Stochastic self-sustained oscillations of non-autonomous systems

In the present minireview, we analyze autonomous and non-autonomous oscillations of dynamical and stochastic systems in the framework of common concepts. We introduce the definition of an attractor for a non-autonomous system. We also propose the definition of self-sustained oscillations, which can be applied for both autonomous and non-autonomous systems. We consider noise-induced oscillations and formulate the definition of stochastic self-sustained oscillations for this case. All the statements made in this work are illustrated by particular examples.     

Rabi oscillations of optical modes in a waveguide with dynamic modulation

We investigate the optical Rabi oscillations in a dual-mode slab waveguide that undergoes a spatial–temporal refractive index modulation. Frequency conversion is induced during Rabi oscillations since dynamic modulation is employed. We also show that the contrast of Rabi oscillations can be controlled by the initial phase of dynamic modulation, which can be tuned arbitrarily from zero to unity as the phase varies. The contrast of zero corresponds to the dynamic supermodes and unity refers to complete Rabi oscillations. It suggests that the phase can be a new degree of freedom to control optical Rabi oscillations. This study may find applications in optical sensors, switches and mode converters.     

Self-modulation regimes of a phase-locked loop with the second-order filter

We study self-modulation oscillations of a phase-locked loop with the second-order filter. The regions in parameter space in which the modulated oscillations are generated and the properties of modulating oscillations are analyzed. Special attention is paid to studying chaotically modulated oscillations. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 49, No. 4, pp. 357–368, April 2006.     

Quantum corrections to the conductivity and periodic orbits: Integrable systems

We extend the recently developed semiclassical theory for the conductivity to periodic semiconductor structures whose classical dynamics is integrable. We find that the conductivity of integrable systems exhibits quantum oscillations as function of magnetic field and Fermi energy which are closely related to both the Shubnikov-de Haas oscillations and the oscillations observed in the conductivity of antidot lattices. A general expression for the quantum oscillations is derived which is analogous to the Berry-Tabor formula for the spectral density of integrable systems.     

Effect of the lattice alignment on Bloch oscillations of a Bose-Einstein condensate in a square optical lattice

We consider a Bose-Einstein condensate of ultracold atoms loaded into a square optical lattice and subject to a static force. For vanishing atom-atom interactions the atoms perform periodic Bloch oscillations for arbitrary direction of the force. We study stability of these oscillations for non-vanishing interactions, which is shown to depend on an alignment of the force vector with respect to the lattice crystallographic axes. If the force is aligned along any of the axes, the mean field approach can be used to identify the stability conditions. On the contrary, for a misaligned force one has to employ the microscopic approach, which predicts periodic modulation of Bloch oscillations in the limit of a large forcing.     

Magnetomotive cooling and excitation of carbon nanotube oscillations under voltage bias

We explore a semi-classical scheme for cooling and excitation of mechanical oscillations of a suspended carbon nanotube which is incorporated as a deflection-sensitive resistor in an LRC-circuit. The active feedback consists of a magnetically induced Lorentz force. We show that, for feasible experimental parameters, we can obtain self-sustained oscillations or cooling by several orders of magnitude.     

Temperature effects on the magnetoplasmon spectrum of a weakly modulated graphene monolayer

In this work, we determine the effects of temperature on the magnetoplasmon spectrum of an electrically modulated graphene monolayer as well as a two-dimensional electron gas (2DEG). The intra-Landau band magnetoplasmon spectrum within the self-consistent field approach is investigated for both the aforementioned systems. Results obtained not only exhibit Shubnikov-de Haas (SdH) oscillations but also commensurability oscillations (Weiss oscillations). These oscillations are periodic as a function of inverse magnetic field. We find that both the magnetic oscillations, SdH and Weiss, have a greater amplitude and are more robust against temperature in graphene compared to a conventional 2DEG. Furthermore, there is a π phase shift between the magnetoplasmon oscillations in the two systems which can be attributed to Dirac electrons in graphene acquiring a Berry's phase as they traverse a closed path in a magnetic field.     

Current to frequency conversion in a Josephson circuit

The voltage oscillations which occur in an ideally current-biased Josephson junction were proposed to make a current standard for metrology. We demonstrate similar oscillations in a more complex Josephson circuit derived from the Cooper pair box: the quantronium. When a constant current I is injected in the gate capacitor of this device, oscillations develop at the frequency f(B)=I/2e, with e the electron charge. We detect these oscillations through the sidebands induced at multiples of f(B) in the spectrum of a microwave signal reflected on the circuit, up to currents I exceeding 100 pA. We discuss the potential interest of this current-to-frequency conversion experiment for metrology.     

