical simulations are performed to confirm the theoretical results such as Hopf bifurcations and, further, demonstrate a wide variety of dynamics ranging from simple limit cycle to the existence of multi-stability and the birth of chaotic dynamics.

Dynamic Models of Pollution Penalties and Rewards with Time Delays

In cases of nonpoint pollution sources, the regulator can observe the total emission but unable to distinguish between the firms. The regulator then selects an environmental standard. If the total emission level is higher than the standard, then the firms are uniformly punished, and if lower, then uniformly awarded. This environmental regulation is added to n-firm dynamic oligopolies, and the asymptotical behavior of the corresponding dynamic systems is examined. Two particular models are considered with linear and hyperbolic price functions. Without delays, the equilibrium is always (locally) asymptotically stable. It is shown how the stability can be lost if time delays are introduced in the output quantities of the competitors as well as in the firms’ own output levels. Complete stability analysis is presented for the resulting one- and two-delay models including the derivations of stability thresholds, stability switching curves, and directions of the stability switches.