MESOSCOPIC DESCRIPTION OF CHEMICAL SUPERCRITICAL HOPF BIFURCATION

2004 ◽  
Vol 14 (07) ◽  
pp. 2393-2397 ◽  
Author(s):  
QIAN SHU LI ◽  
RUI ZHU
Keyword(s):  
Hopf Bifurcation ◽  
Small Amplitude ◽  
Model System ◽  
Limit Cycle Oscillation ◽  
Hopf Bifurcation Point ◽  
Supercritical Hopf Bifurcation ◽  
New Type ◽  
Internal Signal ◽  
Cycle Oscillation ◽  
Oregonator Model

The mesoscopic dynamic behavior of the Oregonator model of the Belousov–Zhabotinsky chemical reaction is investigated as the model system experiences a supercritical Hopf bifurcation from focus to limit cycle oscillation. The study is performed by stochastically simulating the corresponding chemical master equation. Comparing the mesoscopic dynamic results with those obtained by the macroscopic dynamics, we find in the mesoscopic description a new type of oscillating state, in which large-amplitude oscillations and small-amplitude oscillations appear randomly alternately. This new state comes out spontaneously within a certain region called Hopf bifurcation range by us. In the mesoscopic description, the Hopf bifurcation point cannot be shown, being replaced by a Hopf bifurcation range. Furthermore, the applications of this new oscillating state to internal signal stochastic resonance are pointed out.

Stability and Hopf Bifurcation in a Predator–Prey Model with the Cost of Anti-Predator Behaviors

2019 ◽  
Vol 29 (13) ◽  
pp. 1950185 ◽  
Author(s):  
Ting Qiao ◽  
Yongli Cai ◽  
Shengmao Fu ◽  
Weiming wang
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle Oscillation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Existence And Stability ◽  
Cycle Oscillation ◽  
Predator Model ◽  
The Stability ◽  
The Cost

In this paper, we investigate the influence of anti-predator behavior in prey due to the fear of predators with a Beddington–DeAngelis prey–predator model analytically and numerically. We give the existence and stability of equilibria of the model, and provide the existence of Hopf bifurcation. In addition, we investigate the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also prevent the occurrence of limit cycle oscillation and increase the stability of the system.

Dynamics of a delayed competitive system affected by toxic substances with imprecise biological parameters

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5271-5293
Author(s):  
A.K. Pal ◽  
P. Dolai ◽  
G.P. Samanta
Keyword(s):  
Equilibrium Points ◽  
Stable Limit Cycle ◽  
Limit Cycle Oscillation ◽  
Toxic Substances ◽  
Stable Limit ◽  
Biological Parameters ◽  
Delay Model ◽  
Competitive System ◽  
Interval Numbers ◽  
Cycle Oscillation

In this paper we have studied the dynamical behaviours of a delayed two-species competitive system affected by toxicant with imprecise biological parameters. We have proposed a method to handle these imprecise parameters by using parametric form of interval numbers. We have discussed the existence of various equilibrium points and stability of the system at these equilibrium points. In case of toxic stimulatory system, the delay model exhibits a stable limit cycle oscillation. Computer simulations are carried out to illustrate our analytical findings.

scholarly journals Reducing parametric uncertainty in limit-cycle oscillation computational models

2017 ◽  
Vol 121 (1241) ◽  
pp. 940-969 ◽  
Author(s):  
R. Hayes ◽  
R. Dwight ◽  
S. Marques
Keyword(s):  
Limit Cycle ◽  
Computational Models ◽  
Structural Parameters ◽  
Parametric Uncertainty ◽  
Limit Cycle Oscillation ◽  
Posterior Distributions ◽  
Input Parameters ◽  
Data Points ◽  
Cycle Oscillation ◽  
True Values

ABSTRACTThe assimilation of discrete data points with model predictions can be used to achieve a reduction in the uncertainty of the model input parameters, which generate accurate predictions. The problem investigated here involves the prediction of limit-cycle oscillations using a High-Dimensional Harmonic Balance (HDHB) method. The efficiency of the HDHB method is exploited to enable calibration of structural input parameters using a Bayesian inference technique. Markov-chain Monte Carlo is employed to sample the posterior distributions. Parameter estimation is carried out on a pitch/plunge aerofoil and two Goland wing configurations. In all cases, significant refinement was achieved in the distribution of possible structural parameters allowing better predictions of their true deterministic values. Additionally, a comparison of two approaches to extract the true values from the posterior distributions is presented.

scholarly journals Modelling HIV dynamics with cell-to-cell transmission and CTL response

2021 ◽  
Author(s):  
Zirui Zhu ◽  
Ranchao Wu ◽  
Yu Yang ◽  
Yancong Xu
Keyword(s):  
Hopf Bifurcation ◽  
Cell Interaction ◽  
Backward Bifurcation ◽  
Hiv Treatment ◽  
Hiv Model ◽  
Ctl Response ◽  
Hiv Dynamics ◽  
Subcritical Hopf Bifurcation ◽  
Cell Transmission ◽  
New Type

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell-to-cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so-called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov-Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell-to-cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so-called new type bifurcation persist. Finally, some conclusions and discussions are also given.

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