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A model of three bidirectionally coupled phase oscillators in a ring is studied at the transition to a complete frequency synchronization. Analytic expressions for the critical coupling strengths, at which oscillators synchronize to a common frequency, are obtained. These expressions are determined for cases when the initial oscillators' frequencies are arranged arbitrarily or they are assigned according to fixed separations. Three unidirectionally coupled phase oscillators are synchronized in an analogous manner to the bidirectional system. This finding allows us to find an analytic equation for the critical coupling strength in the case of the model of the unidirectionally coupled phase oscillators. The bifurcation diagrams show excellent agreement between the analytic formulas and the numerical solutions of the differential equations that describe the models. Key words: frequency synchronization, Kuramoto model, coupled phase oscillators. Nous etudions un modele de trois oscillateurs bidirectionnels couples en phase dans un anneau a la transition vers une complete synchronisation en frequence. Il nous permet d'obtenir des expressions analytiques pour les forces de couplage critiques, auxquelles les oscillateurs se synchronisent a la meme frequence. Ces expressions sont determinees pour les cas ou les frequences initiales sont arbitraires ou avec des separations fixes. De la meme facon, trois oscillateurs unidirectionnels couples en phase sont egalement synchronises. Ce resultat permet de trouver une equation analytique pour les forces de couplage critiques dans le cas du modele des oscillateurs unidirectionnels couples en phase. Les diagrammes de bifurcation montrent un excellent accord entre les formules analytiques et les solutions numeriques des equations differentielles qui decrivent les modeles. [Traduit par la Redaction] Mots-cles: synchronisation en frequence, modele de Kuramoto, oscillateurs couples, dynamique non lineaire, methodes mathematiques en physique., 1. Introduction Coupled oscillators are used to understand the behavior of systems of interacting elements in physics, chemistry, biology, social relations, engineering, and technology [1-5]. Although all these systems have [...]

In this technical note we study the problem of exponential synchronization for one of the most popular models of coupled phase oscillators, the Kuramoto model. We consider the special case of finite oscillators with distinct, bounded natural frequencies. Our first result derives a lower bound on the coupling gain which is necessary for the onset of synchronization. This bound improves the one derived by Jadbabaie et al. [8]. We then calculate a lower bound on the coupling gain that is sufficient to guarantee oscillator synchronization and derive further sufficient conditions to ensure exponential synchronization of the angular frequencies of all oscillators to the mean natural frequency of the group. We also characterize the coupling gain that is sufficient for the oscillator phase differences to approach any desired compact set in finite time. Index Terms--Cooperative control, exponential stabilization, Kuramoto oscillators, multi-agent systems, synchronization.

Existing algorithms for computing the nth-order frequency response functions of the Duffing oscillator have helped promote frequency domain analysis of nonlinear systems but still have a number of practical difficulties. A very efficient algorithm to enable the nth-order symmetric generalized frequency response functions (GFRFs) to be written down directly in terms of the coefficients of the generalized higher order Duffing oscillator model is developed in this paper. The analytical expression derived in this paper shows that the key procedure in the determination of the higher order GFRFs is the computation of the Stirling set of the second kind. This enables the structure of the higher order GFRFs to be seen more clearly. Index Terms--Higher order frequency response functions, nonlinear systems, Volterra series.

A nonlinear perturbation model of a complementary LC-tuned voltage-controlled oscillator is derived, which consists of two mutually-coupled second-order differential equations. The first-order approximate periodic solution of the describing equations is found, obtaining closed-fore expressions for both the amplitude and the harmonics of oscillation, as well as for the correction of the oscillation frequency due to the nonlinear effect of varactors. This allows us to find the tuning curve in explicit form. The accuracy of presented formulas was validated by circuit simulations. Index Terms--Analog integrated circuits, nonlinear analysis, oscillators, sensitivity analysis, tuning curve, varactors, voltage-controlled oscillators (VCOs).

This article describes how to use smooth solvers for simulation of a class of piecewise smooth systems of ordinary differential equations, called Filippov systems, with discontinuous vector fields. In these systems constrained motion along a discontinuity surface (so-called sliding) is possible and requires special treatment numerically. The introduced algorithms are based on an extension to Filippov's method to stabilise the sliding flow together with accurate detection of the entrance and exit of sliding regions. The methods are implemented in a general way in MATLAB and sufficient details are given to enable users to modify the code to run on arbitrary examples. Here, the method is used to compute the dynamics of three example systems, a dry-friction oscillator, a relay feedback system and a model of an oil well drill-string. Categories and Subject Descriptors: G.1.7 [Numerical Analysis]: Ordinary Differential Equations Initial value problems; G.1.0 [Numerical Analysis]: General--Numerical algorithms; J.2] Computer Applications]: Physical Sciences and Engineering General Terms: Algorithms Additional Key Words and Phrases: Numerical simulation, piecewise-smooth differential equations, Filippov systems, sliding solutions ACM Reference Format: Piiroinen, P. T. and Kuznetsov, Y. A. 2008. An event-driven method to simulate Filippov systems with accurate computing of sliding motions. ACM Trans. Math. Softw., 34, 3, Article 13 (May 2008), 24 pages. DOI = 10.1145/1356052.1356054 http://doi.acm.org/10.1145/1356052.1356054