Quantum dynamics of a mechanical resonator driven by a cavity

We explore the quantum dynamics of a mechanical resonator whose position is coupled to the frequency of an optical (or microwave) cavity mode. When the cavity is driven at a frequency above resonance the mechanical resonator can gain energy and for sufficiently strong coupling this results in limit-cycle oscillations. Using a truncated Wigner function approach, which captures the zero-point fluctuations in the system, we develop an approximate analytic treatment of the resonator dynamics in the limit-cycle regime. We find that the limit-cycle oscillations produced by the cavity are associated with rather low levels of energy fluctuations in the resonator. Compared to a resonator at the same temperature which is driven by a pure harmonic drive to a given average energy, the cavity-driven oscillations can have much lower energy fluctuations. Furthermore, at sufficiently low temperatures, the cavity can drive the resonator into a non-classical state which is number-squeezed.     

Quantization of plasma density irregularities under the action of a powerful electromagnetic wave: The spectrum of upper hybrid oscillations self-consistently trapped in density cavities

We present a theoretical model of the self-localization of the upper hybrid (UH) oscillations in plasma density depletions due to thermal nonlinearities driven by a homogeneous and monochromatic pump electric field. The Bohr-Sommerfeld condition for the trapped UH oscillations demands that the parameters of the density cavity be quantized. The depth and square of the depletion width across the magnetic field is proportional to an integer. The depth of the parabolically shaped cavity is proportional to the square of its width. The characteristic relative value of the density minimum is a few percent and the width is of the order of one meter for the pump wave amplitudes used in the ionospheric F-region experiments. We consider also the parametric decay of primary, localized UH oscillations trapped in the quantized plasma density depletions into secondary UH oscillations and lower-hybrid waves. We calculated the spectrum of the non-linear stabilized secondary UH oscillations which are also self-consistently trapped in the same density cavity. The spectrum of the UH oscillations is consistent with the observed spectrum of the downshifted (DM) and upshifted (UM) maximum in the stimulated electromagnetic emissions (SEE). Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 7, pp. 641–650, July 1999.     

Experimental characterization of DFB and FP chaotic lasers with strong incoherent optical feedback

We have experimentally investigated chaotic power oscillations in single-longitudinal mode DFB and multi-longitudinal mode FP lasers as a function of incoherent optical feedback strengths of up to 42%. We have demonstrated the existence of chaos in the output oscillations of both laser types using classical experimental tools such as RF spectrum, standard deviation, and maximum Lyapunov exponent, which all increase with increasing of feedback strength for both in single-longitudinal mode DFB lasers and multi-longitudinal mode FP lasers. It is also shown that power switching among longitudinal modes of multimode FP semiconductor laser is a considerable portion of the chaotic power oscillations for both strong and weak incoherent optical feedback.     

Interaction-induced beats of Friedel oscillations in quantum wires

We analyze the spectrum of electron density oscillations in an interacting one-dimensional electron system with an impurity. The system's inhomogeneity is characterized by different values of Fermi wave vectors kF=k L/R on left or right side of the scatterer, leading to a Landauer dipole formation. We demonstrate, that while in the noninteracting system the Friedel oscillations possess only one periodicity related to the local kF, say kL on the left side, the interplay of the interactions and the Landauer dipole generates an additional peak in the spectrum of density oscillations at the counterpart kR. Being only present in correlated systems, the position and shape of this spectral feature, which in coordinate space is observable as a beating pattern in the Friedel oscillations, reveals many important details about the nature of interactions. Thus it has a potential to become an investigation tool in condensed matter physics.     

Low frequency resonant dispersion of sound in bubble media

Presented is a theory of a new type of resonant dispersion of sound in a gas-liquid bubble media based on the use of effective dynamic density. We show that because of spheroidal-translational oscillations of the bubbles the dynamic density of the gas-liquid media has a resonance dependence on the frequency, which is manifested in wave process as a low-frequency resonant dispersion of sound. This dispersion is significantly different from the known high-frequency resonant dispersion, which is due to resonance in the volume oscillations of the bubbles. The results of the experiments confirming the existence of the resonant dispersion of sound at a frequency equal to half the natural frequency of the spheroidal oscillations of bubbles are provided.     

