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

Wang, Zaihua; Hu, Haiyan

doi:10.1007/s10409-008-0170-1

Cited by 10 publications

(3 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition to the HALE and LW approaches, we consider a modified method that can be found in Wang and Hu [58]. This last method was applied on a delayed Van Der Pol oscillator and showed some success in retaining the complex system dynamics in the averaged equations and maintaining at the same time a simpler formulation.…”

Section: The Methods Of Wang and Hu (Wh Method)mentioning

confidence: 99%

On the importance of time delay and noise in thermoacoustic modeling

Gant¹,

Ghirardo²,

Bothien

2021

Journal of Sound and Vibration

View full text Add to dashboard Cite

Section: The Methods Of Wang and Hu (Wh Method)mentioning

confidence: 99%

On the importance of time delay and noise in thermoacoustic modeling

Gant¹,

Ghirardo²,

Bothien

2021

Journal of Sound and Vibration

View full text Add to dashboard Cite

“…It is widely known that past history as well as current conditions can affect population dynamics, and such interactions have motivated the introduction of time delays in population growth models [8][9][10][11][12]. And time delay is ubiquitous and has critical effect on the dynamics of the dynamical systems [13][14][15]. In fact, there is a time delay from obtaining food to giving birth to a new life for each species.…”

Section: Introductionmentioning

confidence: 99%

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

Wang

Pei

2011

Acta Mech Sin

View full text Add to dashboard Cite

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 fluctuate 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.

show abstract

“…Furthermore, much effort has been paid to investigate the stability and bifurcation of periodic solution. For example, Wang and Hu [2] presented a modified averaging scheme with application to the secondary Hopf bifurcation of a delayed van der Pol oscillator. Gan and He [3] studied the structural safety in a kind of excited Duffing oscillator.…”

mentioning

confidence: 99%