Subharmonic Synchronization Research Articles

A spectral force representation and its physical implication for vortex shedding past a stationary or an oscillating circular cylinder at low Reynolds number

Vortex shedding is an ubiquitous phenomenon behind a bluff body (such as circular cylinder) and becomes more complicated when the body is also in oscillation. It is apparent that periodic behavior must be accompanied by the time-varying force, such as lift and drag (coefficients) with known distinguished cases (say, at Re=200) of low-frequency modulation (LFM), sub-harmonic synchronization (SHS), and normal harmonic synchronization (NHS). In a classical analysis, the force spectrum is often analyzed by the Fourier transform or some more recent methods, and typically, a quite complex frequency spectrum is obtained owing to the inherent nonlinearity in the flow system. In the present study, we extend the principal frequency analysis [Lu et al., “An EMD-based principal frequency analysis with applications to nonlinear mechanics,” Mech. Syst. Signal Process. 150, 107300 (2021)] to the principal spectrum analysis (PSA) with both its amplitude and phase in a composite functional form and provide a spectral representation (SR) of the force coefficients only in terms of the characteristic frequencies. In particular, we consider the unsteady laminar flow past a stationary circular cylinder or an oscillating circular cylinder (with frequency f0), while the resulting vortex shedding frequency is denoted by fVS. The spectral representation via the proposed PSA can reveal nonlinear interactions of the two characteristic frequencies (f0 and fVS) in influencing the force coefficients and distinguish direct and interactive modes in which f0 and fVS interact with each other. As a matter of fact, the successively shed vortices are not identical in the strength (amplitude) nor in the phase function. The spectral representation further enables us to identify complicated vorticity activity near around the bluff body: the periodicity of the strength of the shed vortices and the phase shift in the successive vortex shedding—all at the integer multiples of the greatest common-divisor (gcd) of the (two) characteristic frequencies. The gcd frequency of ⟨f0, fVS⟩ is identified as the genuine (slow, long-term) frequency of the entire vortex shedding process in contrast to the (fast, short-term) vortex shedding frequency. It turns out in this scheme of classification by the PSA-SR that all the distinguished types of the above-mentioned LFM, SHS, and NHS can be considered to be gcd-frequency synchronization.

Synchronization of the internal dynamics of optical soliton molecules

Compact bound states of light pulses in ultrafast lasers are known as optical soliton molecules. They constitute nonlinear superstructures of choice to investigate complex dynamical phenomena that manifest similarly in a wide range of nonlinear systems. Akin to matter molecules, optical soliton molecules can feature vibrational motions between their internal constituents. However, these vibrations are intrinsically nonlinear, with oscillation frequencies sensitive to system parameters. Therefore, vibrating soliton molecules present an opportunity for control. We here investigate the precise control of their oscillation frequencies through the universal mechanism of synchronization between master and slave oscillators. Self-oscillating soliton molecules are prepared within a passively mode-locked fiber laser. We experimentally demonstrate the synchronization of the internal vibrations of soliton molecules through the optical injection of a master oscillator signal. Direct observation of the synchronization process is enabled by balanced optical cross-correlation detection, a technique allowing real-time detection of intramolecular separation with femtosecond temporal resolution. We show efficient sub-harmonic, fundamental, and super-harmonic synchronization, forming a pattern of Arnold tongues with respect to the injection strength. Numerical simulations support experimental observations. By retrieving these universal synchronization features, the role of the soliton molecule as a nonlinear dynamical system of chief importance is further highlighted.

Aerodynamic stiffness effect and sub-harmonic synchronization of aeroelastic system in the presence of dynamic stall

A computational study of forced aeroelastic system of an airfoil is performed under various free-stream velocities. The results show that, as the free-stream velocity increases, the natural frequency of the system is increasing due to the aerodynamic stiffness effect except for a certain velocity range where the sub-harmonic synchronization occurs. To explore the characteristics of aerodynamic stiffness alone, the free oscillation response under the same conditions are simulated. The simulation shows that the aerodynamic stiffness presents a non-monotonic characteristic with the increase of the free-stream velocity. Further investigation indicates that the trend of aerodynamic stiffness varying with free-stream velocity is closely related to different areas at which the minimum angle of attack is located, which are greater than zero, less than zero or less than the dynamic stall angle in opposite direction. Equivalent linearization analysis qualitatively explains the reasons for the different aerodynamic stiffness characteristics in the above three phases. Comparing the results of the forced and the free system, it is found that the natural frequency under the influence of aerodynamic stiffness in the second stage is close to the 1/2 sub-harmonic of forcing frequency, resulting in the sub-harmonic synchronization. Moreover, the occurrence of dynamic stall in the opposite direction is considered to be the mechanism of unlock from the 1/2 sub-harmonics of forcing frequency.

