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Journal Article•DOI•

A Study of Locking Phenomena in Oscillators

R. Adler
01 Jun 1946-Vol. 34, Iss: 6, pp 351-357
TL;DR: In this paper, a differential equation is derived which gives the oscillator phase as a function of time, and with the aid of this equation, the transient process of "pull-in" as well as the production of distorted beat note are described in detail.
Abstract: Impression of an external signal upon an oscillator of similar fundamental frequency affects both the instantaneous amplitude and instantaneous frequency. Using the assumption that time constants in the oscillator circuit are small compared to the length of one beat cycle, a differential equation is derived which gives the oscillator phase as a function of time. With the aid of this equation, the transient process of "pull-in" as well as the production of a distorted beat note are described in detail. It is shown that the same equation serves to describe the motion of a pendulum suspended in a viscous fluid inside a rotating container. The whole range of locking phenomena is illustrated with the aid of this simple mechanical model.
Citations
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Journal Article•DOI•
A study of injection locking and pulling in oscillators

[...]

Behzad Razavi1•
University of California, Los Angeles1
30 Aug 2004-IEEE Journal of Solid-state Circuits
TL;DR: In this paper, an identity obtained from phase and envelope equations is used to express the requisite oscillator nonlinearity and interpret phase noise reduction, and the behavior of phase-locked oscillators under injection pulling is also formulated.
Abstract: Injection locking characteristics of oscillators are derived and a graphical analysis is presented that describes injection pulling in time and frequency domains. An identity obtained from phase and envelope equations is used to express the requisite oscillator nonlinearity and interpret phase noise reduction. The behavior of phase-locked oscillators under injection pulling is also formulated.

1,159 citations

Journal Article•DOI•
Synchronization in complex networks of phase oscillators: A survey

[...]

Florian Dörfler1, Francesco Bullo2•
ETH Zurich1, University of California, Santa Barbara2
01 Jun 2014-Automatica
TL;DR: This survey reviews the vast literature on the theory and the applications of complex oscillator networks, focusing on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology.

1,021 citations

Cites background from "A Study of Locking Phenomena in Osc..."

  • ...the work by Huygens (1893) on ‘‘an odd kind of sympathy’’ between coupled pendulum clocks, mutual influence of organ pipes (Rayleigh, 1896), locking phenomena in circuits and radio technology (Adler, 1946; Appleton, 1922; Van Der Pol, 1927), the analysis of brainwaves and self-organizing systems (Wiener, 1948, 1958), and it still fascinates the scientific community nowadays (Strogatz, 2003; Winfree, 2001). We refer to Blekhman (1988) and Pikovsky, Rosenblum, and Kurths (2003) for a detailed historical account of synchronization studies. A variation of the considered coupled oscillator model (1) was first proposed by Winfree (1967). Winfree considered general (not necessarily sinusoidal) interactions among the oscillators....

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  • ...the work by Huygens (1893) on ‘‘an odd kind of sympathy’’ between coupled pendulum clocks, mutual influence of organ pipes (Rayleigh, 1896), locking phenomena in circuits and radio technology (Adler, 1946; Appleton, 1922; Van Der Pol, 1927), the analysis of brainwaves and self-organizing systems (Wiener, 1948, 1958), and it still fascinates the scientific community nowadays (Strogatz, 2003; Winfree, 2001). We refer to Blekhman (1988) and Pikovsky, Rosenblum, and Kurths (2003) for a detailed historical account of synchronization studies....

    [...]

  • ...the work by Huygens (1893) on ‘‘an odd kind of sympathy’’ between coupled pendulum clocks, mutual influence of organ pipes (Rayleigh, 1896), locking phenomena in circuits and radio technology (Adler, 1946; Appleton, 1922; Van Der Pol, 1927), the analysis of brainwaves and self-organizing systems (Wiener, 1948, 1958), and it still fascinates the scientific community nowadays (Strogatz, 2003; Winfree, 2001). We refer to Blekhman (1988) and Pikovsky, Rosenblum, and Kurths (2003) for a detailed historical account of synchronization studies. A variation of the considered coupled oscillator model (1) was first proposed by Winfree (1967). Winfree considered general (not necessarily sinusoidal) interactions among the oscillators. He discovered a phase transition from incoherent behavior with dispersed phases to synchrony with aligned frequencies and coherent (i.e., nearby) phases. Winfree found that this phase transition depends on the trade-off between the heterogeneity of the oscillator population and the strength of the mutual coupling, which he could formulate by parametric thresholds. However, Winfree’s model was too general to be analytically tractable. Inspired by these works, Kuramoto (1975) simplified Winfree’s model and arrived at the coupled oscillator dynamics (1) with a complete interaction graph and uniform weights aij = K/n:...

