This article concerns a symmetrical Lotka-Volterra predator-prey system with delays. By analyzing the associated characteristic equation of the original system at the positive equilibrium and choosing the delay as the bifurcation parameter, the local stability and Hopf bifurcation of the system are investigated. Using the normal form theory, we also establish the direction and stability of the ...

We study analytically the (N-1)-fold degenerate Hopf bifurcation at which N stationary modes with identical parameters become unstable in a model of a solid-state laser with intracavity second harmonic generation. We use the normal form method and exploit the symmetries of the problem. Up to N=3, stable periodic antiphased solutions emerge from the Hopf bifurcation. For N=4, stable quasiperiodi...

About a three-dimensional autonomous continuous Lorenz-like system. Through the full analysis of equilibrium points, we can derive the parameter condition of Hopf bifurcation; Next we adopt a Washout filter controller designed independently to control the Hopf bifurcation of the system, and the effect of the controller’s parameters on the position of bifurcation point, the bifurcation type, and...

This paper discusses a problem of the Hopf bifurcation control for a mathematical model of intracellular calcium oscillations by calculating the curvature coefficient of limit cycle and the bifurcation control theory. We find that the appearance and disappearance of calcium oscillations in this system are due to the supercritical and subcritical Hopf bifurcation of equilibrium points, respectiv...

Whenever there is a time delay in a dynamical system, the study of stability becomes an infinite-dimensional problem. The centre manifold theorem, together with the classical Hopf bifurcation, is the most valuable approach for simplifying the infinite-dimensional problem without the assumption of small time delay. This dimensional reduction is illustrated in this paper with the delay versions o...

We explore the nature of the transitions between stick slip and steady sliding in models for boundary lubrication introduced in J. M. Carlson and A. A. Batista, Phys. Rev. E 53, 4153 ~1996!. The models are based on the rate and state approach which has been very successful in characterizing the behavior of dry interfaces @A. Ruina, J. Geophys. Res. 88, 10 359 ~1983!#. Our models capture the key...

In this paper, global dynamics of an SIR model are investigated in which the incidence rate is being considered as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated). Analytical study of the model shows that the model has two equilibrium points (diseasefree equilibrium (DFE) and endemic equilibrium (EE)). The disease-free equilibrium (DFE) is locally asymptotically ...

In this paper, we consider a diffusive predator–prey system with strong Allee effect and two delays. First, explore the stability region of positive constant steady state by calculating switching curves. Then derive Hopf double bifurcation theorem via crossing directions Moreover, calculate normal forms near singularities taking delays as parameters. We carry out some numerical simulations for ...

Abstract In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of corresponding kinetic system, existence stability conditions equilibrium type studied. Via center manifold theorem, sufficient in limit cycles obtained, respectively. addition, we find that strip patterns mainly induced by spot numerical simulations. These provide c...

A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stabil...