共查询到19条相似文献,搜索用时 62 毫秒
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时滞Lienard非线性系统的Hopf分岔 总被引:3,自引:0,他引:3
本文研究了Lienard非线性时滞系统的线性稳定性和Hopf分岔,考究了特征方程随两参数变化时根的分布,应用中心流形和范式分析失去线性稳定性出现的Hopf分岔及其稳定性。 相似文献
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以Chen系统为研究对象,研究其加入扰动项后的动力学特性。分析了时滞扰动Chen系统的平衡点及其稳定性等问题,计算特征根分布及其对应Hopf分岔特征量。根据Hopf分岔定理判断分岔类型及方向,给出分岔发生的条件并运用Matlab仿真验证了当系统经过临界点时会发生Hopf分岔现象。针对时滞扰动带来的Hopf分岔设计线性控制器并将其添加到系统方程中,对其进行滞后控制,仿真结果表明,该控制器使系统在不改变平衡点的前提下,将分岔临界点由0.1609延迟至0.1860,可对扰动引起的Hopf分岔进行有效控制。 相似文献
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van der Pol型时滞系统的两参数余维一Hopf分岔及其稳定性 总被引:5,自引:0,他引:5
研究具有三次非线性时滞项的van der Pol型时滞系统随两参数(时滞量和增益系数)余维一Hopf分岔,说明了线性化特性方程随两参数变化时的根的分布和Hopf分岔存在性;通过构造中心流形并且使用范式方法确定出Hopf分岔的方向以及周期解的稳定性;分析了时滞量对所论系统发生Hopf分岔的影响。 相似文献
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神经网络时滞系统非共振双Hopf分岔及其广义同步 总被引:2,自引:0,他引:2
本文建立了具有自连接和抑制-兴奋型他连接的两个同性神经元模型。其中自连接是由于兴奋型的突触产生,而他连接则分别对应于两神经元兴奋、抑制型的突触。发现如果有兴奋型自连接就会有双Hopf分岔,而没有时滞自连接时双Hopf分岔就会消失,因此自连接引起了双Hopf分岔。作为一个例子,通过变动连接中的时滞和他连接中的比重,1/√2双Hopf分岔得到了详细研究。通过中心流形约化,分岔点邻域内各种不同的动力学行为得到了分类,并以解析形式表出。神经元活动的分岔路径得以表明。从得到的解析近似解可以发现,本文所研究的具有兴奋一抑制型他连接的两相同神经元的节律不能完全同步而只能广义同步。时滞也可以使其节律消失,两神经元变为非活动的。这些结果在控制神经网络关联记忆和设计人工神经网络方面有着潜在的应用。 相似文献
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对超混沌系统进行分岔反控制的研究已成为当前一个重要研究方向,常采用线性控制器实现反控制。首先,对一个四维超混沌系统的Hopf分岔特性进行了分析,利用高维分岔理论推导出分岔特性与参数之间的关系式,以此判断系统的分岔类型。然后,设计一个由线性与非线性组合成的混合控制器对系统进行分岔反控制,控制参数取值不同时,系统会呈现出不同的分岔特性。通过分析得出,调控线性控制器参数可以使系统Hopf分岔提前或延迟发生;同时,调控混合控制器的两个控制参数,可以改变系统Hopf分岔特性,实现分岔反控制。 相似文献
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糖尿病治疗模型中技术时滞诱发的双Hopf分岔 总被引:1,自引:0,他引:1
本文研究了利用外部辅助设备来治疗糖尿病的生理模型,其中存在着两个时滞;辅助设备的技术时滞τ1和肝脏的生理时滞τ2。发现由于技术时滞τ1的出现,使模型存在着共振和非共振的双Hopf分岔。应用非线性动力学理论,对由此产生的非共振分岔的动力学行为进行了分类。结果表明,随着技术时滞τ1和糖尿病人患病程度α的变化,利用该模型可以预测不同的医疗结果:血糖稳定(康复)、简单的和复杂的血糖波动。结果对分析、预测、优化糖尿病治疗方案的医疗结果、评估该方案的医疗风险和可行性等有着潜在的应用价值。本文结果的意义在于针对糖尿病患者患病的不同程度,可以定性的调节辅助设备的技术时滞τ1,以达到更好的治疗效果。 相似文献
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The Hopfbifurcation for the Brusselator ordinary-differential-equation (ODE) model and the corresponding partial-differential-equation (PDE) model are investigated by using the Hopf bifurcation theorem. The stability of the Hopf bifurcation periodic solution is discussed by applying the normal form theory and the center manifold theorem. When parameters satisfy some conditions, the spatial homogenous equilibrium solution and the spatial homogenous periodic solution become unstable. Our results show that if parameters are properly chosen, Hopf bifurcation does not occur for the ODE system, but occurs for the PDE system. 相似文献
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This paper investigates the dynamics of a TCP system described by a first- order nonlinear delay differential equation. By analyzing the associated characteristic transcendental equation, it is shown that a Hopf bifurcation sequence occurs at the pos- itive equilibrium as the delay passes through a sequence of critical values. The explicit algorithms for determining the Hopf bifurcation direction and the stability of the bifur- cating periodic solutions are derived with the normal form theory and the center manifold theory. The global existence of periodic solutions is also established with the method of Wu (Wu, J. H. Symmetric functional differential equations and neural networks with memory. Transactions of the American Mathematical Society 350(12), 4799-4838 (1998)). 相似文献
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The general Brusselator system is considered under homogeneous Neumann boundary conditions. The existence results of the Hopf
bifurcation to the ordinary differential equation (ODE) and partial differential equation (PDE) models are obtained. By the