Stochastic Discrete Time Crystals: Entropy Production and Subharmonic Synchronization

Discrete time crystals are periodically driven systems that display spontaneous symmetry breaking of time translation invariance in the form of indefinite subharmonic oscillations. We introduce a thermodynamically consistent model for a discrete time crystal and analyze it using the framework of stochastic thermodynamics. In particular, we evaluate the rate of energy dissipation of this many-body system of interacting noisy subharmonic oscillators in contact with a heat bath. The mean-field model displays the phenomenon of subharmonic synchronization, which corresponds to collective subharmonic oscillations of the individual units. The 2D model does not display synchronization but it does show a time-crystalline phase, which is characterized by a power-law behavior of the number of coherent subharmonic oscillations with system size. This result demonstrates that the emergence of coherent oscillations is possible even in the absence of synchronization.

Higher-order and long-range synchronization effects for classification and computing in oscillator-based spiking neural networks

In the circuit of two thermally coupled VO2 oscillators, we studied a higher order synchronization effect, which can be used in object classification techniques to increase the number of possible synchronous states of the oscillator system. We developed the phase-locking estimation method to determine the values of subharmonic ratio and synchronization effectiveness. In our experiment, the number of possible synchronous states of the oscillator system was twelve, and subharmonic ratio distributions were shaped as Arnold's tongues. In the model, the number of states may reach the maximum value of 150 at certain levels of coupling strength and noise. The long-range synchronization effect in a one-dimensional chain of oscillators occurs even at low values of synchronization effectiveness for intermediate links. We demonstrate a technique for storing and recognizing vector images, which can used for reservoir computing. In addition, we present the implementation of analog operation of multiplication, the synchronization based logic for binary computations, and the possibility to develop the interface between spike neural network and a computer. Based on the universal physical effects, the high order synchronization can be applied to any spiking oscillators with any coupling type, enhancing the practical value of the presented results to expand spike neural network capabilities.

Observation of Arnold Tongues in Coupled Soliton Kerr Frequency Combs.

We demonstrate various regimes of synchronization in systems of two coupled cavity soliton-based Kerr frequency combs. We show subharmonic, harmonic, and harmonic-ratio synchronization of coupled microresonators, and reveal their dynamics in the form of Arnold tongues, structures that are ubiquitous in nonlinear dynamical systems. Our experimental results are well corroborated by numerical simulations based on coupled Lugiato-Lefever equations. This Letter illustrates the newfound degree of flexibility in synchronizing Kerr combs across a wide range of comb spacings and could find applications in time and frequency metrology, spectroscopy, microwave photonics, optical communications, and astronomy.

Thermal coupling and effect of subharmonic synchronization in a system of two VO2 based oscillators

We explore a prototype of an oscillatory neural network (ONN) based on vanadium dioxide switching devices. The model system under study represents two oscillators based on thermally coupled VO2 switches. Numerical simulation shows that the effective action radius RTC of coupling depends both on the total energy released during switching and on the average power. It is experimentally and numerically proved that the temperature change ΔT commences almost synchronously with the released power peak and T-coupling reveals itself up to a frequency of about 10 kHz. For the studied switching structure configuration, the RTC value varies over a wide range from 4 to 45 μm, depending on the external circuit capacitance C and resistance Ri, but the variation of Ri is more promising from the practical viewpoint. In the case of a “weak” coupling, synchronization is accompanied by attraction effect and decrease of the main spectra harmonics width. In the case of a “strong” coupling, the number of effects increases, synchronization can occur on subharmonics resulting in multilevel stable synchronization of two oscillators. An advanced algorithm for synchronization efficiency and subharmonic ratio calculation is proposed. It is shown that of the two oscillators the leading one is that with a higher main frequency, and, in addition, the frequency stabilization effect is observed. Also, in the case of a strong thermal coupling, the limit of the supply current parameters, for which the oscillations exist, expands by ∼10%. The obtained results have a universal character and open up a new kind of coupling in ONNs, namely, T-coupling, which allows for easy transition from 2D to 3D integration. The effect of subharmonic synchronization hold promise for application in classification and pattern recognition.

