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1.
针对Rössler系统平衡点的Hopf分岔,以Washout滤波器为控制器,详细讨论了控制器参数对Hopf分岔点位置、分岔类型以及周期解振幅的控制问题.首先根据Routh-Hurwitz判据计算了受控系统的参数空间稳定域,找出了对应的Hopf分岔边界,并由此分析了滤波器时间常数、线性控制增益对分岔点位置的影响.然后,引入Normal Form直接法方便地求出系统Hopf分岔Normal Form系数,由此确定出改变分岔类型和周期解振幅的控制器非线性增益选择原则.最后用数值计算验证了本文的结论.
关键词:
Rö
ssler系统
Washout滤波器
Hopf分岔
Normal Form 相似文献
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Bifurcation is investigated with the full velocity difference traffic model. Applying the Hopt theorem, an analytical Hopf bifurcation calculation is performed and the critical road length is determined for arbitrary numbers of vehicles. It is found that the Hopf bifurcation critical points locate on the boundary of the linear instability region. Crossing the boundary, the uniform traffic flow loses linear stability via Hopf bifurcation and the oscillations appear. 相似文献
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Linear stability of solitary waves near transcritical bifurcations is analyzed for the generalized nonlinear Schrödinger equations with arbitrary forms of nonlinearity and external potentials in arbitrary spatial dimensions. Bifurcation of linear-stability eigenvalues associated with this transcritical bifurcation is analytically calculated. Based on this eigenvalue bifurcation, it is shown that both solution branches undergo stability switching at the transcritical bifurcation point. In addition, the two solution branches have opposite linear stability. These analytical results are compared with the numerical results, and good agreement is obtained. 相似文献
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P. SHAHRZADM. MAHZOON 《Journal of sound and vibration》2002,256(2):213-225
The limit cycle flutter of a two-dimensional wing with non-linear pitching stiffness is investigated. For modelling the aerodynamic forces of the wing steady linear and non-linear models as well as an unsteady model were used. The flutter speed was calculated using the harmonic balance method and by predicting Hopf bifurcation. Analytical solutions based on the centre manifold theory and normal forms were obtained as were results given by the harmonic balance method. The analytical solutions were compared with those obtained by numerical integration. The results show that the harmonic balance method can forecast flutter speed with a good accuracy while analytical solutions based on centre manifold theorem are accurate only in a small neighbourhood of the bifurcation point. The oscillation of the airfoil after flutter for two different models, linear and non-linear pitching stiffness were compared with each other and the flutter speeds for two linear steady and an unsteady aerodynamic model calculated. The obtained results show that flutter analysis based on the linear steady model is conservative only for the ratios of plunge frequency to pitch frequency lower than 1. 相似文献
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对称双弹簧振子受迫、有阻尼横振动的混沌行为 总被引:4,自引:1,他引:3
对受周期外力驱动的对称双弹簧振子进行了研究,建立了系统的动力学方程,用线性稳定性分析方法讨论了平衡点附近邻域的稳定性,利用数值计算并结合多种分析方法,求解非线性方程和判断解的性质.通过改变系统参数,画出时域图、相图及分岔图等.计算分析和数值实验发现,这个简单的力学系统存在十分丰富的动力学行为(分岔、混沌).理论分析和数值实验结果一致. 相似文献
7.