A model is presented for an injection-locked oscillator. It is derived from a perturbation of the free-running oscillator steady state, which is accurately determined from a harmonic balance (HB) simulation. As a first application, the proposed model is used to trace the existing solutions of the injection-locked system and their stability. It is then extended to include the effect of noise on the stability. Finally, the injection-locked system performance is studied under phase modulation. Sinusoidal phase modulation is considered and the stability of the modulated solutions is examined by Floquet theory. The validity of the model is verified for a 4.422-GHz oscillator, by HB and envelope transient simulations, as well as measurements. Index Terms--Angle modulation, envelope transient (ET), harmonic balance (HB), injection-locked oscillator, stochastic differential equations.

A novel Lorenz-type system of nonlinear differential equations is proposed. Unlike the original Lorenz system, where the chaotic dynamics remain confined to the positive half-space with respect to the Z state variable due to a limiting threshold effect, the proposed system enables bipolar swing of this state variable. In addition, the classical set of parameters (a, b, c) controlling the behavior of the Lorenz system are reduced to a single parameter, namely a. Two possible modes of operation are admitted by the system; switching between these two modes results in the creation of a complex butterfly chaotic attractor. Numerical simulations and results from an experimental setup are presented. Index Terms--Chaos, chaotic oscillators, Lorenz system.

The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) 'channeling' of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system. [DOI: 10.1115/1.1345524]

We study energy pumping in an impulsively excited, two-degrees-of-freedom damped system with essential (nonlinearizable) nonlinearities by means of two analytical techniques. First, we transform the equations of motion using the action-angle variables of the underlying Hamiltonian system and bring them into the form where two-frequency averaging can be applied. We then show that energy pumping is due to resonance capture in the 1:1 resonance manifold of the system, and perform a perturbation analysis in an 0 ([square root of ([epsilon])) neighborhood of this manifold in order to study the attracting region responsible for the resonance capture. The second method is based on the assumption of 1:1 internal resonance in the fast dynamics of the system, and utilizes complexification and averaging to develop analytical approximations to the nonlinear transient responses of the system in the energy pumping regime. The results compare favorably to numerical simulations. The practical implications of the energy pumping phenomenon are discussed. [DOI: 10.1115/1.1345525]

The time-domain modeling of the operation of a 94-GHz pulsed silicon IMPATT oscillator, based on a physical approach, is described in this paper. It relies on the coupling of electrical, thermal, electromagnetic, and carrier transport physical models. The model has been used to study the high-power stable operation of a 94-GHz oscillator. The results of a comparison between simulations, using two different types of passive radio-frequency load circuits, including experimental measurements, are presented and discussed. They tend to demonstrate that it is now possible to develop accurate millimeter-wave-circuit predictive models even for application based on a nonlinear thermal and electrical transient operation such as IMPATT oscillators. Index Terms - Millimeter-wave pulsed IMPATT oscillator, time-domain global modeling.

A novel approach for determining the modal power distribution in a stable multimode laser cavity is used to analyze a class of axially symmetric flat-topped transverse intensity laser beams. The algorithm yields closed-form expressions for the modal weights and for the M(super 2) factor of the radiated beam and eliminates the need for coherence measurements. It is shown that the recovering algorithm for the mode weights produces satisfactory results, which appear to be insensitive to fluctuations of the intensity values.

The nonlinear distortion of frequency or phase modulation systems composed by a voltage-controlled oscillator (VCO) modulator and a phase-locked loop (PLL) discriminator is addressed. Volterra series nonlinear transfer functions up to third order, for an LC Colpitts type modulator and a PLL with simultaneous nonlinear phase-detector and VCO, are combined as a cascade of mildly nonlinear systems. This approach gives a fully analytical description of that complex arrangement, which allowed the derivation of theoretical linearization conditions for that FM/PM modem. Index Terms - Frequency modulation, intermodulation distortion, phase-locked loops, Volterra series.

This letter proposes a timing recovery scheme for transmission systems with baud-rate sampling allowing the use of an all digital phase-locked loop. The impact of the numerically controlled oscillator on the attainable signal-to-noise ratio and jitter under conditions of the European Telecommunications Standardization Institute (ETSI) HDSL standard is discussed. Index Terms - Digital communication, synchronization, timing, timing jitter.