Free Vibrations of a Square Cylinder of Varying Mass Ratios

Two-dimensional space-time finite-element simulations are carried out to study the effect of oscillator mass ratio, m* on free in-line and transverse vibrations of a rigid square cylinder. The effect of viscous damping or resistance is not considered and thus cylinders of mass ratio 1, 5, 10 and 20 execute free undamped vibrations. Results are presented for 50≤Re≤250 where Re is the Reynolds number. For m* = 1, synchronization between cylinder oscillation and vortex-shedding is 1:1 (periodic flow). Thus, the Cl-Y phase portraits must be single looped closed curves. For m* = 5-20, synchronization is 1:1 (periodic flow) prior to galloping. For m* = 5, the synchronization during galloping is quite different from 1:3. Along ‘increasing Re’, the entire galloping branch of m* = 5 cylinder is quasi-periodic. However, along ‘decreasing Re’, it is mostly quasi-periodic and it is periodic for Re≤190. An 1:3 sub-harmonic synchronization is seen for the m* = 10 and 20 cylinders only in the periodic regime of galloping occurring at higher Re. For m *= 1, only one kink is seen in the response curve signifying transition from initial to lower branch. Two kinks are captured for m* = 5 cylinder. For m*= 10 and 20, an additional third kink is resolved in the galloping branch that signifies transition from quasi-periodic flow/body motion to periodic flow/body motion.

Subharmonic synchronization ofself-oscillations in discrete time

Self-oscillations in the system oscillating in discrete time and being under the influence of an external harmonious signal are investigated. As an object of researches the option of discrete map of the oscillator of Van der Pol offered earlier by authors is brought to attention. The modes of synchronous self-oscillations are analysed by means of methods of numerical experiment and harmonic balance. It is established that the effect of subharmonic synchronization in discrete time can be realized at any relation of frequencies of synchronized self-oscillations and the synchronizing signal.

Fluid–structure interaction of a square cylinder at different angles of attack

AbstractThis study investigates the free transverse flow-induced vibration (FIV) of an elastically mounted low-mass-ratio square cylinder in a free stream, at three different incidence angles: ${{\alpha }}=0^\circ $, $20^\circ $ and $45^\circ $. This geometric setup presents a body with an angle of attack, sharp corners and some afterbody, and therefore is a generic body that can be used to investigate a wide range of FIV phenomena. A recent study by Nemes et al. (J. Fluid Mech., vol. 710, 2012, pp. 102–130) provided a broad overview of the flow regimes present as a function of both the angle of attack ${{\alpha }}$ and reduced flow velocity ${U^{*}}$. Here, the focus is on the three aforementioned representative angles of attack: ${{\alpha }}=0^\circ $, where the FIV is dominated by transverse galloping; ${{\alpha }}=45^\circ $, where the FIV is dominated by vortex-induced vibration (VIV); and an intermediate value of ${{\alpha }}=20^\circ $, where the underlying FIV phenomenon has previously been difficult to determine. For the ${{\alpha }}=0^\circ $ case, the amplitude of oscillation increases linearly with the flow speed except for a series of regimes that occur when the vortex shedding frequency is in the vicinity of an odd-integer multiple of the galloping oscillation frequency, and the vortex shedding synchronizes to this multiple of the oscillation frequency. It is shown that only odd-integer multiple synchronizations should occur. These synchronizations explain the ‘kinks’ in the galloping amplitude response for light bodies first observed by Bearman et al. (J. Fluids Struct., vol. 1, 1987, pp. 19–34). For the ${{\alpha }}=45^\circ $ case, the VIV response consists of a number of subtle, but distinctly different regimes, with five regimes of high-amplitude oscillations, compared to two found in the classic VIV studies of a circular cylinder. For the intermediate ${{\alpha }}=20^\circ $ case, a typical VIV ‘upper branch’ occurs followed by a ‘higher branch’ of very large-amplitude response. The higher branch is caused by a subharmonic synchronization between the vortex shedding and the body oscillation frequency, where two cycles of vortex shedding occur over one cycle of oscillation. It appears that this subharmonic synchronization is a direct result of the asymmetric body. Overall, the FIV of the square cylinder is shown to be very rich, with a number of distinct regimes, controlled by both ${{\alpha }}$ and ${U^{*}}$. Importantly, ${{\alpha }}$ controls the underlying FIV phenomenon, as well as controlling the types of possible synchronization between the oscillation and vortex shedding.