V. V. Gubernov A. V. Kolobov A. A. Polezhaev H. S. Sidhu 《Combustion Theory and Modelling》2013,17(3):385-407
In this paper we investigate the properties and the linear stability of premixed combustion waves in a non-adiabatic thermal-diffusive model with a two-step chain-branching reaction mechanism. Here we focus only on the emergence of the pulsating instabilities, and the stability analysis is carried out for Lewis numbers for fuel greater than one, and various values of Lewis number for radicals. We consider the problem in two spatial dimensions to allow perturbations of a multidimensional nature. It is demonstrated that the flame speed as a function of the parameters is a double-valued C-shaped function, i.e. for a given set of parameter values there are either two solutions, fast and slow solution branches, propagating with different speed, or the combustion wave does not exist. The extinction of combustion waves occurs at finite values of the parameters and non-zero flame speed. The flame structure demonstrates a slow recombination regime behaviour with negligible fuel leakage for the fast solution branch away from the extinction condition. For parameter values close to the extinction condition and on the slow solution branch, the fuel leakage is significant and a fast recombination regime is observed. It is demonstrated that two types of instabilities emerge in the model: the uniform planar and the travelling instability. The slow solution branch is always unstable due to the uniform perturbations. The fast solution branch is either stable or loses stability due to the travelling or uniform perturbations. The switching between the onset of various regimes of instability is due to the bifurcation of co-dimension two. In the adiabatic limit this bifurcation is found for Lewis number for fuel equal to one, whereas in the non-adiabatic case it moves towards values above unity. The properties of the travelling instability are studied in detail. 相似文献
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We investigate the bifurcation diagram of a laser with saturable absorber in the low and medium intensity regimes. The linear stability of the stationary solutions corresponding to these regimes is studied. In the low intensity domain, a Hopf bifurcation point is determined from which a time-periodic solution emerges. This solution is contructed and its stability is analyzed in the vicinity of the bifurcation point. It is shown that this time-periodic solution is stable in a finite domain of the parameter space. 相似文献
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Karl Heinz Hoffmann 《Zeitschrift für Physik B Condensed Matter》1982,49(3):245-252
The behaviour of the Hopf bifurcation under the influence of external noise is investigated by means of a twodimensional model which uses Gaussian white noise as input. The model includes the case of multiplicative and/or additive noise. Applying the Birkhoff transformation the model is transformed to the coordinates normally used to discuss the deterministic Hopf bifurcation. Then the stationary solution of the model is calculated as an expansion for weak noise: The Hopf bifurcation under the influence of noise exhibits a bifurcation interval with width and position depending on the noise power. Moreover, a class of the systems described by the model can perform noise driven bifurcations. 相似文献
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The steady state response and bifurcation of nonlinear random business cycle model to random narrow-band excitation with time delay state feedback are studied in this paper. The method of multiple scales is used to determine the business cycle model of modulation of amplitude and phase. The effects of delay, detuning, bandwidth and magnitude of random excitation on dynamics of the business cycle system are investigated. The results show that the complex dynamics such as bifurcation, jump domain and so on are induced by time delay and the phenomena that multiple solution or bifurcation is induced by noise. 相似文献
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Stability and Neimark-Sacker bifurcation analysis of food-limited population model with time delay 下载免费PDF全文
In this paper, a kind of discrete delay food-limited model obtained by Euler method is investigated, where the discrete delay τ is regarded as a parameter. By analyzing the associated characteristic equation, the linear stability of this model is studied. It is shown that Neimark-Sacker bifurcation occurs when τ crosses some critical values. The explicit formulae which determine the stability, direction, and other properties of bifurcating periodic solution are derived by means of the theory of center manifold and normal form. Finally, numerical simulations are performed to verify the analytical results. 相似文献
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This paper undertakes a nonlinear analysis of a model for a maglev system with time-delayed feedback. Using linear analysis, we determine constraints on the feedback control gains and the time delay which ensure stability of the maglev system. We then show that a Hopf bifurcation occurs at the linear stability boundary. To gain insight into the periodic motion which arises from the Hopf bifurcation, we use the method of multiple scales on the nonlinear model. This analysis shows that for practical operating ranges, the maglev system undergoes both subcritical and supercritical bifurcations, which give rise to unstable and stable limit cycles respectively. Numerical simulations confirm the theoretical results and indicate that unstable limit cycles may coexist with the stable equilibrium state. This means that large enough perturbations may cause instability in the system even if the feedback gains are such that the linear theory predicts that the equilibrium state is stable. 相似文献
