Hopf-Bifurkation bei Lasern

The semiclassical equations describing a ring laser show two successive bifurcations, one stationary and one Hopf bifurcation. This phenomenon is analyzed mathematically. The initial value problem for the laser equations and the stability of the stationary solutions are discussed in detail. The transition to ultrashort laser pulses is shown to be a Hopf bifurcation. The direction of the bifurcation is determined for a numerical example. It turns out that it depends on the parameters of the system.     

弹性支承-刚性转子系统同步全周碰摩的分岔响应

基于航空发动机转子系统的结构特点,将航空发动机转子系统简化为一个非线性弹性支承的刚性转子系统．根据Lagrange方程建立了弹性支承-刚性不对称转子系统同步全周碰摩的运动方程;采用平均法进行求解,得到了关于系统振幅的分岔方程;根据两状态变量约束分岔理论,分别给出了系统在无碰摩和碰摩阶段参数平面的转迁集和分岔图,讨论了转子偏心、阻尼对系统分岔行为的影响;应用Liapunov稳定性理论分析了系统碰摩周期解的稳定性和失稳方式,给出了系统参数——转速平面上周期解的稳定范围;该文的研究结果对航空发动机转子系统的设计有一定的理论意义．     

On the stability and transition for the Navier-Stokes-alpha model

The main aim of this paper is to investigate the stability and transition of the Navier-Stokes-alpha model. By using the continued-fraction method, combining with the dynamic transition theory, we show the existence of a Hopf bifurcation in this model as Reynolds number crosses a critical value. Upon deriving the explicit expression of a non-dimensional number P called transition number, which is a function of the critical Reynolds number and the aspect ratio, we further analyze the transition associated with the Hopf bifurcation. More precisely, it is shown that the modeled flow exhibits either a continuous or catastrophic transition at the critical Reynolds number, whose specific type of the transition is determined by the sign of the real part of P at the critical Reynolds number, and the spatio-temporal structure of the limit cycle bifurcated that corresponds to a wave that propagates slowly westward and is symmetric about the mid-axis of the channel.     

Bursting oscillations, bifurcation and synchronization in neuronal systems

This paper investigates bursting oscillations and related bifurcation in the modified Morris-Lecar neuron. It is shown that for some appropriate parameters, the modified Morris-Lecar neuron can exhibit two types of fast-slow bursters, that is “circle/fold cycle” bursting and “subHopf/homoclinic” bursting with class 1 and class 2 neural excitability, which have different neuro-computational properties. By means of the analysis of fast-slow dynamics and phase plane, we explore bifurcation mechanisms associated with the two types of bursters. Furthermore, the properties of some crucial bifurcation points, which can determine the type of the burster, are studied by the stability and bifurcation theory. In addition, we investigate the influence of the coupling strength on synchronization transition and the neural excitability in two electrically coupled bursters with the same bursting type. More interestingly, the multi-time-scale synchronization transition phenomenon is found as the coupling strength varies.     

旋转流动的低模分析及仿真研究

为了探讨Couette-Taylor流从层流到湍流过渡的方式以及流动发展到湍流之后混沌吸引子的某些特征等问题,采用低模分析方法研究了Couette-Taylor流的部分动力学行为及仿真问题,讨论了Couette-Taylor流三模态类Lorenz型方程组的动力学行为,包括定态的失稳、极限环的出现、分岔与混沌的演变和全局稳定性分析等。通过线性稳定性分析和数值模拟等方法给出了此三维模型分岔与混沌等动力学行为及其演化历程,并借此解释了Couette-Taylor流试验中观察到的部分涡流的演化过程.基于系统的分岔图、Lyapunov指数谱、功率谱、Poincaré(庞加莱)截面和返回映射等揭示了系统混沌行为的普适特征.     