Byline: David Taylor (1), Philip Holmes (2) Keywords: Key words:aBifurcation; Coupling; Oscillator; Network Abstract: aChains of coupled oscillators of simple 'rotator' type have been used to model the central pattern generator (CPG) for locomotion in lamprey, among numerous applications in biology and elsewhere. In this paper, motivated by experiments on lamprey CPG with brainstem attached, we investigate a simple oscillator model with internal structure which captures both excitable and bursting dynamics. This model, and that for the coupling functions, is inspired by the Hodgkin--Huxley equations and two-variable simplifications thereof. We analyse pairs of coupled oscillators with both excitatory and inhibitory coupling. We also study traveling wave patterns arising from chains of oscillators, including simulations of 'body shapes' generated by a double chain of oscillators providing input to a kinematic musculature model of lamprey.. Author Affiliation: (1) Iomega Corporation, 800 Tasman Drive, Milpitas, CA 95035, USA, US (2) Program in Applied and Computational Mathematics and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544-5263, USA, US Article note: Received: 25 November 1996 / Revised version: 9 December 1997

A new type of 94-GHz pulsed silicon impact avalanche and transit time (IMPATT) oscillator numerical modeling is described. It consists of a set of three models of increasing complexity; namely, a pure sine model, time-domain isothermal model, and time-domain electro-thermal model, which basically rely on a diode one-dimensional bipolar drift-diffusion model embedded in a time-domain circuit modeling. In this paper, they are first used to investigate the 94-GHz diode intrinsic operation and performance. Secondly, the load-impedance level has been optimized. In each case, the thermal behavior is especially considered. Thirdly, pulse-operation-simulation results are compared with experiments performed at Thomson CSF, Radars et Contre-Mesures, Elancourt, France. Finally, some improvements of the present modeling are discussed in Section VI. Index Terms - Macroscopic modeling, millimeter waves, pulsed IMPATT oscillator.

A nonlinear model of the ring oscillator is proposed. The oscillations are related to the irregularity or the kink-formed solitary wave, that is, an elementary phenomenon in a nonlinear system. According to the proposed model, we found that the higher harmonic generation cannot exist when the total number of stages is five or less. Index Terms - MOSFET oscillators, nonlinear wave propagation, very large scale integration.

The modified Bethe theory is applied to the waveguide-to-resonator transition, which is used for output coupling in an orotron oscillator designed for operation at 94 GHz. The finite thickness of the coupling hole between output waveguide and resonator is taken into account. The approach results in an equivalent circuit, which describes the reflection-type resonator, with respect to the output waveguide. Its equivalent circuit parameters are given in closed-form expressions. The calculated reflection coefficients are compared to experimental results. The assumed single-mode operation in the resonator is demonstrated by field-profile measurements. Index Terms: Electromagnetic coupling, optical resonator, Q measurements.

Previous investigations of noise in mutually synchronized coupled-oscillator systems are extended to include the effects of phase noise introduced by externally injected signals. The analysis is developed for arbitrarily coupled arrays and an arbitrary collection of coherent injected signals, and is illustrated with the specific case of linear chains of nearest neighbor coupled oscillators either globally locked (locking signal applied to each array element) or with the locking signal applied to a single-array element. It is shown that the general behavior is qualitatively similar to a single injection-locked oscillator, with the output noise tracking the injected noise near the carrier, and returning to the free-running array noise far from the carrier, with intermediate behavior significantly influenced by the number of array elements and injection strength. The theory is validated using a five-element GaAs MESFET oscillator array operating at X-band. Index Terms: Array noise, globally locked, injection locked, injection strength, noise admittance, noise offset frequency, phase noise, power spectral density, single element locked, synchronization.

In this letter, we show that the two-region third-order piecewise-linear dynamics of the chaotic Colpitts oscillator may be mapped to a Chua's oscillator with an asymmetric nonlinearity.

The topological resemblance of the Colpitts oscillator to that of Chua's circuit suggests that the two circuits may also be similar in their dynamic behavior. The similarity in the dynamics is demonstrated in the present communication by resorting to computer simulation. We deal in particular with the unknown chaotic behavior of the Colpitts oscillator. It is shown by referring to recently published articles that the Colpitts oscillator and Chua's circuit are in fact topologically conjugate one to another.

Voltage controlled oscillator model neurons (VCONs) are integrated circuits similar to phase locked loops that have been modified to simulate neuro-physiological phase relationships. VCON networks facilitate computer experiments on and modeling of neural networks. Aside from their use in mathematics, the models allow better understanding of physiological networks and have been used in neuro-biology experiments and research.

