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The article about solution of system of nonlinear differential equations that are almost impossible to solve by analytical methods by constructing a mathematical model of nonlinear oscillations occurring in three-phase ferroresonance circuits. A system of nonlinear differential equations was formed by approximating the volt-ampere characteristics of a ferromagnetic element in a ferroresonance circuit. Mathematical models for solving technical problems characterizing subharmonic oscillation processes in three-phase ferroresonance circuits and systems using the finite-difference method are expressed in the form of differential equations without appropriate initial conditions. Amathematical model of a system of equations representing subharmonic oscillations in ferroresonance connections depending on the value of the selected parameters in the field of change of any variable is considered.

A frequency transmission band of the nonlinear energy harvester shall be studied numerically. For the analysis, the nonlinear piezoelectric energy harvesting system based on a cantilever elastic beam has been applied to harvest the kinetic energy of the moving frame. We used a double-well potential induced by permanent magnets for a ferromagnetic beam resonator. A piezoelectric patch attached along the beam was used as a transducer of the mechanical into electrical energy. It occurred that the system could work in a wide interval of frequency beyond the linear resonance. Besides the response with a period of excitation, solutions with dominating sub-harmonics of the harmonic inertial force excitation have been found. Particular solutions were illustrated, classified, and discussed using phase portraits and Fourier spectra of the output signals.

This paper studies intermittent instabilities in parallel power system. Taking parallel buck converters with coupled interference signals as an example, model and actual system are established. By numerical simulation, the nonlinear behaviours are captured under some conditions. Characteristics and mechanism of intermittency are depicted by nonlinear theory. In this paper, mappings time to parameter are established and discrete iterative mappings for nonlinear systems are derived. All the simulation results are consistent with the theoretical analysis results. The research method adopted in the paper is applicable for the other parallel powersystems. The conclusion can provide a reference for the parallel converters to maintain stable operation and serves as a theoretical basis for further optimization design and system control.

In this paper we study several settings of the -subharmonic extension problem on open Riemann surfaces. The problem is completely solved (for all ) for so-called Runge-type extensions. Several (in some sense sharp) sufficient conditions and counterexamples are found also for Walsh-type extensions. As applications, these results allow us to prove the existence of -subharmonic extensions, automorphic with respect to some appropriate groups of automorphisms of an open Riemann surface. Bibliography: 22 titles.

We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with superharmonic drift, and a congested crowd motion model originally proposed by [MRS]- [MRSV]. We first use viscosity solutions to show that the porous medium equation solutions converge to the Hele- Shaw solution as the exponent tends to infinity. Next, using of the gradient flow structure of both the porous medium equation and the crowd motion model, we prove that the porous medium equation solutions also converge to the congested crowd motion. Combining these results lets us deduce that in the case where the initial data to the crowd motion model is given by a patch, or characteristic function, the solution evolves as a patch that is the unique solution to the Hele-Shaw problem. While proving our main results we also obtain a comparison principle for solutions to the minimizing movement scheme based on the Wasserstein metric, of independent interest., Comment: 36 pages, 5 Figures

Objective. This study aims at investigating the potential mechanism of period doubling (PD) (subharmonic generation) in human steady-state visual evoked potentials (SSVEPs) by using a mathematical model. Approach. Robinson's Corticothalamic Model, which includes three main neuronal populations (cortical, thalamic reticular, and thalamic relay neurons) was employed. SSVEP experiments were simulated using this model and dependence of PD behavior in relation to the values of model parameters was investigated. The feedback loop in the model that is responsible for the generation of subharmonic components was thus identified, and this loop was isolated from the rest of the model and analyzed with a describing function approach. Main Results. It has been found in general, for a wide range of parameter values, that if the excitation frequency or half of it is close to the native oscillation frequency of the system, the native oscillation ceases to exist and oscillations at either the excitation frequency or half of it are observed. This observation is in line with the experimental findings except for some discrepancies which are also discussed. The intrathalamic feedback loop is identified to be the potential source of subharmonic oscillations. When isolated from the rest of the model and simulated by itself, it has been found that this feedback loop can show a resonance phenomenon at the subharmonic frequency. By deriving a set of equations based on the necessary conditions for a resonance phenomenon, a semi-analytical method was developed by which one can predict the existence of subharmonic generation for a given set of parameters and stimulus frequency. Significance. This study is the first model-based investigation of the mechanism of subharmonic oscillations. The proposed semi-analytical method can replace extensive time and memory consuming parameter sweep studies.

