Synchronized Hopf bifurcation analysis in a neural network model with delays

We consider the synchronized periodic oscillation in a ring neural network model with two different delays in self-connection and nearest neighbor coupling. Employing the center manifold theorem and normal form method introduced by Hassard et al., we give an algorithm for determining the Hopf bifurcation properties. Using the global Hopf bifurcation theorem for FDE due to Wu and Bendixson's criterion for high-dimensional ODE due to Li and Muldowney, we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large.     

Stability and Hopf bifurcation analysis on Goodwin model with three delays

In this paper, a class of Goodwin models with three delays is dealt. The dynamic properties including stability and Hopf bifurcations are studied. Firstly, we prove analytically that the addressed system possesses a unique positive equilibrium point. Moreover, using the Cardano’s formula for the third degree algebra equation, the distribution of characteristic roots is proposed. And then, the sum of the delays is chosen as the bifurcation parameter and it is demonstrated that the Hopf bifurcation would occur when the delay exceeds a critical value. Finally, a numerical simulation for justifying the theoretical results is also provided.     

Hopf bifurcation in a three-species system with delays

A kind of three-species system with Holling II functional response and two delays is introduced. Its local stability and the existence of Hopf bifurcation are demonstrated by analyzing the associated characteristic equation. By using the normal form method and center manifold theorem, explicit formulas to determine the direction of the Hopf bifurcation and the stability of bifurcating periodic solution are also obtained. In addition, the global existence results of periodic solutions bifurcating from Hopf bifurcations are established by using a global Hopf bifurcation result. Numerical simulation results are also given to support our theoretical predictions.     

Bifurcation and synchronization of synaptically coupled FHN models with time delay

This paper presents an investigation of dynamics of the coupled nonidentical FHN models with synaptic connection, which can exhibit rich bifurcation behavior with variation of the coupling strength. With the time delay being introduced, the coupled neurons may display a transition from the original chaotic motions to periodic ones, which is accompanied by complex bifurcation scenario. At the same time, synchronization of the coupled neurons is studied in terms of their mean frequencies. We also find that the small time delay can induce new period windows with the coupling strength increasing. Moreover, it is found that synchronization of the coupled neurons can be achieved in some parameter ranges and related to their bifurcation transition. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behavior are clarified.     

Global Hopf bifurcation analysis on a BAM neural network with delays

A BAM neural network with three neurons is considered. Sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large. Numerical simulations are presented to support the theoretical results found.     

Hopf bifurcation analysis of two neurons with three delays

Using the system parameter instead of the delays as the bifurcation parameter, linear stability and Hopf bifurcation including its direction and stability of a two-neuron network with three delays are investigated in this paper. The main tools to obtain our results are the normal form method and the center manifold theory introduced by Hassard. Simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior.     

Hopf bifurcation analysis for congestion control with heterogeneous delays

In this paper, a congestion control algorithm with heterogeneous delays is considered. Local stability of the equilibrium solution of this algorithm is investigated based on analyzing the corresponding transcendental characteristic equation. Especially, using one of the system parameters of congestion control algorithm as the bifurcation parameter, when the system parameter exceeds a critical value, the congestion control algorithm undergoes a supercritical Hopf bifurcation, and the explicit formulae determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al.’s approaches. Finally, some numerical simulations are performed to verify the theoretical results.     

Hopf bifurcation and stability for a neural network model with mixed delays

A generalized model of the two-neuron network with mixed delays is studied. The main purpose of this paper is to explore the linear stability of the trivial solution and Hopf bifurcation of a two-neuron network with continuous and discrete delays. The general formula of the direction, the estimation formula of period and stability of bifurcated periodic solutions are also studied. Finally, the numerical simulations are given to illustrate the theoretical analysis.     

Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays

In this paper, the Leslie-Gower predator-prey system with two delays is investigated. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of periodic solutions.     

两时滞广义Logistic模型的Hopf分支与混沌

构建了具有两个时滞的广义Logistic模型,分情况讨论了系统正平衡点发生局部Hopf分支和稳定性切换的条件,分析了分支点关于系统参数的单调性和极限性质.数值模拟佐证了理论结果,展示了周期振动,倍周期分支,混沌等复杂的动力学行为.     

Hopf bifurcation analysis of a density predator-prey model with crowley-martin functional response and two time delays

In this paper, a delayed density dependent predator-prey model with Crowley-Martin functional response and two time delays for the predator is considered. By analyzing the corresponding characteristic equations, the local stability of each of the feasible equilibria of the system is addressed and the existence of Hopf bifurcation at the coexistence equilibrium is established. With the help of normal form method and center manifold theorem, some explicit formulas determining the direction of Hopf bifurcation and the stability of bifurcating period solutions are derived. Finally, numerical simulations are given to illustrate the theoretical results.     

Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay

In this paper, the Bautin bifurcation of synchronous solution of a coupled FHN neural system with delay is investigated. Firstly, the method of Lyapunov functional is used to obtain the synchronization conditions of the neural system, and then distributions of the roots of the characteristic equation associated with the linearization of the synchrosystem are discussed. Center manifold and normal form are employed to calculate its Lyapunov coefficients. A group of sufficient conditions are given to present Bautin bifurcation of the synchrosystem by applying the Bautin bifurcation theorem of delay differential equations developed by Anca-Veronica Ion. The Bautin bifurcation diagram in the physical parameter space is provided to illustrate the correctness of our theoretical analysis.     

Stability and Hopf bifurcation analysis on a ring of four neurons with delays

We consider a four-neuron ring with self-feedback and delays. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. Numerical simulations are presented to illustrate the results.     

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 four-neuron BAM neural network with distributed delays

In this paper, a four-neuron BAM neural network with distributed delays is considered, where kernels are chosen as weak kernels. Its dynamics is studied in terms of local stability analysis and Hopf bifurcation analysis. By choosing the average delay as a bifurcation parameter and analyzing the associated characteristic equation, Hopf bifurcation occurs when the bifurcation parameter passes through some exceptive values. The stability of bifurcating periodic solutions and a formula for determining the direction of Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, numerical simulation results are given to validate the theorem obtained.     

Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model

In this paper, we study the Hopf bifurcation phenomenon of a one-dimensional Schnakenberg reaction-diffusion model subject to the Neumann boundary condition. Our results reveal that both spatially homogeneous periodic solutions and spatially heterogeneous periodic solution exist. Moreover, we conclude that the spatially homogeneous periodic solutions are locally asymptotically stable and the spatially heterogeneous periodic solutions are unstable. In addition, we give specific examples to illustrate the phenomenon that coincides with our theoretical results.     

Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays

In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.     

Hopf bifurcation and stability crossing curves in a cobweb model with heterogeneous producers and time delays

We study a continuous time cobweb model with discrete time delays where heterogeneous producers behave as adapters in the market. Specifically, they partially adjust production (which is subject to some gestation lags) towards the profit-maximising quantity under static expectations. The dynamics of the economy is described by a two-dimensional system of delay differential equations. We characterise stability properties of the stationary state of the system and show the emergence of Hopf bifurcations. We also apply some recent mathematical techniques (stability crossing curves) to show how heterogeneous time delays affect the stability of the economy.     

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.