Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Stability and Hopf bifurcation for a regulated logistic growth model with discrete and distributed delays

In this paper, we investigate the stability and Hopf bifurcation of a new regulated logistic growth with discrete and distributed delays. By choosing the discrete delay τ as a bifurcation parameter, we prove that the system is locally asymptotically stable in a range of the delay and Hopf bifurcation occurs as τ crosses a critical value. Furthermore, explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Finally, an illustrative example is also given to support the theoretical results.     

Hopf bifurcation in a environmental defensive expenditures model with time delay

In this paper a three-dimensional environmental defensive expenditures model with delay is considered. The model is based on the interactions among visitors V, quality of ecosystem goods E, and capital K, intended as accommodation and entertainment facilities, in Protected Areas (PAs). The tourism user fees (TUFs) are used partly as a defensive expenditure and partly to increase the capital stock. The stability and existence of Hopf bifurcation are investigated. It is that stability switches and Hopf bifurcation occurs when the delay t passes through a sequence of critical values, τ₀. It has been that the introduction of a delay is a destabilizing process, in the sense that increasing the delay could cause the bio-economics to fluctuate. Formulas about the stability of bifurcating periodic solution and the direction of Hopf bifurcation are exhibited by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the results.     

LIMIT CYCLE PROBLEM OF QUADRATIC SYSTEM OF TYPE （ Ⅲ ）m=0, （ Ⅲ ）

To continue the discussion in （Ⅰ ） and （ Ⅱ ）,and finish the study of the limit cycle problem for quadratic system （ Ⅲ ）m=0 in this paper. Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation,the number of limit cycles depends on the different situations of separatrix cycle to be formed around O. If it is a homoclinic cycle passing through saddle S1 on 1 ＋ax-y = 0,which has the same stability with the limit cycle created by Hopf bifurcation,then the uniqueness of limit cycles in such cases can be proved. If it is a homoclinic cycle passing through saddle N on x= 0,which has the different stability from the limit cycle created by Hopf bifurcation,then it will be a case of two limit cycles. For the case when the separatrix cycle is a heteroclinic cycle passing through two saddles at infinity,the discussion of the paper shows that the number of limit cycles will change from one to two depending on the different values of parameters of system.     

Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics

In this paper, we consider the growth of densities of two kinds of typical HAB algae: diatom and dianoflagellate on some coasts of China’s mainland. Since there exist many random factors that cause the change of the algae densities, we shall develop a new nonlinear dynamical model with stochastic excitations on the algae densities. Applying a stochastic averaging method on the model, we obtain a two-dimensional diffusion process of averaged amplitude and phase. Then we investigate the stability and the Hopf bifurcation of the stochastic system with FPK (Fokker Planck–Kolmogorov) theory and obtain the stationary transition probability density of the process. We obtain the critical values of parameters for the occurrences of Hopf bifurcation in terms of probability. We also investigate numerically the effects of various parameters on the stationary transition probability density of the occurrences of Hopf bifurcation. The numerical results are in good correlation with the analysis. We draw the conclusion that if the Hopf bifurcation occurs with a radius large enough, i.e., if the densities of the HAB algae reach a high value, the HAB will take place with comparatively high probability.     

Stability of flow past cylinder cascade

The global stability analysis of the two–dimensional incompressible unsteady flow past a circular clinder cascade is studied using linear stability analysis. A new numerical technique is used to find the value of critical Reynolds number that causes the flow to undergo Hopf bifurcation. An attempt has also been made to study the blockage effect on critical Reynolds number and associated Strouhal number. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability and bifurcation analysis in a viral model with delay

In this paper, we consider a three‐dimensional viral model with delay. We first investigate the linear stability and the existence of a Hopf bifurcation. It is shown that Hopf bifurcations occur as the delay τ passes through a sequence of critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit formulaes that determine the stability, the direction, and the period of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the validity of the main results. Finally, some brief conclusions are given. Copyright © 2012 John Wiley & Sons, Ltd.     

Hopf bifurcation control in the XCP for the Internet congestion control system

In this paper, we investigated the Hopf bifurcation of the eXplicit Control Protocol (XCP) for the Internet congestion control system. These bifurcation behaviors may cause heavy oscillation of average queue length and induce network instability. A time-delayed feedback control method was proposed for controlling Hopf bifurcation in the XCP system. Numerical simulation results are presented to show that the time-delayed feedback controller is efficient in controlling Hopf bifurcation.     

Dynamics of a competitive Lotka-Volterra system with three delays

In this paper, a competitive Lotka-Volterra system with three delays is investigated. By choosing the sum τ of three delays as a bifurcation parameter, we show that in the above system, Hopf bifurcation at the positive equilibrium can occur as τ crosses some critical values. And we obtain the formulae determining direction of Hopf bifurcation and stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Stability and Hopf bifurcation for a competition and cooperation model of two enterprises with delay

In this paper, a retarded competition and cooperation model of two enterprises is investigated. We first prove that the existence of the unique positive equilibrium for the mentioned model. By choosing the delay $\tau$ as a bifurcation parameter, we show that Hopf bifurcation at the positive equilibrium of the system can occur as $\tau$ crosses some critical values. Further, using the normal form theory and center manifold theorem, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, some numerical simulations supporting our theoretical results and the economic meaning of model are also included.     