Spontaneous Slow Oscillations and Sequential Patterns Due to Short-Term Plasticity in a Model of the Cortex

We study a realistic model of a cortical column taking into account short-term plasticity between pyramidal cells and interneurons. The simulation of leaky integrate-and-fire neurons shows that low-frequency oscillations emerge spontaneously as a result of intrinsic network properties. These oscillations are composed of prolonged phases of high and low activity reminiscent of cortical up and down states, respectively. We simplify the description of the network activity by using a mean field approximation and reduce the system to two slow variables exhibiting some relaxation oscillations. We identify two types of slow oscillations. When the combination of dynamic synapses between pyramidal cells and those between interneurons accounts for the generation of these slow oscillations, the end of the up phase is characterized by asynchronous fluctuations of the membrane potentials. When the slow oscillations are mainly driven by the dynamic synapses between interneurons, the network exhibits fluctuations of membrane potentials, which are more synchronous at the end than at the beginning of the up phase. Additionally, finite size effect and slow synaptic currents can modify the irregularity and frequency, respectively, of these oscillations. Finally, we consider possible roles of a slow oscillatory input modeling long-range interactions in the brain. Spontaneous slow oscillations of local networks are modulated by the oscillatory input, which induces, notably, synchronization, subharmonic synchronization, and chaotic relaxation oscillations in the mean field approximation. In the case of forced oscillations, the slow population-averaged activity of leaky integrate-and-fire neurons can have both deterministic and stochastic temporal features. We discuss the possibility that long-range connectivity controls the emergence of slow sequential patterns in local populations due to the tendency of a cortical column to oscillate at low frequency.

Subharmonic Synchronization of Picosecond Yb Fiber Laser to Picosecond Ti:Sapphire Laser for Stimulated Raman Scattering Microscopy

We demonstrate a subharmonically synchronized picosecond laser system and its application to stimulated Raman scattering (SRS) microscopy for label-free biological imaging with high sensitivity and high spectral resolution. Experimentally, 5.5-ps Yb fiber laser pulses at a repetition rate of 38 MHz were successfully synchronized to 4.8-ps Ti:Sapphire laser pulses at a repetition rate of 76 MHz, by using a two-photon detector, an intracavity electro-optic modulator, and a piezoelectric transducer. The temporal jitter was measured to be 120 fs. These pulses were applied to SRS microscopy. We confirmed the shot-noise-limited sensitivity, and succeeded in SRS spectroscopy of a polymer bead as well as three-dimensional SRS imaging of a HeLa cell.

Subharmonic Synchronization of Picosecond Yb Fiber Laser to Picosecond Ti:Sapphire Laser for Stimulated Raman Scattering Microscopy

Subharmonic Synchronization of Picosecond Yb Fiber Laser to Picosecond Ti:Sapphire Laser for Stimulated Raman Scattering Microscopy

Experimental investigation of harmonic and subharmonic synchronization of 40 GHz mode-locked quantum-dash laser diodes

Synchronization of a 40 GHz quantum-dash mode-locked (ML) Fabry-Perot laser diode with optically injected pulse streams is experimentally studied. Injected signals consist of nonmodulated and modulated trains of 1.6 ps pulses at various repetition rates, ranging from 10 to 160 GHz and 10 to 160 Gbps, respectively. Subharmonic, fundamental, and harmonic synchronization of the ML laser allows retrieval of stable 40 GHz clock pulses featuring a width of 1.8 ps. Frequency components at 10 and 20 GHz do not create any amplitude modulation on the recovered 40 GHz clock pulses when injecting signals at 10 and 20 GHz/Gbps. In addition, external synchronization of the laser with pulse streams at 80 and 160 GHz/Gbps is sustained despite the absence of significant components at or below 40 GHz.

Vibration in a self-oscillating vocal fold model with left-right asymmetry in body-layer stiffness

This study compares the phonatory behavior of an asymmetric vocal fold model to that of each individual vocal fold model in a hemi-configuration. Although phonation frequencies of the two folds in hemi-configurations had a ratio close to 1:3, a subharmonic synchronization between the two folds was not observed in the asymmetric model. Instead, the vibratory behavior was dominated by the dynamics of one fold only, and the other fold was enslaved to vibrate at the same frequency. Increasing subglottal pressure induced a shift in relative dominance between the two folds, leading to abrupt changes in both vibratory pattern and frequency.

Subharmonic stochastic synchronization and resonance in neuronal systems.

We study the response of a model neuron, driven simultaneously by noise and at least two weak periodic signals. We focus on signals with frequencies components kf(0),(k+1)f(0),...(k+n)f(0) with k>1. The neuron's output is a sequence of pulses spaced at random interpulse intervals. We find an optimum input noise intensity for which the output pulses are spaced approximately 1/f(0), i.e., there is a stochastic resonance (SR) at a frequency missing in the input. Even higher noise intensities uncover additional, but weaker, resonances at frequencies present in the input. This is a different form of SR whereby the most robust resonance is the one enhancing a frequency, which is absent in the input, and which is not possible to recover via any linear processing. This can be important in understanding sensory systems including the neuronal mechanism for perception of complex tones.