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  • ...the work by Huygens (1893) on ‘‘an odd kind of sympathy’’ between coupled pendulum clocks, mutual influence of organ pipes (Rayleigh, 1896), locking phenomena in circuits and radio technology (Adler, 1946; Appleton, 1922; Van Der Pol, 1927), the analysis of brainwaves and self-organizing systems (Wiener, 1948, 1958), and it still fascinates the scientific community nowadays (Strogatz, 2003; Winfree, 2001)....

    [...]

  • ...…of sympathy’’ between coupled pendulum clocks, mutual influence of organ pipes (Rayleigh, 1896), locking phenomena in circuits and radio technology (Adler, 1946; Appleton, 1922; Van Der Pol, 1927), the analysis of brainwaves and self-organizing systems (Wiener, 1948, 1958), and it still…...

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Journal Article•DOI•
Observation of discrete time-crystalline order in a disordered dipolar many-body system

[...]

Soonwon Choi1, Joonhee Choi1, Renate Landig1, Georg Kucsko1, Hengyun Zhou1, Junichi Isoya2, Fedor Jelezko3, Shinobu Onoda, Hitoshi Sumiya4, Vedika Khemani1, Curt von Keyserlingk5, Norman Y. Yao6, Eugene Demler1, Mikhail D. Lukin1 •
Harvard University1, University of Tsukuba2, University of Ulm3, Sumitomo Electric Industries4, Princeton University5, University of California, Berkeley6
08 Mar 2017-Nature
TL;DR: This work observes long-lived temporal correlations, experimentally identifies the phase boundary and finds that the temporal order is protected by strong interactions, which opens the door to exploring dynamical phases of matter and controlling interacting, disordered many-body systems.
Abstract: Understanding quantum dynamics away from equilibrium is an outstanding challenge in the modern physical sciences Out-of-equilibrium systems can display a rich variety of phenomena, including self-organized synchronization and dynamical phase transitions More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter; for example, the interplay between periodic driving, disorder and strong interactions has been predicted to result in exotic 'time-crystalline' phases, in which a system exhibits temporal correlations at integer multiples of the fundamental driving period, breaking the discrete time-translational symmetry of the underlying drive Here we report the experimental observation of such discrete time-crystalline order in a driven, disordered ensemble of about one million dipolar spin impurities in diamond at room temperature We observe long-lived temporal correlations, experimentally identify the phase boundary and find that the temporal order is protected by strong interactions This order is remarkably stable to perturbations, even in the presence of slow thermalization Our work opens the door to exploring dynamical phases of matter and controlling interacting, disordered many-body systems

760 citations

Cites background from "A Study of Locking Phenomena in Osc..."

  • ...First, the linewidth Γ (equation (1)) of the subharmonic peak should be quadratically sensitive to the deviation of θ from π ....

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  • ...Second, according to the dephasing model (equation (1)), the lifetime of the 3T-periodic DTC order is expected to be longer than that of the 2T-periodic DTC order owing to enhanced dephasing (from a lack of spin-locking) in the bare basis28....

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Journal Article•DOI•
Nonlinear Auto-Oscillator Theory of Microwave Generation by Spin-Polarized Current

[...]

Andrei Slavin1, Vasyl S. Tiberkevich1•
University of Rochester1
16 Mar 2009-IEEE Transactions on Magnetics
TL;DR: In this paper, a general analytic approach to the theory of microwave generation in magnetic nano-structures driven by spin-polarized current was proposed. But the proposed approach is based on the universal model of an auto-oscillator with negative damping and nonlinear frequency shift.
Abstract: This paper formulates a general analytic approach to the theory of microwave generation in magnetic nano-structures driven by spin-polarized current and reviews analytic results obtained in this theory. The proposed approach is based on the universal model of an auto-oscillator with negative damping and nonlinear frequency shift. It is demonstrated that this universal model, when applied to the case of a spin-torque oscillator (STO) based on a current-driven magnetic nano-pillar or nano-contact, gives adequate description of most of the experimentally observed properties of STO. In particular, the model describes the power and frequency of the generated microwave signal as functions of the bias current and magnetic field, predicts the magnitude and properties of the generation linewidth, and explains the STO behavior under the influence of periodic and stochastic external signals: frequency modulation, phase-locking to external signals, mutual phase-locking in an array of STO, broadening of the generation linewidth near the generation threshold, etc. The proposed nonlinear auto-oscillator theory is rather general and can be used not only for the development of practical nano-sized STO, but, also, for the description of nonlinear auto-oscillating systems of any physical nature.