We study bifurcations from one-parameter families of symmetric periodic orbits in reversible systems and give simple criteria for subharmonic symmetric periodic orbits to be born from the one-parameter families. Our result is illustrated for a generalization of the Henon–Heiles system. In particular, it is shown that there exist infinitely many families of symmetric periodic orbits bifurcating from a family of symmetric periodic orbits under a general condition. Numerical computations for these bifurcations and symmetric periodic orbits are also given.

Ultrasound microbubble contrast agents have been demonstrated to scatter subharmonic energy at one-half the driving frequency. At ultrasound frequencies in the 20-40 MHz range, the subharmonic offers the potential to differentiate the blood in the microcirculation from the surrounding tissue. It is unknown whether current contrast agents, manufactured to be resonant between 2 and 12 MHz, are ideal for subharmonic imaging at higher frequencies. We performed numerical simulations of the Keller-Miksis model for the behavior of a single bubble and experimental investigations of Definity microbubbles in water. The results supported the hypothesis that off-resonant bubbles, excited at their second harmonic, may be primarily responsible for the observed subharmonic energy. For frequencies between 20 and 32 MHz and 32 and 40 MHz, the optimal bubble diameters for the generation of subharmonics in vitro were determined experimentally to be 1.2-5 microm and less than 1.2 microm, respectively. Definity may be a suitable ultrasound contrast agent for subharmonic imaging at 20 MHz with peak-negative pressures between 380 and 590 kPa and pulses greater than or equal to four cycles in duration.

We investigate the impact of granular particles at the bottom of the container, which is vibrating harmonically in the vertical direction. Experiments demonstrate that the impact strength, controlled by the normalized vibration acceleration, undergoes a series of subharmonic bifurcations, in the sequence of period-doubling, period-quadrupling, chaos, period-tripling, period-sextupling, chaos, period-quadrupling, period-octupling, and so on. It is observed that accompanying the impact bifurcations there always exists a condensed state of particles at the lower part of the bed. By using the completely inelastic bouncing-ball model, a possible explanation for the observed bifurcation phenomena is presented.

Given and any (Jordan) -domain in , we prove that any function of class that is subharmonic in can be extended to a function of class that is subharmonic on the whole and give an estimate of the -norm of its gradient. The corresponding assertion for is false even for discs. These results also hold for balls in , . We also obtain some corollaries, including the corresponding assertions on the -extension of subharmonic functions.

It is proved that if there exists a periodic solution for a class of non-autonomous differential dynamic systems, it can only be subharmonic, ultra-subharmonic periodic solution is impossible. Moreover, the existence of R-type ultra-subharmonic periodic solution defined for a specified planar system is also denied. As an application of the above conclusions, through investigating some typical examples, it is pointed out that the existence of ultra-subharmonic periodic orbits in a planar perturbation system cannot be determined by second-order Melnikov method. An explanation is also provided.

The streak instability is examined experimentally by artificially generating spanwise-periodic low-speed streaks in a laminar boundary layer on a flat plate. Fundamental and subharmonic modes are excited for each of the sinuous and varicose instabilities and their development is compared with the corresponding result of a single low-speed streak. The development of subharmonic sinuous mode does not strongly depend on the streak spacing and it grows with almost the same growth rate as that for the single streak. By contrast, the development of fundamental sinuous mode is very sensitive to the streak spacing and is completely suppressed when the streak spacing is smaller than a critical value, about 2.5 times the streak width for the low-speed streaks examined. On the varicose instability, the fundamental mode is less amplified than the subharmonic mode, but the growth of both modes is weak compared with the case of the single streak.