center manifold theory and the normal form method, the bifurcation direction and stability of periodic solutions are established.
Moreover, some numerical simulations are shown to support the analytical results. At the same time, the positive steady-state
solutions and spatially inhomogeneous periodic solutions are graphically shown to supplement the analytical results. 相似文献
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甘春标 《Acta Mechanica Sinica》2004,20(5):558-566
A new procedure is developed to study the stochastic Hopf bifurcation in quasiintegrable-Hamiltonian systems under the Gaussian white noise excitation. Firstly, the singular boundaries of the first-class and their asymptotic stable conditions in probability are given for the averaged Ito differential equations about all the sub-system‘s energy levels with respect to the stochastic averaging method. Secondly, the stochastic Hopf bifurcation for the coupled sub-systems are discussed by defining a suitable bounded torus region in the space of the energy levels and employing the theory of the torus region when the singular boundaries turn into the unstable ones. Lastly, a quasi-integrable-Hamiltonian system with two degrees of freedom is studied in detail to illustrate the above procedure.Moreover, simulations by the Monte-Carlo method are performed for the illustrative example to verify the proposed procedure. It is shown that the attenuation motions and the stochastic Hopf bifurcation of two oscillators and the stochastic Hopf bifurcation of a single oscillator may occur in the system for some system‘s parameters. Therefore, one can see that the numerical results are consistent with the theoretical predictions. 相似文献
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We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems. 相似文献
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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. 相似文献
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A new approach for the computation of Hopf bifurcation points 总被引:4,自引:1,他引:3
叶瑞松 《应用数学和力学(英文版)》2000,21(11):1300-1307
IntroductionConsiderthefollowingparameterdependentnonlinearproblemf(x,λ) =0 , f:X×R →X ,( 1 )whereX=Rn,λisrealparameter,f∈Cr(r≥ 2 ) .Theoriginalproblemcouldbeasystemofdifferentialequation ,butherewewillassumethatasuitablediscretizationhasbeenmadeandtheproblemhasth… 相似文献
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Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback 总被引:2,自引:0,他引:2
This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed
velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system
may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical
value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges
in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating
periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes
of the bifurcating periodic solutions.
The project supported by the National Natural Science Foundation of China (19972025) 相似文献
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In this paper we studied a non-ideal system with two degrees of freedom consisting of a dumped nonlinear oscillator coupled to a rotatory part. We investigated the stability of the equilibrium point of the system and we obtain, in the critical case, sufficient conditions in order to obtain an appropriate Normal Form. From this, we get conditions for the appearance of Hopf Bifurcation when the difference between the driving torque and the resisting torque is small. It was necessary to use the Bezout Theorem, a classical result of Algebraic Geometry, in the obtaining of the foregoing results. 相似文献