A modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator

In this paper, a modified averaging scheme is presented for a class of time-delayed vibration systems with slow variables. The new scheme is a combination of the averaging techniques proposed by Hale and by Lehman and Weibel, respectively. The averaged equation obtained from the modified scheme is simple enough but it retains the required information for the local nonlinear dynamics around an equilibrium. As an application of the present method, the delay value for which a secondary Hopf bifurcation occurs is successfully located for a delayed van der Pol oscillator.

Bifurcation behavior and coexisting motions in a time-delayed power system

With the increase of system scale, time delays have become unavoidable in nonlinear power systems, which add the complexity of system dynamics and induce chaotic oscillation and even voltage collapse events. In this paper, coexisting phenomenon in a fourth-order time-delayed power system is investigated for the first time with different initial conditions.With the mechanical power, generator damping factor, exciter gain, and time delay varying, the specific characteristic of the time-delayed system, including a discontinuous ＂jump＂ bifurcation behavior is analyzed by bifurcation diagrams, phase portraits, Poincar′e maps, and power spectrums. Moreover, the coexistence of two different periodic orbits and chaotic attractors with periodic orbits are observed in the power system, respectively. The production condition and existent domain of the coexistence phenomenon are helpful to avoid undesirable behavior in time-delayed power systems.

Stability and Neimark-Sacker bifurcation analysis of a food-limited population model with a time delay

In this paper,a kind of discrete delay food-limited model obtained by the Euler method is investigated,where the discrete delay τ is regarded as a parameter.By analyzing the associated characteristic equation,the linear stability of this model is studied.It is shown that Neimark-Sacker bifurcation occurs when τ crosses certain critical values.The explicit formulae which determine the stability,direction,and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form.Finally,numerical simulations are performed to verify the analytical results.

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

Chaos Control and Bifurcation Behavior for a Sprott E System with Distributed Delay Feedback

In this paper, the problem of controlling chaos in a Sprott E system with distributed delay feedback is considered. By analyzing the associated characteristic transcendental equation, we focus on the local stability and Hopf bifurcation nature of the Sprott E system with distributed delay feedback. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using the normal form theory and center manifold theory. Numerical simulations for justifying the theoretical analysis are provided.