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In this paper, the dynamical behaviour of a linear impulsive system is discussed both theoretically and numerically. The existence and the stability of period-one solution are discussed by using a discrete map. The conditions of existence for flip bifurcation are derived by using the centre manifold theorem and bifurcation theorem. The bifurcation analysis shows that chaotic solutions appear via a cascade of period-doubling in some interval of parameters. Moreover, the periodic solutions, the bifurcation diagram, and the chaotic attractor, which show their consistence with the theoretical analyses, are given in an example. 相似文献
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M. Grmela 《Journal of statistical physics》1971,3(3):347-364
An exact mathematical discussion of the linearized Enskog-Vlasov equation is given. A criterion for the occurrence of the linear instability is related to a criterion for the occurrence of the bifurcation of the equilibrium stationary solution to the nonlinear Enskog-Vlasov equation. Mathematical results are interpreted physically in connection with phase transitions. 相似文献
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Effect of metal oxide arrester on the chaotic oscillations in the voltage transformer with nonlinear core loss model using chaos theory 下载免费PDF全文
In this paper, controlling chaos when chaotic ferroresonant oscillations occur in a voltage transformer with nonlinear core loss model is performed. The effect of a parallel metal oxide surge arrester on the ferroresonance oscillations of voltage transformers is studied. The metal oxide arrester(MOA) is found to be effective in reducing ferroresonance chaotic oscillations. Also the multiple scales method is used to analyze the chaotic behavior and different types of fixed points in ferroresonance of voltage transformers considering core loss. This phenomenon has nonlinear chaotic dynamics and includes sub-harmonic, quasi-periodic, and also chaotic oscillations. In this paper, the chaotic behavior and various ferroresonant oscillation modes of the voltage transformer is studied. This phenomenon consists of different types of bifurcations such as period doubling bifurcation(PDB), saddle node bifurcation(SNB), Hopf bifurcation(HB), and chaos. The dynamic analysis of ferroresonant circuit is based on bifurcation theory. The bifurcation and phase plane diagrams are illustrated using a continuous method and linear and nonlinear models of core loss. To analyze ferroresonance phenomenon, the Lyapunov exponents are calculated via the multiple scales method to obtain Feigenbaum numbers. The bifurcation diagrams illustrate the variation of the control parameter. Therefore, the chaos is created and increased in the system. 相似文献
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J.C. JI 《Journal of sound and vibration》2003,259(4):845-856
The effect of time delays occurring in the feedback control loop on the linear stability of a simple magnetic bearing system is investigated by analyzing the associated characteristic transcendental equation. It is found that a Hopf bifurcation can take place when time delays pass certain values. The direction and stability of the Hopf bifurcation are determined by constructing a center manifold and by applying the normal form method. It is also found that a codimension two bifurcation can occur through a Hopf and a steady state bifurcation interaction. Finally, numerical simulations are performed to verify the analytical predictions. 相似文献
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通过线性与非线性状态反馈, 实现了对四维Qi系统零平衡点的Hopf分岔反控制.首先确定产生Hopf分岔的线性控制项,得到线性控制增益的选取原则.然后,利用稳定性分析,借助于对线性受控Qi系统的Jordan标准型的直接控制以及适当的变换,确定影响Hopf分岔稳定性的非线性控制项,得到非线性控制增益的选取原则.针对所考虑分岔参数的不同,给出不同的控制方案.最后通过数值模拟验证了理论分析结果的正确性.
关键词:
Qi系统
Hopf分岔
反控制
稳定性 相似文献
20.
Matthew O. Williams Jon WilkeningEli Shlizerman J. Nathan Kutz 《Physica D: Nonlinear Phenomena》2011,240(22):1791-1804
We apply the adjoint continuation method to construct highly accurate, periodic solutions that are observed to play a critical role in the multi-pulsing transition of mode-locked laser cavities. The method allows for the construction of solution branches and the identification of their bifurcation structure. Supplementing the adjoint continuation method with a computation of the Floquet multipliers allows for explicit determination of the stability of each branch. This method reveals that, when gain is increased, the multi-pulsing transition starts with a Hopf bifurcation, followed by a period-doubling bifurcation, and a saddle-node bifurcation for limit cycles. Finally, the system exhibits chaotic dynamics and transitions to the double-pulse solutions. Although this method is applied specifically to the waveguide array mode-locking model, the multi-pulsing transition is conjectured to be ubiquitous and these results agree with experimental and computational results from other models. 相似文献