Experiment on Unsymmetrical Subharmonic Oscillations in a Symmetrical Three-Phase Circuit

This paper presents several results of experiments on the generation of 1/3-subharmonic oscillations in a nonlinear symmetrical three-phase circuit. Particular attention is paid to the generation of unsymmetrical modes. Furthermore we describe a switching phase controller essential to the generation of the subharmonic oscillations in the experimental circuits

Periodic phase-locking and phase slips in active rotator systems

The Adler equation with time-periodic frequency modulation is studied. A series of resonances between the period of the frequency modulation and the time scale for the generation of a phase slip is identified. The resulting parameter space structure is determined using a combination of numerical continuation, time simulations and asymptotic methods. Regions with an integer number of phase slips per period are separated by regions with noninteger numbers of phase slips, and include canard trajectories that drift along unstable equilibria. Both high and low frequency modulation is considered. An adiabatic description of the low frequency modulation regime is found to be accurate over a large range of modulation periods

[Research Pertaining to Physics, Space Sciences, Computer Systems, Information Processing, and Control Systems]

Research project reports pertaining to physics, space sciences, computer systems, information processing, and control system

Self-oscillation

Physicists are very familiar with forced and parametric resonance, but usually not with self-oscillation, a property of certain dynamical systems that gives rise to a great variety of vibrations, both useful and destructive. In a self-oscillator, the driving force is controlled by the oscillation itself so that it acts in phase with the velocity, causing a negative damping that feeds energy into the vibration: no external rate needs to be adjusted to the resonant frequency. The famous collapse of the Tacoma Narrows bridge in 1940, often attributed by introductory physics texts to forced resonance, was actually a self-oscillation, as was the swaying of the London Millennium Footbridge in 2000. Clocks are self-oscillators, as are bowed and wind musical instruments. The heart is a "relaxation oscillator," i.e., a non-sinusoidal self-oscillator whose period is determined by sudden, nonlinear switching at thresholds. We review the general criterion that determines whether a linear system can self-oscillate. We then describe the limiting cycles of the simplest nonlinear self-oscillators, as well as the ability of two or more coupled self-oscillators to become spontaneously synchronized ("entrained"). We characterize the operation of motors as self-oscillation and prove a theorem about their limit efficiency, of which Carnot's theorem for heat engines appears as a special case. We briefly discuss how self-oscillation applies to servomechanisms, Cepheid variable stars, lasers, and the macroeconomic business cycle, among other applications. Our emphasis throughout is on the energetics of self-oscillation, often neglected by the literature on nonlinear dynamical systems.Comment: 68 pages, 33 figures. v4: Typos fixed and other minor adjustments. To appear in Physics Report

Periodically Disturbed Oscillators

By controlling the timing of events and enabling the transmission of data over long distances, oscillators can be considered to generate the "heartbeat" of modern electronic systems. Their utility, however, is boosted significantly by their peculiar ability to synchronize to external signals that are themselves periodic in time. Although this fascinating phenomenon has been studied by scientists since the 1600s, models for describing this behavior have seen a disconnect between the rigorous, methodical approaches taken by mathematicians and the design-oriented, physically-based analyses carried out by engineers. While the analytical power of the former is often concealed by an inundation of abstract mathematical machinery, the accuracy and generality of the latter are constrained by the empirical nature of the ensuing derivations. We hope to bridge that gap here. In this thesis, a general theory of electrical oscillators under the influence of a periodic injection is developed from first principles. Our approach leads to a fundamental yet intuitive understanding of the process by which oscillators lock to a periodic injection, as well as what happens when synchronization fails and the oscillator is instead injection pulled. By considering the autonomous and periodically time-varying nature that underlies all oscillators, we build a time-synchronous model that is valid for oscillators of any topology and periodic disturbances of any shape. A single first-order differential equation is shown to be capable of making accurate, quantitative predictions about a wide array of properties of periodically disturbed oscillators: the range of injection frequencies for which synchronization occurs, the phase difference between the injection and the oscillator under lock, stable vs. unstable modes of locking, the pull-in process toward lock, the dynamics of injection pulling, as well as phase noise in both free-running and injection-locked oscillators. The framework also naturally accommodates superharmonic injection-locked frequency division, subharmonic injection-locked frequency multiplication, and the general case of an arbitrary rational relationship between the injection and oscillation frequencies. A number of novel insights for improving the performance of systems that utilize injection locking are also elucidated. In particular, we explore how both the injection waveform and the oscillator's design can be modified to optimize the lock range. The resultant design techniques are employed in the implementation of a dual-moduli prescaler for frequency synthesis applications which features low power consumption, a wide operating range, and a small chip area. For the commonly used inductor-capacitor (LC) oscillator, we make a simple modification to our framework that takes the oscillation amplitude into account, greatly enhancing the model's accuracy for large injections. The augmented theory uniquely captures the asymmetry of the lock range as well as the distinct characteristics exhibited by different types of LC oscillators. Existing injection locking and pulling theories in the available literature are subsumed as special cases of our model. It is important to note that even though the veracity of our theoretical predictions degrades as the size of the injection grows due to our framework's linearization with respect to the disturbance, our model's validity across a broad range of practical injection strengths are borne out by simulations and measurements on a diverse collection of integrated LC, ring, and relaxation oscillators. Lastly, we also present a phasor-based analysis of LC and ring oscillators which yields a novel perspective into how the injection current interacts with the oscillator's core nonlinearity to facilitate injection locking.</p

Oscillatory stimuli differentiate adapting circuit topologies

This is the author accepted manuscript. The final version is available from Springer Nature via the DOI in this record.Biology emerges from interactions between molecules, which are challenging to elucidate with current techniques. An orthogonal approach is to probe for 'response signatures' that identify specific circuit motifs. For example, bistability, hysteresis, or irreversibility are used to detect positive feedback loops. For adapting systems, such signatures are not known. Only two circuit motifs generate adaptation: negative feedback loops (NFLs) and incoherent feed-forward loops (IFFLs). On the basis of computational testing and mathematical proofs, we propose differential signatures: in response to oscillatory stimulation, NFLs but not IFFLs show refractory-period stabilization (robustness to changes in stimulus duration) or period skipping. Applying this approach to yeast, we identified the circuit dominating cell cycle timing. In Caenorhabditis elegans AWA neurons, which are crucial for chemotaxis, we uncovered a Ca2+ NFL leading to adaptation that would be difficult to find by other means. These response signatures allow direct access to the outlines of the wiring diagrams of adapting systems.The work was supported by US National Institutes of Health grant 5RO1-GM078153-07 (F.R.C.), NRSA Training Grant CA009673-36A1 (S.J.R.), a Merck Postdoctoral Fellowship at The Rockefeller University (S.J.R.), and the Simons Foundation (S.J.R.). J.L. was supported by a fellowship from the Boehringer Ingelheim Fonds. E.D.S. was partially supported by the US Office of Naval Research (ONR N00014-13-1-0074) and the US Air Force Office of Scientific Research (AFOSR FA9550-14-1-0060)