Analysis and synthesis of perturbed Duffing oscillators

Analysis and synthesis of perturbed Duffing oscillators have been presented. The oscillations in such systems are regarded as limit cycles in perturbed Hamiltonian systems under polynomial perturbations of sixth degree and are analysed by using the Melnikov function. It has been proved that there exists a polynomial perturbation depending on the zeros of the Melnikov function so that the system considered can have either two simple limit cycles, or one limit cycle of multiplicity 2, or one simple limit cycle. A synthesis of such oscillators based on the Melnikov's theory has been proposed. Copyright © 2006 John Wiley & Sons, Ltd.     

Exact solutions of cyclically symmetric oscillator equations with non-linear coupling

Kaplan and Yardeni have found very simple exact limit cycle solutions in cyclically symmetric systems of N oscillator equations with linear coupling in zero order of a perturbation parameter and non-linear coupling in first order. In contrast with such solutions in other non-linear systems, each of these limit cycles is a normal mode of the unperturbed equations, with no change in frequency. the sources of this simple behaviour are studied here with the equations expressed in terms of the normal mode co-ordinates of the unperturbed system rather than in the original co-ordinates. It is found that the simplicity of the Kaplan-Yardeni solutions arises partly from an additional symmetry of the perturbation terms beyond the cyclic symmetry and partly from the specific choices of the perturbations. Extension of their systems to arbitrary N leads to the result that all such sets of equations have similar simple limit cycles. More general cyclically symmetric sets of equations are also discussed, with limit cycle solutions whose frequencies are shifted from the zero-order values by easily calculated amounts or with solutions which are linear combinations of zero-order normal modes.     

Exact solutions of cyclically symmetric oscillator equations with non-linear coupling. Part II: Coupling with phase shift

Several authors have found very simple exact limit cycle solutions in cyclically symmetric systems of N oscillator equations with linear coupling in zero order of a perturbation parameter and non-linear coupling in first order. In contrast with such solutions in most other non-linear systems, each of these limit cycles is an exact normal mode of the unperturbed equations with no change in frequency or addition of higher harmonics. In Part I of this paper it was shown that the construction and analysis of such systems of equations are substantially simplified if the equations are expressed in terms of the normal mode co-ordinates of the unperturbed system. the effects of the cyclic symmetry, as well as those of a higher symmetry shared by previous authors' models, were studied. It is shown here that similar results can be obtained in systems if the coupling involves a phase shift. the phase shift places added conditions on the systems, so that some sets of equations, shown to have the simple limit cycle solutions, no longer have them after shift is introduced. the methods of the earlier paper, however, can be used to find families of systems with phase shifts which have such solutions. A result in Part I, that frequencies in a system with the higher symmetry mentioned above are unchanged from those of the unperturbed system, is not valid if phase shifts are introduced.     

Fully nonlinear oscillator noise analysis: an oscillator with no asymptotic phase

Oscillators exist in many systems. Detailed and correct characterization and comprehension of noise in autonomous systems such as oscillators is of utmost importance. Previous approaches to oscillator noise analysis are based on some kind of perturbation analysis, some linear and some nonlinear. However, the derivations of the equations for perturbation analysis are all based on information that is produced by a linearization of the oscillator equations around the periodic steady‐state solution, where it is assumed that the oscillator is orbitally stable and it has the so‐called asymptotic phase property. In this paper, we first discuss these notions from a qualitative perspective, and demonstrate that the asymptotic phase property is crucial in validating all of the previous approaches. We then present the case of a simple oscillator that is orbitally stable but without asymptotic phase, for which previous approaches fail. We then present a fully nonlinear noise analysis of this oscillator. We derive and compute nonlinear, non‐stationary and non‐Gaussian stochastic characterizations for both amplitude and phase noise. We arrive at results that are distinctly different when compared with the ones obtained previously for oscillators with asymptotic phase. We compare and verify our analytical results against extensive Monte Carlo simulations. Copyright © 2006 John Wiley & Sons, Ltd.     