Acoustic analogue of electronic BLOCH oscillations and resonant Zener tunneling in ultrasonic superlattices

We demonstrate the existence of Bloch oscillations of acoustic fields in sound propagation through a superlattice of water cavities and layers of methyl methacrylate. To obtain the acoustic equivalent of a Wannier-Stark ladder, we employ a set of cavities with different thicknesses. Bloch oscillations are observed as time-resolved oscillations of transmission in a direct analogy to electronic Bloch oscillations in biased semiconductor superlattices. Moreover, for a particular gradient of cavity thicknesses, an overlap of two acoustic minibands occurs, which results in resonant Zener-like transmission enhancement.     

Glory oscillations in the index of refraction for matter waves

We have measured the index of refraction for sodium de Broglie waves in gases of Ar, Kr, Xe, and N2 over a wide range of sodium velocities. We observe glory oscillations--a velocity-dependent oscillation in the forward scattering amplitude. An atom interferometer was used to observe glory oscillations in the phase shift caused by the collision, which are larger than glory oscillations observed in the cross section. The glory oscillations depend sensitively on the shape of the interatomic potential, allowing us to discriminate among various predictions for these potentials, none of which completely agrees with our measurements.     

Chaotic oscillations in a map-based model of neural activity

We propose a discrete time dynamical system (a map) as a phenomenological model of excitable and spiking-bursting neurons. The model is a discontinuous two-dimensional map. We find conditions under which this map has an invariant region on the phase plane, containing a chaotic attractor. This attractor creates chaotic spiking-bursting oscillations of the model. We also show various regimes of other neural activities (subthreshold oscillations, phasic spiking, etc.) derived from the proposed model.     

The Evolution of Acoustic Radiation by an Ensemble of Vortex Rings in Air

The evolution of acoustic radiation emitted by an ensemble of vortex rings in air is studied on the basis of nonstationary Navier–Stokes equations. We use the expansions of required functions into a power series of the initial vorticity which is a small value. The Navier–Stokes equation system reduces to a parabolic system with constant coefficients for the higher derivatives. The problem is posed as follows. The vorticity is defined inside the toroid at t = 0. The other parameters of the gas are assumed to be constant throughout the space at the initial instant of time. The solution is expressed in terms of multiple integrals, which are calculated using Korobov grids. The density oscillations were investigated. The results show that the frequency spectrum depends on time; high-frequency oscillations are observed at small times and low-frequency oscillations then occur. At the same time, the amplitude of high-frequency oscillations decreases in comparison with low-frequency oscillations. Thus, a transition of energy from the high-frequency spectrum to the lowfrequency spectrum occurs. These results can be useful for modeling decaying grid turbulence.     

Ultrafast Fiske effect in semiconductor superlattices

The current flowing across a semiconductor superlattice in tilted electric and magnetic fields is known to exhibit resonant enhancement, when Landau states of neighboring wells align at certain ratios of the field strengths. We show that the ultrafast version of this effect, in which coherent electron wave packets are involved, has a profound analogy to the Fiske effect in superconductor Josephson junctions and superfluid weak links, in that the coupling of the tunneling-induced charge oscillations (magneto-Bloch versus Josephson oscillations) to another oscillator (in-plane cyclotron oscillations versus external oscillator modes) opens an elastic rectifying transport channel. We explore the superlattice effect both theoretically and experimentally, and find that the transient self-induced current can be adequately modeled if the damping of both types of coupled electron oscillations is properly taken into account.     

经由脉冲式爆炸连接的复合式张弛振荡

张弛振荡现象普遍存在于自然科学以及工程技术的各个领域,探索张弛振荡的可能路径是张弛振荡研究的重要问题之一.最近,一种名为"脉冲式爆炸"(pulse-shaped explosion,PSE)的可以诱发张弛振荡的新机制被相继报道.PSE意味着平衡点和极限环表现出了与参数变化相关的脉冲式急剧量变,这导致系统出现急剧转迁现象,进而诱发张弛振荡.本文以多频激励Mathieu-van der Pol-Duffing系统为例,探讨了复合式的张弛振荡现象.当参数激励和外部激励存在相位差时,快子系统包含了两个不同的向量场部分,由此得到了系统的双稳定特性.特别地,在狭小的参数范围内,分岔会随着PSE的产生而产生,这使得PSE更具复杂性.基于此,揭示了两种复合式的张弛振荡,其特征是每一周期的演化过程包含了由PSE连接的两个张弛振荡簇.我们的研究深化了对PSE及张弛振荡复杂动力学行为的理解.