Hopf bifurcation in multiparty political systems with time delay in switching

A model of a political system with three political parties and a group of nonaffiliated voters is considered. The model consists of four coupled ordinary differential equations in which members of the third party change allegiance and after a time delay become an active member of one of the other parties. Equilibrium and stability analysis are carried out. Taking the time delay as a bifurcation parameter, it is shown that a Hopf bifurcation can occur.     

Hopf bifurcation in REM algorithm with communication delay

The purpose of this paper is to study bifurcation of an Internet congestion control algorithm, namely REM (Random Exponential Marking) algorithm, with communication delay. By choosing the delay constant as a bifurcation parameter, we prove that REM algorithm exhibits Hopf bifurcation. The formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions are obtained by applying the center manifold theorem and the normal form theory. Finally, a numerical simulation is present to verify the theoretical results.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Hopf bifurcation analysis of integro-differential equation with unbounded delay

The complexity of a nonlinear dynamical system is controllable via a selection of system parameters. One representative behavior of such a complex system can be illustrated by Hopf bifurcation. This paper presents a Hopf bifurcation analysis of a kind of integro-differential equations with unbounded delay. Based on the Hopf bifurcation principle, a set of relationships among system parameters are obtained when a periodic orbit exists in the system. A numerical analysis is applied to solve the integro-differential delay equation. This paper proves the existence of Hopf bifurcation in the corresponding difference equations under the same system parameters as that in the integro-differential delay equations.     

时滞微分方程广义Hopf 分岔

本文运用Lyapunov-Schmidt 约化方法研究了一般时滞微分方程的分岔情况, 具体分析了当参数达到一个临界值时, 系统的无穷小生成元具有一对k 重非半单纯虚特征值的情形, 得到了判定分岔周期解存在性和分岔方向的判据, 而且该判据明显依赖于系统参数, 并通过对van der Pol 方程的详细分析进一步验证了我们的结果.     

一类具时滞和阶段结构的捕食模型的稳定性与Hopf分支

研究一类具有时滞和阶段结构的捕食模型的稳定性和Hopf分支的存在性问题.通过分析特征方程,得到了正平衡点局部稳定的条件.同时,应用中心流形定理和规范型理论,得到了确定Hopf分支方向和分支周期解的稳定性的计算公式.最后对所得理论结果进行了数值模拟.     

Hopf bifurcation analysis in a tri-neuron network with time delay

A neural network model with three neurons and a single delay is considered. The existence of local Hopf bifurcations is first considered and then explicit formulas are derived by using the normal form method and center manifold theory to determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. A global Hopf bifurcation theorem due to Wu and a Bendixson's criterion for high-dimensional ODE due to Li and Muldowney are used to obtain a group of conditions for the system to have multiple periodic solutions when the delay is sufficiently large. Finally, numerical simulations are carried out to support the theoretical analysis of the research.     

Hopf bifurcation of a tumor immune model with time delay

In this paper, a tumor immune model with time delay is studied. First, the stability of nonnegative equilibria is analyzed. Then the time delay τ is selected as a bifurcation parameter and the existence of Hopf bifurcation is proved. Finally, by using the canonical method and the central manifold theory, the criteria for judging the direction and stability of Hopf bifurcation are given.     

Hopf bifurcation analysis of a food-limited population model with delay

The dynamics of a food-limited population model with delay are investigated. We prove that a sequence of Hopf bifurcations occur at the positive equilibrium as the delay increases. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived, using the theory of normal form and center manifold. Global existence of periodic solutions is established by using a global Hopf bifurcation result due to [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799–4838].     

Stability and Hopf bifurcation analysis on a spruce-budworm model with delay

In this paper, the dynamics of a spruce-budworm model with delay is investigated. We show that there exists Hopf bifurcation at the positive equilibrium as the delay increases. Some sufficient conditions for the existence of Hopf bifurcation are obtained by investigating the associated characteristic equation. By using the theory of normal form and center manifold, explicit expression for determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are presented.     

Global stability and Hopf bifurcation of a plankton model with time delay

A differential delay equation model with a discrete time delay and a distributed time delay is introduced to simulate zooplankton–nutrient interaction. The differential inequalities’ methods and standard Hopf bifurcation analysis are applied. Some sufficient conditions are obtained for persistence and for the global stability of the unique positive steady state, respectively. It was shown that there is a Hopf bifurcation in the model by using the discrete time delay as a bifurcation parameter.     

Hopf bifurcation in a two-competitor, one-prey system with time delay

In this paper, we consider a delayed two-competitor/one-prey system in which both two competitors exhibit Holling II functional response. By choosing the time delay as a bifurcation parameter, it is found that the Hopf bifurcation occurs when the delay passes through a certain critical value. Numerical simulations are performed to illustrate the analytical results.     

