Global Hopf Bifurcation Theory Research Articles

Global Hopf bifurcation for differential equations with multiple threshold‐type state‐dependent delays

Using the ‐equivariant degree, we develop a global Hopf bifurcation theory for system of differential equations with multiple threshold‐type state‐dependent delays whose prototype is the human respiratory system with multiple blood transport time delays. To establish a theoretic framework for modeling practices of periodic breathing, we further investigate the periodic oscillations of carbon dioxide concentrations in the lung, brain, and tissue compartments and conduct a local and global Hopf bifurcation analysis for the model when varying the commensurate scale of the multiple delays in a transformed system. Such a global Hopf bifurcation will indicate the onset and persistence of the periodic oscillations.

Hopf Bifurcation Analysis of a Housefly Model with Time Delay

The oscillatory dynamics of a delayed housefly model is analyzed in this paper. The local and global stabilities at the non-negative equilibria are obtained via analyzing the distribution of eigenvalues and Lyapunov–LaSalle invariance principle, and the model undergoes the supercritical Hopf bifurcation and the transient oscillation. Based on Wu’s global Hopf bifurcation theory, the existence of the global bifurcation is established under certain conditions.

DYNAMICS OF A TWO-PATCH NICHOLSON'S BLOWFLIES MODEL WITH RANDOM DISPERSAL

<abstract> The global dynamics of the Nicholson's blowfly reaction-diffusion model with zero Dirichlet boundary condition is less understood. In this paper, we provide a discrete version of diffusive Nichlson's blowflies equation with zero Dirichlet boundary condition. Local and global stability of the equilibria are obtained by some comparison arguments, fluctuation method and the theory of exponential ordering. Hopf bifurcation at the positive equilibrium and the global existence of the periodic solutions are studied by local and global Hopf bifurcation theory. </abstract>

Global dynamics of a diffusive phytoplankton-zooplankton model with toxic substances effect and delay.

This paper examines a diffusive toxic-producing plankton system with delay. We first show the global attractivity of the positive equilibrium of the system without time-delay. We further consider the effect of delay on asymptotic behavior of the positive equilibrium: when the system undergoes Hopf bifurcation at some points of delay by the normal form and center manifold theory for partial functional differential equations. Global existence of periodic solutions is established by applying the global Hopf bifurcation theory.

Global Continuation of Periodic Oscillations to a Diapause Rhythm

We consider a scalar delay differential equation $$\dot{x}(t)=-dx(t)+f((1-\alpha )\rho x(t-\tau )+\alpha \rho x(t-2\tau ))$$ with an instant mortality rate $$d>0$$, the nonlinear Rick reproductive function f, a survival rate during all development stages $$\rho $$, and a proportion constant $$\alpha \in [0, 1]$$ with which population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within $$(3\tau , 6\tau )$$ for a wide range of parameter values. We show this existence of periodic solutions not only for the delay $$\tau $$ near the first critical value $$\tau ^*$$ when a local Hopf bifurcation takes place near the positive equilibrium, but for all $$\tau >\tau ^*$$. We obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li–Muldowney technique to rule out periodic solutions with period $$3\tau $$. We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium.

Local and global Hopf bifurcation in a neutral population model with age structure

In this paper, an age‐structured population model with the form of neutral functional differential equation is studied. We discuss the stability of the positive equilibrium by analyzing the characteristic equation. Local Hopf bifurcation results are also obtained by choosing the mature delay as bifurcation parameter. On the center manifold, the normal form of the Hopf bifurcation is derived, and explicit formulae for determining the criticality of bifurcation are theoretically given. Moreover, the global continuation of Hopf bifurcating periodic solutions is investigated by using the global Hopf bifurcation theory of neutral equations. Finally, some numerical examples are carried out to support the main results.

Global Hopf Bifurcation for Differential-Algebraic Equations with State-Dependent Delay

We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the $S^1$-equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation of the periodic oscillations.

Global Hopf bifurcations of neutral functional differential equations with state-dependent delay

A global Hopf bifurcation theory for a system of neutral functional differential equations (NFDEs) with state-dependent delay is investigated by applying the \begin{document}$S^{1}$\end{document} -equivariant degree theory. We use the information about the characteristic equation of the formal linearization with frozen delay to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system. The results are important in studying bifurcations of NFDEs with state-dependent delay.

Hopf bifurcation for neutral-type neural network model with two delays

In this paper, the dynamics of a neutral neural network model with two delays is investigated. The condition to ensure the stability of the zero solution of the system is decided by choosing τ1 and τ2 as parameters, respectively. Then the Hopf bifurcation is discussed by using the center manifold theory and normal form method introduced by Hassard and Kazarinoff. Global existence of periodic solution is studied by using the global Hopf bifurcation theory. Finally, some numerical simulations are carried out to illustrate the analytical results.

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-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.

Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme

Global Hopf bifurcation analysis is carried out on a six-dimensionalFitzHugh-Nagumo (FHN) neural network with a time delay. First, theexistence of local Hopf bifurcations of the system is investigatedand the explicit formulae which can determine the direction of thebifurcations and the stability of the periodic solutions are derivedusing the normal form method and the center manifold theory. Thenthe sufficient conditions for the system to have multiple periodicsolutions when the delay is far away from the critical values ofHopf bifurcations are obtained by using the Wu's global Hopfbifurcation theory and the Bendixson's criterion. Especially, asynchronized scheme is used during the analysis to reduce thedimension of the system. Finally, example numerical simulations aregiven to support the theoretical analysis.

Bifurcation Analysis for a Physiological Control System with Delay

In this paper, a delayed physiological control system is investigated. The sufficient conditions for stability of positive equilibrium and existence of local Hopf bifurcation are derived. Furthermore, global existence of periodic solutions is established by using the global Hopf bifurcation theory. Finally, numerical examples are given to support the theoretical analysis. Keywords—Physiological control system; global Hopf bifurcation; periodic solutions.

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 S 1 -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.

Global Continua of Rapidly Oscillating Periodic Solutions of State-Dependent Delay Differential Equations

We apply our recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with state-dependent delay. We give sufficient geometric conditions to ensure the uniform boundedness of periodic solutions, obtain an upper bound of the period of non-constant periodic solutions in a connected component of Hopf bifurcation, and establish the existence of rapidly oscillating periodic solutions.

LOCAL AND GLOBAL HOPF BIFURCATION IN A TWO-NEURON NETWORK WITH MULTIPLE DELAYS

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.

LOCAL AND GLOBAL HOPF BIFURCATION IN A DELAYED HEMATOPOIESIS MODEL

In this paper, we consider the following nonlinear differential equation [Formula: see text] We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second critical value of the delay τ. Finally, numerical simulation results are given to support the theoretical predictions.

S1-degree and global Hopf bifurcation theory of functional differential equations

The recently developed S 1-degree and bifrucation theory are applied to provide a purely topological argument of a global Hopf bifurcation theory for functional differential equations of mixed type. In the special case where the equation is of retarded type, the established result represents an analog of Alexander and Yorke's global Hopf bifurcation theorem which has been obtained by Chow, Fiedler, Mallet-Paret, and Nussbaum, using different approaches.