In this paper, we report the generation of a high-repetition-rate optical pulse train with a mode-locked laser diode (MLLD) and its control techniques. We discuss key technologies to increase the mode-locking frequency and reveal that passive mode-locking based on an intracavity semiconductor saturable absorber is a promising way to achieve mode-locking at a very high repetition rate exceeding hundreds of GHz. Generation of a transform-limited 480 GHz optical pulse train was successfully achieved using a short-cavity (174 µm) MLLD. We also propose and demonstrate two novel techniques, subharmonic optical synchronous mode-locking and an all-optical down-converting method, to stabilize such an ultrafast MLLD and to evaluate the jitter characteristics accurately by all-optical methods without limitations imposed by an electronic device's speed. Stabilization of ultrafast MLLDs (160 and 480 GHz passive MLLDs) and the jitter measurements were successfully demonstrated.

Development of ultrasound transducers with double-peak-type frequency characteristics for harmonic imaging and subharmonic imaging is reported in this paper. The peak on the low-frequency side in the frequency characteristics is used to transmit ultrasound into tissues or microbubble ultrasound contrast agents, and another peak on the high-frequency side is used to receive the second harmonic component in the ultrasound transducer with double-peak-type frequency characteristics for harmonic imaging. On the other hand, the peak on the high-frequency side in the frequency characteristics is used to transmit ultrasound and another peak on the low-frequency side is used to receive the 1/2 subharmonic component in the ultrasound transducer with double-peak-type frequency characteristics for subharmonic imaging. The results of the transducer design based on numerical calculation are reported in this paper.

We have investigated the magnetization of a triangular microhole lattice on a Pb film by means of a SQUID magnetometer under small dc fields (|H| ≤ 4.7 Oe). We find sharp magnetization peaks at the integral multiples of the magnetic matching field at a temperature near the critical temperature Tc. Anomalously, the matching peaks are symmetric and reversible against the field reversal. Therefore, neither the Little–Parks effect nor the preferential pinning effect at the matching fields can explain this phenomenon. The half-integer subharmonic peaks are also observed while they are very weak. We have also carried out magnetization measurements under high dc fields (|H| ≤ 1000 Oe) to investigate a phase diagram covering the superconducting regime completely. We find the irreversibility line in the superconducting state.

Novel effects emerge from an interplay between multiple Andreev reflections and Coulomb interaction in quantum dot coupled to superconducting leads and subject to a finite potential bias $V$. Combining an intuitive physical picture with rigorous path integral formalism we evaluate the current $I$ through the dot and find that the interaction shifts the subharmonic pattern of the $I-V$ curve is shifted toward higher $V$. For sufficiently ^M strong interaction the subgap current (at $eV < 2 \Delta$) is virtually suppressed., Comment: 4 two column pages including 1 eps figure

The subharmonic gap structure (SGS) in long-overlap Nb-AlOx-Nb Josephson tunnel junctions has been investigated. The experimental results show peaks in the differential conductance at both odd and even integer fractions of the gap voltage, VG. Furthermore, the conductance peaks at VG/2 has been observed to split into two peaks with different characteristics. At high magnetic fields, the I-V characteristics approach a single curve, while retaining the SGS conductance peaks. The gap structure and the SGS show the same temperature dependence. The SGS can be explained by a Josephson self-coupling effect or multiple Andreev reflections. The results are important for the development of the flux flow oscillator used a local oscillator in integrated all-superconducting 200-800 GHz receivers.

Spatially growing wakes behind a circular cylinder at 140 ≤ Re ≤ 1000 are simulated numerically. Two-dimensional, incompressible Navier–Stokes equations are solved using a splitting finite difference method. Special attention is paid to the development of the vortical structures in far wakes. Effects of subharmonic forcing on the vortical structures are also examined by perturbing the uniform velocity of a free-stream. In both unforced and forced wakes, the transition from the primary Karman vortex street in near wakes to the secondary vortex street in far wakes is observed for all of the Reynolds number cases examined. In unforced cases, rapid decay of the Karman vortices is observed, but clear evidence of merging of the Karman vortices in the transition process was not found. On the other hand, in forced wakes, merging of the Karman vortices is found to play a central role in the transition process. The effects of the Reynolds number as well as the forcing parameters on the vortical structures in far wakes are also examined.

