We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation wer...

A delayed three-species food chain system with two types of functional responses, Holling type and Beddington–DeAngelis type, is investigated. By analyzing the distribution of the roots of the associated characteristic equation, we get the sufficient conditions for the stability of the positive equilibrium and the existence of Hopf bifurcation. In particular, the properties of Hopf bifurcation ...

This paper discusses feed-forward chains near points of synchrony-breaking Hopf bifurcation. We show that at synchrony-breaking bifurcations the center manifold inherits a feed-forward structure and use this structure to provide a simplified proof of the theorem of Elmhirst and Golubitsky that there is a branch of periodic solutions in such bifurcations whose amplitudes grow at the rate of λ 1 ...

As new applications of Schrödinger type inequalities appearing in Jiang (J. Inequal. Appl. 2016:247, 2016), we first investigate the existence and uniqueness of a Schrödingerean equilibrium. Next we propose a tritrophic Hastings-Powell model with two different Schrödingerean time delays. Finally, the stability and direction of the Schrödingerean Hopf bifurcation are also investigated by using t...

This paper presents a detailed analysis on the dynamics of a ring network with short-cut. We first investigate the absolute synchronization on the basis of Lyapunov stability approach, and then discuss the linear stability of the trivial solution by analyzing the distribution of zeros of the characteristic equation. Based on the equivariant branching lemma, we not only obtain the existence of p...

Abstract: This paper is concerned with a two species Leslie-Gower predator-prey system with time delay. By regarding the delay τ as the bifurcation parameter, the stability of positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcating periodic solutions are determined by applying the norm form theory and the cent...

A kind of fourth-order delay differential equation is considered. Firstly, the linear stability is investigated by analyzing the associated characteristic equation. It is found that there are stability switches for time delay and Hopf bifurcations when time delay cross through some critical values. Then the direction and stability of the Hopf bifurcation are determined, using the normal form me...

In this paper, we provide a short proof for the Hopf bifurcation theorem in Gurtin-MacCamy’s population dynamics model. Here use Crandall and Rabinowitz’s approach, based on implicit function theorem. Compared with previous methods, here require age-specific birth rate to be slightly smoother (roughly of bounded variation), but have huge gain length proof.