Hopf Bifurcation Analysis of the Halvorsen System

Nonlinear dynamics in a Solow model with delay and non-convex technology

In this paper, we study a classical Gierer-Meinhardt model with some degenerate conditions. The stability of the constant steady-state solution is first obtained. After that, the Hopf bifurcation and steady-state bifurcation are discussed in detail, especially under the degenerate cases such as invalid simple eigenvalue condition and violated crossing condition. Meanwhile, the bifurcation direction of simple steady-state bifurcation is given, and the local bifurcation can be extended to the global bifurcation. Finally, through numerical simulations, we clearly show that degenerate bifurcations can lead to richer patterns, such as the coupling of two eigenfunction modes. It is shown that the usual pitchfork Hopf and steady-state bifurcations are transformed into the transcritical bifurcations under the degenerate conditions. The results obtained on the degenerate cases can enrich the theoretical and numerical results on the Gierer-Meinhardt model in the existing work.

In recent years, neutral-type differential-difference equations have been applied extensively in the field of engineering, and their dynamical behaviors are more complex than that of the delay differential-difference equations. In this paper, the equations used to describe a neutral-type neural network system of differential difference equation with two delays are studied (i.e. neutral-type differential equations). Firstly, by selecting [Formula: see text], [Formula: see text] respectively as a parameter, we provide an analysis about the local stability of the zero equilibrium point of the equations, and sufficient conditions of asymptotic stability for the system are derived. Secondly, by using the theory of normal form and applying the theorem of center manifold introduced by Hassard et al., the Hopf bifurcation is found and some formulas for deciding the stability of periodic solutions and the direction of Hopf bifurcation are given. Moreover, by applying the theorem of global Hopf bifurcation, the existence of global periodic solution of the system is studied. Finally, an example is given, and some computer numerical simulations are taken to demonstrate and certify the correctness of the presented results.

This paper investigates a toxic phytoplankton–zooplankton model with Michaelis–Menten type phytoplankton harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, fold, Hopf, fold-Hopf and double Hopf bifurcation, when the parameters change and go through some of the critical values, the dynamical properties of the system will change also, such as the stability, equilibrium points and the periodic orbit. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations, and the completion bifurcation set by calculating the universal unfoldings near the double Hopf bifurcation point are given by the normal form theory and center manifold theorem. We obtained that the stable coexistent equilibrium point and stable periodic orbit alternate regularly when the digestion time delay is within some finite value. That is, we derived the pattern for the occurrence, and disappearance of a stable periodic orbit. Furthermore, we calculated the approximation expression of the critical bifurcation curve using the digestion time delay and the harvesting rate as parameters, and determined a large range in terms of the harvesting rate for the phytoplankton and zooplankton to coexist in a long term.

Hopf bifurcation and spatio-temporal patterns in a hierarchical network with delays and [formula omitted] symmetry

Bifurcations and chaos in a system with impacts

This paper is concerned with a delayed diffusive predator–prey model with Michaelis–Menten-type prey harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, transcritical saddle-node, Hopf bifurcations, which can cause the behavior mutations of the system, such as the change of the stability, the change of the number of equilibria, and the emergence of the periodic orbits. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations near the positive steady state are given by the normal form theory and center manifold theorem. Meanwhile, the approximate expressions of periodic orbits are given. Furthermore, we derive the sufficient condition for global existence of the spatial homogeneous periodic orbits. Finally, some numerical simulations are carried out to support our analytical results.

In this paper, two sunflower equations are considered. Using delay τ as a parameter and applying the global Hopf bifurcation theorem, we investigate the existence of global Hopf bifurcation for the sunflower equation. Furthermore, we analyze the local Hopf bifurcation of the modified equation with nonlinear relation about stem's increase, including the occurrence, the bifurcation direction, the stability and the approximation expression of the bifurcating periodic solution using the theory of normal form and center manifold. Finally, the obtained results of these two equations are compared, which finds that the result about the period of their bifurcating periodic solutions is obviously different, while the bifurcation direction and stability are identical.

We have developed a class of viral infection model with cell-to-cell transmission and humoral immune response. The model addresses both immune and intracellular delays. We also constructed Lyapunov functionals to establish the global dynamical properties of the equilibria. Theoretical results indicate that considering only two intracellular delays did not affect the dynamical behavior of the model, but incorporating an immune delay greatly affects the dynamics, i.e. an immune delay may destabilize the immunity-activated equilibrium and lead to Hopf bifurcation, oscillations and stability switches. Our results imply that an immune delay dominates the intracellular delays in the model. We also investigated the direction of the Hopf bifurcation and the stability of the periodic solutions by applying normal form and center manifold theory, and investigated the existence of global Hopf bifurcation by regarding the immune delay as a bifurcation parameter. Numerical simulations are carried out to support the analytical conclusions.

Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model

In this paper, a delayed predator–prey model is considered. The existence and stability of the positive equilibrium are investigated by choosing the delay [Formula: see text] as a bifurcation parameter. We see that Hopf bifurcation can occur as [Formula: see text] crosses some critical values. The direction of the Hopf bifurcations and the stability of the bifurcation periodic solutions are also determined by using the center manifold and normal form theory. Furthermore, based on the global Hopf bifurcation theorem for general function differential equations, which was established by J. Wu using fixed point theorem and degree theory methods, the existence of global Hopf bifurcation is investigated. Finally, numerical simulations to support the analytical conclusions are carried out.

The paper presents a detailed analysis on the dynamics of a two-neuron network with time-delayed connections between the neurons and time-delayed feedback from each neuron to itself. On the basis of characteristic roots method and Hopf bifurcation theorems for functional differential equations, we investigate the existence of local Hopf bifurcation. In addition, the direction of Hopf bifurcation and stability of the periodic solutions bifurcating from the trivial equilibrium are determined based on the normal form theory and center manifold theorem. Moreover, employing the global Hopf bifurcation theory due to [Wu, 1998], we study the global existence of periodic solutions. It is shown that the local Hopf bifurcation indicates the global Hopf bifurcation after the second group critical value of the delay.

Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions

Stability and Hopf bifurcation for a three-component reaction–diffusion population model with delay effect

In this paper, a delayed computer virus propagation model with a saturation incidence rate and a time delay describing temporary immune period is proposed and its dynamical behaviors are studied. The threshold value [Formula: see text] is given to determine whether the virus dies out completely. By comparison arguments and iteration technique, sufficient conditions are obtained for the global asymptotic stabilities of the virus-free equilibrium and the virus equilibrium. Taking the delay as a parameter, local Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stabilities of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, numerical simulations are carried out to illustrate the main theoretical results.