Ultrasonic contrast agents are used to enhance backscatter from blood and thus aid in delineating blood from surrounding tissue. However, behaviour of contrast agents in an acoustic field is nonlinear and leads to harmonic components in the backscattered signal. Various research groups have investigated second-harmonic emissions. In this work, the subharmonic emission from contrast agents is investigated with a view towards potential use in imaging. It is shown that the microbubbles with various surface properties, such as contrast agents, generate significant subharmonics under various insonating conditions. Theoretical results as well as experimental results using Optison indicate the generation of strong subharmonics with burst insonation at twice the resonant frequency of the microbubble. It is suggested that subharmonic imaging may provide a better modality than second-harmonic imaging to delineate blood from tissue and will be of significant importance for imaging deep vessels, such as in echocardiography and vascular diseases, due to the high signal-to-clutter ratio of the subharmonic imaging.

The narrow-tube quasienergy quantization allows an identification of part of the spectrum due to stable elliptic islands of the period map. With its transparent connection to the classical dynamics it also sets an absolute scale to the understanding of the Brillouin zone spectrum of quasienergies. In combination with a recent time-dependent normal-form expansion we show that the applicability and accuracy of the narrow-tube quantization is improved. Its analytic form helps us understand the quasiregular motion centred near periodic responses. We numerically consider harmonic and subharmonic elliptic islands of the weakly nonlinear Duffing oscillator.

It has been proved by Ziglin, for a large class of two-degree-of-freedom (2DOF) Hamiltonian systems, that transverse intersections of the invariant manifolds of saddle fixed points imply infinite branching of solutions in the complex time plane and the non-existence of a second analytic integral of the motion. Here, we follow a similar approach to show the existence of infinitely sheeted solutions for 2DOF. Hamiltonians which exhibit, upon perturbation, subharmonic bifurcations of resonant tori around an elliptic fixed point. Moreover, as shown recently, these Hamiltonian systems are non-integrable if their resonant tori form a dense set.

The quantum statistics of higher harmonics and subharmonics generation processes is studied in an exact way. By means of the described numerical way the Q-distributions and Wigner distributions are calculated and presented. The non-classical effects are examined and corresponding relations are derived.

Auto-oscillations and chaotic behavior were investigated experimentally in a plasma system produced by discharge through mixing He-Ne gases. The glow power spectra and the differential phase-space portraits show period-doubling and Ruelle-Takens routes to chaos. Periodic windows embedded in chaos and multiple subharmonic bifurcations occurring in the windows were observed.

We have investigated the nonlinear characteristics of a microbubble contrast agent Sonazoid (Nycomed, Norway), including the second, third, 1/2-order, 3/2-order and 5/2-order harmonics. We have measured the 1/2-order subharmonic response to different transmission sound pressures. We have found that subharmonic signals cannot be generated until the acoustic pressure reaches a certain value, which is the most different subharmonic from high harmonics. This result is favourable for the further study of the subharmonic in the bubbly liquid. The 3/2-order ultraharmonic response to acoustic pressure was also measured.

It is proved that a subharmonic function is twice differentiable outside a set of small Lebesgue measure.

The phenomena of bifurcation have been observed in experiments on one-dimensional finite-amplitude standing wave. Besides higher harmonics predicted by the theory, subharmonics fractional also appear in the finite-amplitude standing wave field, as the sound pressure gets higher, and eventually a chaotic state is reached.

The authors discuss the dynamics of voltage-driven Josephson junction arrays with negligible capacitances. The array dynamics shows a sensitive dependence on initial phases, leading to current-spike structures in the time-averaged I-V relations. They demonstrate this effect explicitly for the simple system of a two-junction chain, which they also use to illustrate the main features of current response to external driving voltages. They then show that a class of simplified ansatz solutions of two-dimensional voltage-driven Josephson junction arrays under a perpendicular magnetic field can be reduced to two-junction dynamics. In both cases, they find that coherent phase-slip processes can lead to subharmonic lockings with an external RF field.

