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Research on nonautonomous delayed differential equations (DDEs) is crucial and very difficult due to nonautonomy and time delay in many fields. The main work of the present paper is to discuss complex dynamics of nonautonomous DDEs, such as Hopf bifurcation, periodic solutions and multistability. We consider three examples of nonautonomous DDEs with time-varying coefficients: a harmonically forced Duffing oscillator with time delayed state feedback and periodic disturbance, generalized van der Pol oscillator with delayed displacement difference feedback and periodic disturbance, and an electro-mechanical system with delayed and periodic disturbance. Firstly, we obtain the amplitude equations of these three examples by the method of multiple scales (MMS), and then analyze the stability of approximate solutions by the Routh–Hurwitz criterion. The obtained amplitude equations are used to construct the bifurcation diagrams, so that we can observe the occurrence of the Hopf bifurcation and judge its type (super- or subcritical) from the bifurcation diagrams. We discover rich dynamic phenomena of the three systems under consideration, such as Hopf bifurcation, quasi-periodic solutions and the coexistence of multiple stable solutions, and then discuss the impact of some parameter changes on the system dynamics. Finally, we validate the correctness of these theoretical conclusions by software WinPP, and the numerical simulations are consistent with our theoretical findings. Therefore, the MMS we use to analyze the dynamics of nonautonomous DDEs is effective, which is of great significance to the research of nonautonomous DDEs in many fields. [ABSTRACT FROM AUTHOR]

To reduce the global burden of mosquito-borne diseases, e.g. dengue, malaria, the need to develop new control methods is to be highlighted. The sterile insect technique (SIT) and various genetic modification strategies, have a potential to contribute to a reversal of the current alarming disease trends. In our previous work, the ordinary differential equation (ODE) models with different releasing sterile mosquito strategies are investigated. However, in reality, implementing SIT and the releasing processes of sterile mosquitos are very complex. In particular, the delay phenomena always occur. To achieve suppression of wild mosquito populations, in this paper, we reassess the effect of the delayed releasing of sterile mosquitos on the suppression of interactive mosquito populations. We extend the previous ODE models to the delayed releasing models in two different ways of releasing sterile mosquitos, where both constant and exponentially distributed delays are considered, respectively. By applying the theory and methods of delay differential equations, the effect of time delays on the stability of equilibria in the system is rigorously analyzed. Some sustained oscillation phenomena via Hopf bifurcations in the system are observed. Numerical examples demonstrate rich dynamical features of the proposed models. Based on the obtained results, we also suggest some new releasing strategies for sterile mosquito populations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, a model of predator-prey with disease in food chain is investigated - where, prey is infected by bacteria and then the infected prey in turn infects predator, but the disease does not spread among predators. The law for disease development and biodiversity conservation are the focus. Stability and persistence are deduced in terms of system parameters. Next, time required delay is incorporated into the model. Stability and bifurcation analysis of the delay differential equation model are carried out. Furthermore, stability and direction of the bifurcating periodic solutions are performed by the normal form theory and the center manifold argument. Finally, numerical simulations are included for illustrating the theoretical analysis. [ABSTRACT FROM AUTHOR]