约束含参分岔问题的分类

约束边界与分岔参数有关的约束分岔问题,称为约束含参分岔问题．通过引入适当的变换,将约束含参分岔问题转化为新变量的非约束分岔问题,推导出了约束含参分岔问题转迁集的一般形式,结果表明只有约束分岔集受约束含参的影响,其它转迁集与不含参约束分岔的转迁集相同．以含参约束树枝分岔为例分析了此类问题的分岔分类,讨论了约束含参对分岔分类的影响．     

Secondary Bifurcation in Nonlinear Diffusion Reaction Equations

A set of two coupled nonlinear diffusion reaction equations is studied and the existence of secondary bifurcation is shown. Using the method of two-timing, it is found that diffusion reaction equations of this type can exhibit an exchange of stability between distinct nontrivial solutions. This exchange can provide either a smooth or discontinuous transition between stable solutions, and the nontrivial solutions can be either steady or temporally periodic. This analysis is applied to the model biochemical reaction of Prigogine and the types of secondary bifurcation which occur in this model are classified.     

Periodic and nonperiodic oscillatory behavior in a model for activated sludge reactors

A dynamic model for an activated sludge process is proposed to investigate the stability and bifurcation characteristics of this industrially important unit. The model is structured upon two processes: an intermediate participate product formation and active biomass synthesis processes. The growth kinetics expressions are based on substrate inhibition and noncompetitive inhibition of the intermediate product. The bifurcation analysis of the process model shows static as well as periodic behavior over a wide range of model parameters. The model also exhibits other interesting stability characteristics, including bistability and transition from periodic to nonperiodic behavior through period doubling and torus bifurcations. For some range of the reactor residence time the model exhibits chaotic behavior as well. Practical criteria are also derived for the effects of feed conditions and purge fraction on the dynamic characteristics of the bioreactor model.     

Stability and local bifurcation analysis of functionally graded material plate under transversal and in-plane excitations

In this paper, stability and local bifurcation behaviors for a simply supported functionally graded material (FGM) rectangular plate subjected to the transversal and in-plane excitations in the uniform thermal environment are investigated using both analytical and numerical methods. Three kinds of degenerated equilibrium points of the bifurcation response equations are considered, which are characterized by a double zero eigenvalues, a simple zero and a pair of pure imaginary eigenvalues as well as two pairs of pure imaginary eigenvalues in nonresonant case, respectively. With the aid of Maple and normal form theory, the explicit expressions of transition curves are obtained, which may lead to static bifurcation, Hopf bifurcation and 2-D torus bifurcation. Finally, the numerical solutions obtained by using fourth-order Runge–Kutta method agree with the analytic predictions.     

NONLINEAR MATRIX MODELS AND POPULATION DYNAMICS

Nonlinear matrix difference equations are studied as models for the discrete time dynamics of a population whose individual members have been categorized into a finite number of classes. The equations are treated with sufficient generality so as to include virtually any type of structuring of the population (the sole constraint is that all newborns lie in the same class) and any types of nonlinearities which arise from the density dependence of fertility rates, survival rates and transition probabilities between classes. The existence and stability of equilibrium class distribution vectors are studied by means of bifurcation theory techniques using a single composite, biologically meaningful quantity as a bifurcation parameter, namely the inherent net reproductive rate r. It is shown that, just as in the case of linear matrix equations, a global continuum of positive equilibria exists which bifurcates as a function of r from the zero equilibrium state at and only at r = 1. Furthermore the zero equilibrium loses stability as r is increased through 1. Unlike the linear case however, for which the bifurcation is “vertical” (i.e., equilibria exist only for r = 1), the nonlinear equation in general has positive equilibria for an interval of r values. Methods for studying the geometry of the continuum based upon the density dependence of the net reproductive rate at equilibrium are developed. With regard to stability, it is shown that in general the positive equilibria near the bifurcation point are stable if the bifurcation is to the right and unstable if it is to the left. Some further results and conjectures concerning stability are also given. The methods are illustrated by several examples involving nonlinear models of various types taken from the literature.     

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.     