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

In this paper, we concentrate on the spatiotemporal patterns of a delayed reaction‐diffusion Holling‐Tanner model with Neumann boundary conditions. In particular, the time delay that is incorporated in the negative feedback of the predator density is considered as one of the principal factors to affect the dynamic behavior. Firstly, a global Turing bifurcation theorem for τ = 0 and a local Turing bifurcation theorem for τ > 0 are given. Then, further considering the degenerated situation, we derive the existence of Bogdanov‐Takens bifurcation and Turing‐Hopf bifurcation. The normal form method is used to study the explicit dynamics near the Turing‐Hopf singularity. It is shown that a pair of stable nonconstant steady states (stripe patterns) and a pair of stable spatially inhomogeneous periodic solutions (spot patterns) could be bifurcated from a positive equilibrium. Moreover, the Turing‐Turing‐Hopf–type spatiotemporal patterns, that is, a subharmonic phenomenon with two spatial wave numbers and one temporal frequency, are also found and explained theoretically. Our results imply that the interaction of Turing and Hopf instabilities can be considered as the simplest mechanism for the appearance of complex spatiotemporal dynamics.     

Bifurcation analysis for a three-species predator-prey system with two delays

In this paper, a three-species predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we first show that Hopf bifurcation at the positive equilibrium of the system can occur as τ crosses some critical values. Second, we obtain the formulae determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form theory and center manifold theorem. Finally, numerical simulations supporting our theoretical results are also included.     

Stability and Hopf bifurcation of a delayed predator-prey system with nonlocal competition and herd behaviour

In this paper, we investigate the stability and Hopf bifurcation of a diffusive predator-prey system with herd behaviour. The model is described by introducing both time delay and nonlocal prey intraspecific competition. Compared to the model without time delay, or without nonlocal competition, thanks to the together action of time delay and nonlocal competition, we prove that the first critical value of Hopf bifurcation may be homogenous or non-homogeneous. We also show that a double-Hopf bifurcation occurs at the intersection point of the homogenous and non-homogeneous Hopf bifurcation curves. Furthermore, by the computation of normal forms for the system near equilibria, we investigate the stability and direction of Hopf bifurcation. Numerical simulations also show that the spatially homogeneous and non-homogeneous periodic patters.     

Bifurcation analysis for a regulated logistic growth model

In this paper, we consider a regulated logistic growth model. We first consider the linear stability and the existence of a Hopf bifurcation. We show that Hopf bifurcations occur as the delay τ passes through critical values. Then, using the normal form theory and center manifold reduction, we derive the explicit algorithm determining the direction of Hopf bifurcations and the stability of the bifurcating periodic solutions. Finally, numerical simulation results are given to support the theoretical predictions.     

Bifurcations for a predator-prey system with two delays

In this paper, a predator-prey system with two delays is investigated. By choosing the sum τ of two delays as a bifurcation parameter, we show that Hopf bifurcations can occur as τ crosses some critical values. By deriving the equation describing the flow on the center manifold, we can determine the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation result of [J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc. 350 (1998) 4799-4838], we may show the global existence of periodic solutions.     

Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics

A nonlinear stochastic dynamical model on a typical HAB algae diatom and dianoflagellate densities was created and presented in this paper. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. The singular boundary theory of diffusion process and the invariant measure theory were applied in analyzing the bifurcation of stability and the Hopf bifurcation of the stochastic system. The critical value of the stochastic Hopf bifurcation parameter was obtained and the conclusion that the position of Hopf bifurcation drifting with the parameter increase is presented as a result.     

Global Dynamics of a Tuberculosis Epidemic Model and the Influence of Backward Bifurcation

In this paper, we propose and analyze a tuberculosis (TB) model with exogenous re-infection. We assume that treated individual may be again infected by infectious individual. The model exhibits two bifurcations viz. transcritical bifurcation when the basic reproductive number R ₀?=?1 and backward bifurcation where the disease transmission rate β plays as control parameter. The persistent of the model and, the local and global stability criteria of disease-free and endemic equilibria are discussed. By carrying out bifurcation analysis, it is shown that the model exhibits the bistability and undergoes the Hopf bifurcation when immunological memory is everlasting i.e. when σ?=?0. Lastly, some simulations are given to verify our analytical results.     

Complex hopf bifurcation of integral surfaces

The existence of the Hopf bifurcation of a complex ordinary differential equation system in the complex domain is studied in this paper by using the complex qualitative theory. In the complex domain, we conclude that the Hopf bifurcation appears for both directions of the parameter. The formulae of the Hopf bifurcation are also given in this paper.     

Stochastic resonance induced by novel random transitions of motion of FitzHugh–Nagumo neuron model

In contrast to the previous studies which have dealt with stochastic resonance induced by random transitions of system motion between two coexisting limit cycle attractors in the FitzHugh–Nagumo (FHN) neuron model after Hopf bifurcation and which have dealt with the phenomenon of stochastic resonance induced by external noise when the model with periodic input has only one attractor before Hopf bifurcation, in this paper we have focused our attention on stochastic resonance (SR) induced by a novel transition behavior, the transitions of motion of the model among one attractor on the left side of bifurcation point and two attractors on the right side of bifurcation point under the perturbation of noise. The results of research show: since one bifurcation of transition from one to two limit cycle attractors and the other bifurcation of transition from two to one limit cycle attractors occur in turn besides Hopf bifurcation, the novel transitions of motion of the model occur when bifurcation parameter is perturbed by weak internal noise; the bifurcation point of the model may stochastically slightly shift to the left or right when FHN neuron model is perturbed by external Gaussian distributed white noise, and then the novel transitions of system motion also occur under the perturbation of external noise; the novel transitions could induce SR alone, and when the novel transitions of motion of the model and the traditional transitions between two coexisting limit cycle attractors after bifurcation occur in the same process the SR also may occur with complicated behaviors types; the mechanism of SR induced by external noise when FHN neuron model with periodic input has only one attractor before Hopf bifurcation is related to this kind of novel transition mentioned above.