We propose a new tunable linear laser oscillator-amplifier system that incorporates a Fizeau wedge with unequal mirrors as the coupler. Because of asymmetry in reflection and wedge focusing in unequal mirrors, our system acts as a highly efficient selector in the oscillator and practically eliminates reinjection of spontaneous emission in the amplifier. We studied the wedge action by developing a procedure for the calculation of reflected interference patterns at a distance from the wedge. Using this procedure we found the feedback factors for both the oscillator and the amplifier. Background emission levels of less than 1% of the peak intensity in the emitted line were experimentally obtained. OCIS codes: 140.3280, 140.4780, 260.3160, 100.2650.

This paper studies with analog simulators, the dynamics of a system consisting of a van der Pol oscillator coupled to a Duffing oscillator. Amplitude-response curves are obtained in the case of internal resonance. The jump or hysteresis phenomenon is found. Various bifurcafions are observed, and it is found that chaos can appear suddenly, through period-adding or through torus breakdown. The experimental results are compared with the numerical solutions. Index Terms--Analog simulation, coupled nonlinear oscillators, chaos.

This paper deals with the problem of modeling of phase noise in OFDM systems and the impact it may have on the bit error rate performance of such systems subject to a number of system variables and to a number of channel conditions which may be encountered when such systems are deployed for certain applications such as high speed wireless LANs and Digital Television Terrestrial Broadcasting (DTTB). The phase noise processes, the sources of which are the transmitter's and receiver's local oscillator, are modeled using what is believed to be commercially realizable phase noise masks. Such masks represent the long-time averaged .power spectral densities of the local oscillator output signal. Index Terms--COFDM, digital television, DTTB, MQAM, OFDM, phase noise.

A 900-MHz two-stage CMOS voltage-controlled ring oscillator (VCO) with good phase-noise performance is presented. Implemented in a 0.5-/[micro]m CMOS technology and at 2.5-V supply voltage, the VCO has a wide operating frequency range from 661.5 MHz to 1.27 GHz with a peak VCO gain ([K.sub.vco]) of --630 MHz/V. At 900 MHz, the phase noise of the VCO is --105.5 dBc/Hz at 600-kHz frequency offset with low power consumption of 15.4 mW. The gain and phase mismatches are less than 0.25 dB and 0.5 [degrees], respectively, which corresponds to an image rejection of better than 31 dB. The chip area is only 125 X 102 [micro][m.sup.2]. Index Terms--Image rejection, oscillators, phase-locked loop, phase noise, ring oscillators, voltage-controlled oscillators.

In this paper, a new method for autotuning of a phase-lead controller for nonself-regulating or integrating systems is proposed. By connecting the system with a suitable high-order derivative filter and a relay with hysteresis, the frequency data of the system can be obtained and phase-lead controllers can be designed based on some phase margin specifications. Simulations and experimental studies will be included in the paper to illustrate the performance of the proposed algorithm. Index Terms--Autotuning, integrating systems, phase-lead controller, relay oscillation.

This paper investigates the effect of element value deviation on the degenerate mode in rings of three and four coupled van der Pol oscillators. By using the averaging method, we elucidate that the element value deviation has different effects on the rings of three and four coupled oscillators. Namely, for the ring of three coupled oscillators, two single modes which bifurcate from the twofold degenerate mode in the symmetric system can be synchronized when they are close, but they lose synchronization and a quasiperiodic oscillation appears when they are separated to some extent. In contrast, for the ring of four coupled oscillators, the two single modes cannot be synchronized, even if they are close. However, in some rare cases they can be synchronized when they are separated to some extent. We confirm these results of the averaging method by both computer simulation and experiments. Index Terms - Asymmetric system, averaging method, degenerate modes, element value deviation, rings of coupled oscillators.

The steady-state oscillation of a zero-voltage-switching clamped-voltage self-oscillating ballast driven by a saturable transformer is analyzed and simulated. Its self-oscillating operation is divided into six stages according to the hysteresis B-H loop of saturable transformer cores. Stage-wise circuit analysis shows the saturable transformer limits the lamp current and dominates the switching frequency of the ballast. The saturating behavior of driving and switching devices clearly distinguishes this self-oscillating inverter from an external-drive high-quality-factor resonant inverter. Analytical results are verified by mathematical simulation and laboratory experiment. Index Terms - Bipolar transistor oscillators, circuit analysis, circuit simulation, resonant power conversion, RLC circuits.

An analytical model for estimating the gain and performance of CW periodically poled LiNbO3 optical parametric oscillators (PPLN OPO) with strong idler absorption, strong pump depletion and thermally induced dephasing was developed. The proposed model also shows the pump and signal beam focusing effects for several singly resonant CW PPLN OPO devices through its incorporation of the Guha h(sub sm) parameters.