Free boundary surface bifurcation in the phase transition problem of elasticity

The paper is concerns the phase transition problem in an elastic medium lying in a one-parameter force field. The stability and bifurcation problem for the interface between the two phases is studied. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 169–200. Translated by N. A. Karazeeva.     

Stability analysis in a diffusional immunosuppressive infection model with delayed antiviral immune response

In this paper, the diffusion is introduced to an immunosuppressive infection model with delayed antiviral immune response. The direction and stability of Hopf bifurcation are effected by time delay, in the absence of which the positive equilibrium is locally asymptotically stable by means of analyzing eigenvalue spectrum; however, when the time delay increases beyond a threshold, the positive equilibrium loses its stability via the Hopf bifurcation. The stability and direction of the Hopf bifurcation is investigated with the norm form and the center manifold theory. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

一类捕食与被捕食系统的稳定性与分岔分析

本文利用Schur—Cohn—Jury引理及分岔理论讨论了一类捕食与被捕食系统的动力学性质，分析了其正平衡点的稳定性，并讨论了Neimark—Sacker分岔稳定性与方向。通过数值模拟验证了所得结果的正确性。     

Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay

In this paper, we investigated Hopf bifurcation by analyzing the distributed ranges of eigenvalues of characteristic linearized equation. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay have also been derived, and, furthermore, the direction of Hopf bifurcation as well as stability of periodic solution for the exponential RED algorithm with communication delay is studied. We find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, a numerical simulation is presented to verify the theoretical results.     

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.     

On cascades of bifurcations leading to chaos in several nonlinear dissipative systems of ODEs

This paper considers several nonlinear dissipative systems of ordinary differential equations. The studied systems undergo a full analysis of corresponding singular points on a whole set of parameters’ values variation. Specifically, types of singular points, boarders of stability regions, as well as presented local bifurcations, are determined. By using numerical methods a consideration of scenarios of transition to chaos in these systems with one bifurcation parameter variation is held. The aim of this research is a confirmation of a Feigenbaum–Sharkovskii–Magnitskii mechanism of transition to chaos unique for all dissipative systems of ODEs. As the result of analysis of one of the systems the lack of any chaotic behavior is shown with the help of Poincare sections.     

Origin of the division of labour and a stochastic mechanism of differentiation

Nonlinear models are introduced to describe the nonequilibrium dynamics of social evolution. The difference between Western and Oriental culture, and their roles in the origin in the division of labour, are described by a behavioural model in information diffusion and learning competition. It shows a tradeoff between stability and diversity. The stochastic mechanism of social differentiation and the empirical evidence for this is discussed in a stochastic model of multi-staged development. It shows the breakdown of the Gaussian distribution during a transition. Finally, an ideal model of social bifurcation is given.     

压板松动时大型发电机端部绕组的主共振与分岔

研究了大型汽轮发电机定子端部固定绕组的压板松动时,位于两侧压板间某段绕组的振动问题．首先,采用分离变量法,给出了发电机运行时定子端部绕组区域的磁感应强度表达式,并给出了绕组所受电磁力及与松动压板间摩擦力的计算式．其次,建立了研究绕组非线性振动问题的力学分析模型,采用多尺度法对主共振情形进行了解析求解,推得了稳态运动下的幅频响应方程,并对定常解的稳定性及分岔奇异性进行了研究,得到了稳定性的判定条件及分岔方程的转迁集．最后,针对工程实际问题进行了计算,给出了相应的幅频响应曲线图,并进行了分析讨论．     

Stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment

In this paper, the stability and Hopf bifurcation of a delayed viral infection model with logistic growth and saturated immune impairment is studied. It is shown that there exist 3 equilibria. The sufficient conditions for local asymptotic stability of the infection‐free equilibrium and no‐immune equilibrium are given. We also discussed the local stability of positive equilibrium and the existence of Hopf bifurcation. Moreover, the direction and stability of Hopf bifurcation is obtained by using standard form theory and the center manifold theorem. Finally, numerical simulations are performed to verify the theoretical conclusions.