Hopf bifurcation analysis of a general non-linear differential equation with delay

This work represents Hopf bifurcation analysis of a general non-linear differential equation involving time delay. A special form of this equation is the Hutchinson–Wright equation which is a mile stone in the mathematical modeling of population dynamics and mathematical biology. Taking the delay parameter as a bifurcation parameter, Hopf bifurcation analysis is studied by following the theory in the book by Hazzard et al. By analyzing the associated characteristic polynomial, we determine necessary conditions for the linear stability and Hopf bifurcation. In addition to this analysis, the direction of bifurcation, the stability and the period of a periodic solution to this equation are evaluated at a bifurcation value by using the Poincaré normal form and the center manifold theorem. Finally, the theoretical results are supported by numerical simulations.     

Hopf bifurcation in a environmental defensive expenditures model with time delay

In this paper a three-dimensional environmental defensive expenditures model with delay is considered. The model is based on the interactions among visitors V, quality of ecosystem goods E, and capital K, intended as accommodation and entertainment facilities, in Protected Areas (PAs). The tourism user fees (TUFs) are used partly as a defensive expenditure and partly to increase the capital stock. The stability and existence of Hopf bifurcation are investigated. It is that stability switches and Hopf bifurcation occurs when the delay t passes through a sequence of critical values, τ₀. It has been that the introduction of a delay is a destabilizing process, in the sense that increasing the delay could cause the bio-economics to fluctuate. Formulas about the stability of bifurcating periodic solution and the direction of Hopf bifurcation are exhibited by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the results.     

Hopf bifurcation and stability analysis on discrete-time Hopfield neural network with delay

In this paper, a discrete-time Hopfield neural network with delay is considered. We give some sufficient conditions ensuring the local stability of the equilibrium point for this model. By choosing the delay as a bifurcation parameter, we demonstrated that Neimark–Sacker bifurcation (or Hopf bifurcation for map) would occur when the delay exceeds a critical value. A formula for determining the direction bifurcation and stability of bifurcation periodic solutions is given by applying the normal form theory and the center manifold theorem. Some numerical simulations for justifying the theoretical results are also provided.     

Stability switch and Hopf bifurcation for a diffusive prey-predator system with delay

The increasing time delay usually destabilizes any dynamical system. In this paper we give an example that in some special cases the opposite effect can be experienced if the time delay is sufficiently great. We investigate the effect of both the parameter in the time delay kernel and diffusion coefficient on the stability of the positive steady state for a diffusive prey-predator system with delay. We obtain the condition of the occurrence of the stability switches of the positive steady state.     

Global Hopf bifurcation of differential equations with threshold type state-dependent delay

We develop global Hopf bifurcation theory of differential equations with state-dependent delay using the S¹$S^1$-equivariant degree and investigate a two-degree-of-freedom mechanical model of turning processes. For the model of turning processes we show that the extreme points of each vibration component of the non-constant periodic solutions can be embedded into a manifold with explicit algebraic expression. This observation enables us to establish analytical upper and lower bounds of the amplitudes of the periodic solutions in terms of the system parameters and to exclude certain periods. Using the achieved global bifurcation theory we reveal that if the relative frequency between the natural frequency and the turning frequency varies in a certain interval, then generically every bifurcated continuum of periodic solutions must terminate at a bifurcation point. This termination means that the underlying system with parameters in the stability region near the vertical asymptotes of the stability lobes is less subject to chatter. In the process, several sufficient conditions for the non-existence of non-constant periodic solutions are also obtained.     

Hopf bifurcation in numerical approximation for delay differential equations

In this paper we investigate the qualitative behaviour of numerical approximation to a class delay differential equation. We consider the numerical solution of the delay differential equations undergoing a Hopf bifurcation. We prove the numerical approximation of delay differential equation had a Hopf bifurcation point if the true solution does.     

Stability and Hopf bifurcation of a modified delay predator-prey model with stage structure

In this paper, a modified delay predator-prey model with stage structure is established, which involves the economic factor and internal competition of all the prey and predator populations. By the methods of normal form and characteristic equation, we obtain the stability of the positive equilibrium point and the sufficient condition of the existence of Hopf bifurcation. We analyze the influence of the time delay on the equation and show the occurrence of Hopf bifurcation periodic solution. The simulation gives a visual understanding for the existence and direction of Hopf bifurcation of the model.     

Stability and Hopf bifurcation analysis in a three-level food chain system with delay

A class of three level food chain system is studied. With the theory of delay equations and Hopf bifurcation, the conditions of the positive equilibrium undergoing Hopf bifurcation is given regarding τ as the parameter. The stability and direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold argument, and numerical simulations are performed to illustrate the analytical results.