A model for numerical evaluation of the global stability region in power systems determined by a nonlinear limit cycle

A power system is a large dynamic system, which includes many nonlinear elements. According to the nonlinear analyses using Hopf bifurcation theory, it can be detected that a limit cycle exists around an operating point, which may affect the global stability of a power system significantly. The authors have presented a numerical method to analyze the nonlinear characteristics in power systems by observing the power swing after some perturbation where the coefficients of nonlinear terms are determined by the least squares method. In this paper the method is modified for the application to a longitudinally interconnected power system including an excitation system, and the influence of the excitation voltage limiter on the nonlinear phenomena of the whole power system can be detected by some numerical analyses. © 2003 Wiley Periodicals, Inc. Electr Eng Jpn, 144(3): 17–27, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.10193     

基于信号预处理的杜芬振子弱信号检测方法

杜芬振子对周期信号极其敏感,对噪声具有较强免疫力,被研究用于强噪声背景下弱周期信号的检测及其各项参数的测量。为进一步提高杜芬振子检测含噪信号的性能,本文提出了先对待测信号预处理再送入杜芬振子的弱信号检测算法,并在理论上进行了证明。该方法先将待测信号分割、叠加、延拓再送入杜芬振子进行检测,仿真表明该方法在将信号分割成多段、叠加、延拓时可获得较大的信噪比增益,其中分割数目为3时,检测信噪比下限降低约为2 dB,分割数目为5时,检测信噪比下限降低约为3 dB。     

基于图像识别理论的混沌特性判别方法

利用混沌振子可以检测微弱的周期信号，并且具有很多优点。该文针对Duffing混沌振子在信号检测领域中的应用，提出了基于图像识别理论的混沌特性判别方法。该方法具有直观、算法简单及计算量小的优点。文中阐述了所提方法的原理，并给出了相应的判别程序流程图及仿真实验结果。仿真结果表明，利用该方法可以准确快速地判断混沌特性，并能很好地满足Duffing混沌振子信号检测的需要。将该文提出的判别方法用于处理小电流单相接地故障保护的现场试验数据，获得了好的效果。     

A frequency‐domain approach to the analysis of stability and bifurcations in nonlinear systems described by differential‐algebraic equations

A general numerical technique is proposed for the assessment of the stability of periodic solutions and the determination of bifurcations for limit cycles in autonomous nonlinear systems represented by ordinary differential equations in the differential‐algebraic form. The method is based on the harmonic balance (HB) technique, and exploits the same Jacobian matrix of the nonlinear system used in the Newton iterative numerical solution of the HB equations for the determination of the periodic steady state. To demonstrate the approach, it is applied to the determination of the bifurcation curves in the parameters' space of Chua's circuit with cubic nonlinearity, and to the study of the dynamics of the limit cycle of a Colpitts oscillator. Copyright © 2007 John Wiley & Sons, Ltd.     

Piecewise-linear analysis of the Wien Bridge Oscillator

This paper presents a quantitative analysis of the Wien bridge oscillator based on a piecewise-linear approximation of the op-amp characteristic. Usually the circuit parameters are chosen in such a way that the linearized circuit has a pair of complex conjugate natural frequencies which lie slightly to the right of the jw-axis. the frequency of oscillation is assumed to coincide with the value jω₀ at which the poles cross the jω-axis. However, this simplified approach is unable to predict the true frequency of oscillation and the amount of distortion. the aim of the present analysis is to determine the exact waveform, frequency and spectral content corresponding to the steady state behaviour. the phase plane is partitioned into three regions, in each of which the vector field is linear. the limit cycle corresponding to the steady state oscillation is reconstructed by using the Poincaré map method. Expressions for the frequency of oscillation and the Fourier coefficients are given which depend on a critical parameter representing the voltage gain of the amplifier.     

Solutions for a three-dimensional non-linear stabilized oscillator

A system of coupled non-linear equations, describing a three-phase stabilized oscillator, is analysed by introducing ‘cyclotomic’ co-ordinates. We show that this system, under certain conditions, approaches asymptotically non-conservative linear systems; and yet it does have stabilized solutions (limit cycles). The non-linear system is solved analytically for an important class of stabilizing functions. We show that the frequency ω of our oscillator responds instantaneously to changes of certain parameters. This result has useful applications in building quickly responding novel electronic voltage-controlled oscillators.