713 citations

Cites background from "A Study of Locking Phenomena in Osc..."

  • ...For conventional quasilinear auto-oscillators, where , this increase is negligible and the “linear” Adler’s theory of phase-locking [131] works very well....

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  • ...Adler [131] in 1940s for the description of the phase-locking effect in an electrical oscillator based on a standard -circuit (similar to the oscillator considered in Section IV-A), and since then it was extensively used for many other auto-oscillatory systems....

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Journal Article•DOI•
Injection locking of microwave solid-state oscillators

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K. Kurokawa1•
Bell Labs1
01 Oct 1973
TL;DR: Injection locking of microwave solid-state oscillators is discussed in this article, based on the familiar theorem that the total impedance times the current is equal to the applied voltage, based on which the locking range, large-signal injection, locking stability, and AM and FM noise are analyzed.
Abstract: Injection locking of microwave solid-state oscillators is discussed, based on the familiar theorem that the total impedance times the current is equal to the applied voltage. Both quasi-static and dynamic analyses of the locking range, large-signal injection, locking stability, and AM and FM noise are given, and recent experimental work is reviewed briefly. No applications of injection locking are discussed in detail.

654 citations

Cites background from "A Study of Locking Phenomena in Osc..."

  • ...Van der Pol's theory has been gradually refined, extended, or simplified by many authors including Krylov and Bogoliubov [SO], Adler [ 31 ], Huntoon and Weiss [l], Loeb 1321, and...

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  • ...where use is made of (2.8). B. Adler's Equation [ 31 ]...

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References
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Journal Article•DOI•
The Nonlinear Theory of Electric Oscillations

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B. van der Pol1•
Philips1
01 Sep 1934

431 citations

Journal Article•DOI•
A Frequency-Dividing Locked-in Oscillator Frequency-Modulation Receiver

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G.L. Beers
01 Dec 1944
TL;DR: In this article, a new type of frequency-modulation receiving system is described in which a continuously operating local oscillator is frequency-mated by the received signal at one fifth the intermediate frequency.
Abstract: A new type of frequency-modulation receiving system is described in which a continuously operating local oscillator is frequency-modulated by the received signal. In an embodiment of the system which is described, the oscillator is locked in with the received signal at one fifth the intermediate frequency. With this 5:1 relationship between the intermediate frequency and the oscillator frequency, an equivalent reduction in the frequency variations of the local oscillator is obtained. Received signal-frequency variations of ±75 kilocycles are reproduced as ±15-kilocycle variations in the oscillator frequency. The frequency-modulated signal derived from the oscillator is applied to a discriminator which is designed for this reduced range of frequencies. The oscillator is designed to lock in only with frequency variations which occur within the desired-signal channel. The oscillator is, therefore, prevented from following the frequency variations of a signal on an adjacent channel. A substantial improvement in selectivity is thus obtained. The voltage required to lock in the oscillator with a weak signal is approximately one twentieth of the voltage applied to the discriminator. Since this voltage gain is obtained at a different and lower frequency than the intermediate frequency, the stability of the receiver from the standpoint of over-all feedback is materially improved. Other performance advantages and the factors affecting the operation of the sytem are discussed.

15 citations

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A study of injection locking and pulling in oscillators

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Injection locking of microwave solid-state oscillators

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01 Oct 1973
K. Kurokawa
Synchronization: Phase locking and frequency entrainment

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01 Jan 2001
Arkady Pikovsky, Michael Rosenblum, Jürgen Kurths
Synchronization: A Universal Concept in Nonlinear Sciences

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01 Jan 2001
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The Kuramoto model: A simple paradigm for synchronization phenomena

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07 Apr 2005-Reviews of Modern Physics
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