Experiments are described which demonstrate subharmonic Bragg resonances in the context of linear gravity surface waves propagating on a doubly periodic bottom. The experimental results are found to be in good agreement with the predictions of a numerical model based on the full potential theory of linear waves. The importance of evanescent modes in the interference process is demonstrated. In particular, evanescent modes give rise to a large subharmonic effect even for small bottom modulations.

The smoothness of functions subharmonic in a domain D⊂Rn, n ≥ 2, is studied. The method for studying smoothness is based on the Riesz theorem on representation of a subharmonic function as the sum of a harmonic function and the potential Uμ(x) with respect to some Borel measure μ. Bibliography: 16 titles.

In a delayed system excited by low-frequency and high-frequency signals, the necessity of the high-frequency signal on the resonance is discussed. By adjusting the delay time, the resonance occurs in a wide scope of frequencies, including the primary, subharmonic and superharmonic frequencies. Only for very few cases does the high-frequency signal have a positive effect on the resonance. It is the traditional vibrational resonance phenomenon. In most situations, the high-frequency excitation is unnecessary for the resonance. An appropriate delay, rather than the high-frequency signal, is the key factor in improving the weak low-frequency signal.

Takahiro Hayashi and Shiro Biwa. Subharmonic wave generation at interfaces of a thin layer between metal blocks. Japanese Journal of Applied Physics, 2013, 52(7S), 07HC02. https://doi.org/10.7567/JJAP.52.07HC02., Subharmonic waves were significantly generated at the interfaces of aluminum blocks and an aluminum foil subjected to a 6MHz burst incident wave. Although the fundamental wave amplitude was linearly varied with input voltage, the subharmonic waves generated at the interfaces indicated strong nonlinearity. Analyzing the relationship of the subharmonic wave generation, the applied contact pressure, and the stress of the ultrasonic wave has revealed that subharmonic components can be generated when the incident ultrasonic wave has sufficiently large stress to separate the contacting interfaces. At rough surfaces, the subharmonic peak could not be observed because a large true contact pressure prevented the interfaces from separating. © 2013 The Japan Society of Applied Physics.

Patterns such as periodic subharmonic waves and kink-pairs are observed experimentally in a two-dimensional vertically vibrated granular layer located in an annular channel which has a sawtooth-shaped base. Using the thickness of the granular layer and external excitation as control parameters, we find the transitions between different patterns in the layer. The dependence of the kink-pair's velocity on the external excitation is revealed. The local velocity fields of the granular layer indicate that, in addition to horizontal transport in the layer, there is convection in the node area of a kink.

The ultrasound transducer for subharmonic imaging with double-peak frequency characteristics, having a peak on the high-frequency side for illuminating ultrasound and having a peak on the low-frequency side for detecting subharmonic components, was fabricated. Higher peak sensitivity was observed on the high-frequency side during transmitting operation and on the low-frequency side during receiving operation in the fabricated transducer.

Closed stress corrosion cracks (SCCs) have been generated in Ni-based alloy weld metal in nuclear power plants. The ultrasonic inspection is difficult because of the crack closure. For the application of new inspection methods and training/educating of inspection engineers, realistic closed SCC specimens are required. However, there is no means for forming such SCC specimens in a reasonable amount of time. Here, we present a two-step method. The first step is to form an open SCC in chemical solution. The second step is to close the SCC by generating oxide films between the crack faces in high-temperature pressurized water (HTPW). To verify the crack closure, we used a closed-crack imaging apparatus, the subharmonic phased array for crack evaluation (SPACE). Consequently, we found that parts of the SCC after 1321 h immersion were closed in the HTPW. Thus, we verified the two-step method for forming realistic closed SCC specimens in a reasonable amount of time.

A circuit topology for high-order subharmonic (SH) mixers is described. By phase cancellation of idle frequency components, the SH mixer circuit can eliminate the complicated design procedure of idle frequency circuits. Similarly, the SH mixer circuit can achieve a high port isolation by phase cancellation of the leakage LO, RF and idle frequency signals. Based on the high-order SH mixer architecture, a new Ka-band fourth SH mixer is analyzed and designed, it shows the lowest measured conversion loss of 8.3 dB at 38.4 GHz and the loss is lower than 10.3 dB in 34-39 GHz. Measured LO-IF, RF-LO, RF-IF port isolation are better than 30.7 dB, 22.9dB and 46.5 dB, respectively.