Analysis of the tonic vibration reflex: influence of vibration variables on motor unit synchronization and fatigue.

The influence of vibration frequency (40, 80, 100, 120, 150, or 200 Hz) at selected displacement amplitudes (0.2, 0.3 mm) on tonic vibration reflex (TVR) characteristics was investigated. The degree of synchronization of motor unit activity with vibratory stimuli in ten humans was determined using the electromyographic (EMG) activity of the finger and wrist flexor muscles when vibration was applied to the distal tendons of the hand flexor muscles. The EMG spectral analysis indicates that harmonic and subharmonic motor unit synchronization mechanisms contribute to the modulation of the amplitude of the TVR as the vibration frequency increases. Harmonic synchronization decreases while subharmonic synchronization increases as vibration frequency increases. It is suggested that the synchronization process influences muscle fatigue, since it forces the driving of motor units, leading to a decrease in contraction efficiency. This phenomenon most probably results from an impairment of excitation-contraction coupling. High-frequency vibration (> 150 Hz) tends to induce less motor unit synchronization in a frequency range beyond the known mechanical resonance of biological tissues. The findings of this study may be applied to the design of hand-held power tools, since their vibration triggers the TVR in active muscles.

A new approach for a phase controlled self-oscillating mixer

The analytical and experimental demonstration of subharmonic synchronization and phase shifting of a push-pull self-oscillating mixer is presented for the first time. Inherent high mixing gain of the self-oscillating mixer circuit is exploited to generate a strong signal at the same frequency of the reference signal, which is related to the local oscillator's (LO) phase information. A phase error between this signal and the reference signal is extracted in a phase comparator before phase locking. Analytical modeling of frequency and phase stabilization of the push-pull self-oscillating mixer is presented, which is also experimentally verified for a self-oscillating mixer at 12 GHz. This self-oscillating mixer circuit demonstrates efficient phase locking, 0/spl deg/-180/spl deg/ continuous phase shifting capability in addition to the reported large locking range (>10 MHz), low close-in to carrier phase noise ( 17 dB at 17 GHz). The demonstrated subharmonic phase locking approach replaces the need for a frequency multiplier or divider before the phase comparator. The synchronized push-pull self-oscillating mixer circuit is applicable to the millimeter-wave frequency distributed transmitters and receivers, where low-loss phase shifting and efficient subharmonic phase and frequency locking are hard to achieve.

Bifurcation of synchronized oscillations in an oscillator with hard characteristics

AbstractIt is critically important that the synchronization problem be studied as it pertains to oscillators with hard characteristics having two types of limit cycles where periodic external force is applied and oscillations are at a submultiple of the frequency of external force. This paper considers bifurcations of periodic states corresponding to harmonic synchronization and subharmonic synchronization of the order of one‐half and one‐third observed in a Duffing‐Rayleigh equation with hard characteristics. As a result, it was discovered that the synchronization region corresponding to each limit cycle exists independently and, under certain conditions, the two regions have mutual coalescence. Moreover, in the region of subharmonic synchronization of the order of one‐half it was observed that periodic doubling bifurcation intersects the Hopf bifurcation. As a result of analysis using Hopf bifurcations, it was also found that tangent bifurcations correlated with Arnold tongues and the associated resonance phenomena.

Periodic and non-periodic responses of a periodically forced Hodgkin-Huxley oscillator

Membrane potential responses of a Hodgkin-Huxley oscillator to an externally-applied sinusoidal current were numerically calculated with relation to bifurcation parameters of the amplitude and the frequency of the stimulating current. The Hodgkin-Huxley oscillator, or the Hodgkin-Huxley axon in the state of self-sustained oscillation of action potentials, was realized by immersing the axon in calcium-deficient sea water. The forced oscillations were analysed by the stroboscopic plots and/or the Lorenz plots. The results show that the periodically forced Hodgkin-Huxley oscillator exhibits not only periodic motions (harmonic or sub-harmonic synchronization) but also non-periodic motions (quasi-periodic or chaotic oscillation), that the motions were determined by the amplitude and the frequency of the stimulating current, and that the characteristic motions obtained in the present study were in reasonable agreement with those of our previous results, found experimentally in squid giant axons. Also, two kinds of routes to the chaotic oscillations were found; successive period-doubling bifurcations and formation of the intermittently chaotic oscillation from sub-harmonic synchronization.