A phenomenological diagram is presented to explain the interaction between a fundamental wave and its sub-harmonic wave in 2D shear layers based on linear stability theory. These diagrams indicate that there are only four classes of subharmonic interactions, which are symmetric collective interaction, the first class of asymmetric collective interaction, the second class of collective interaction and tearing. Each of them can be determined uniquely by a couple of parameters m and p, where m is the ratio of wavenumber and p is a parameter of phase difference between the fundamental wave and its subharmonic wave. For each class of subharmonic interactions, a couple of parameters m and p have been deduced from the phenomenological diagram. We think that they are significant for accurate flow control of shear layers.

The quantum state transfer from subharmonic frequency to harmonic frequency based on asymmetrically pumped second harmonic generation in a cavity is investigated theoretically. The performance of noise-free frequency up-conversion is evaluated by the signal transfer coefficient and the conversion efficiency, in which both the quadrature fluctuation and the average photon number are taken into consideration. It is shown that the quantum property can be preserved during frequency up-conversion via operating the cavity far below the threshold. The dependences of the transfer coefficient and the conversion efficiency on pump parameter, analysing frequency, and cavity extra loss are also discussed.

In our previous study, we proposed a “triggered high-intensity focused ultrasound (HIFU)” exposure sequence, in which a high-intensity triggering pulse was irradiated just before CW heating waves, to accelerate HIFU treatment. In this study, the second-harmonic was superimposed to the triggering pulse, and the cavitation threshold was investigated by comparing different schemes of second-harmonic superimposition including the case of single-frequency exposure through detecting the subharmonic emission and high-speed camera observation. The experimental results show that cavitation bubbles can be generated most efficiently by using negative-peak-pressure emphasized waves followed by positive-peak-pressure emphasized waves.

Nonlinear ultrasonics plays an important role in the detection of closed cracks in critical structures and machines, such as atomic power plants and airplanes. Although the design and operation of nonlinear measurement require realistic modeling and simulation in two dimensions (2D), only one-dimensional modeling has succeeded in reproducing subharmonic generation. In this study, we propose 2D analysis using a damped double nodes (DDNs) with compression residual stress in the finite-difference time-domain method. We then succeeded in reproducing the essential feature of observed subharmonic generation at closed stress corrosion cracks.

It is shown that the wavelet regression detrended fluctuations of the monthly sunspot number for 1749–2009 years exhibit strong periodicity with a period approximately equal to 3.7 years. The wavelet regression method detrends the data from the approximately 11-years period. Therefore, it is suggested that the one-third subharmonic resonance can be considered as a background for the 11-years solar cycle. It is also shown that the broad-band part of the wavelet regression detrended fluctuations spectrum exhibits an exponential decay that, together with the positive largest Lyapunov exponent, are the hallmarks of chaos. Using a complex-time analytic approach the rate of the exponential decay of the broad-band part of the spectrum has been theoretically related to the Carrington solar rotation period.

This paper gives an overview of the simple yet fundamental bouncing ball problem, which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity. The dynamics is modeled on the basis of a discrete map of difference equations, which numerically solved fully reveals a rich variety of nonlinear behaviors, encompassing irregular non-periodic orbits, subharmonic and chaotic motions, chattering mechanisms, and also unbounded non-periodic orbits.

We formed a stress corrosion crack (SCC) more than 10 mm deep to simulate those generated in atomic power plants. We precisely imaged the formed SCC by a novel imaging method, namely, the subharmonic phased array for crack evaluation (SPACE), and found that it had complex branches. Subsequently, we cut the specimen for the optical observation of the cross sections, and discussed the origin of the SCC extension on the basis of the optical microscopic observation of the cross sections. To examine the open and closed parts of crack in the optical images, we superposed the crack extracted from the optical images onto the SPACE images. We compared the optically and SPACE-measured crack depths, and demonstrated that SPACE is useful in reducing the underestimation of closed-crack depths.