Three-dimensional chaotic autonomous van der pol–duffing type oscillator and its fractional-order form

In this paper, a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator is proposed. The three-dimensional autonomous VdPD oscillator is obtained by replacing the external periodic drive source of two-dimensional chaotic nonautonomous VdPD type oscillator by a direct positive feedback loop. By analyzing the stability of the equilibrium points, the existence of Hopf bifurcation is established. The dynamical properties of proposed three-dimensional autonomous VdPD type oscillator is investigated showing that for a suitable choice of the parameters, it can exhibit periodic behaviors, chaotic behaviors and coexistence between periodic and chaotic attractors. Moreover, the physical existence of the chaotic behavior and coexisting attractors found in three-dimensional proposed autonomous VdPD type oscillator is verified by using Orcard-PSpice software. A good qualitative agreement is shown between the numerical simulations and Orcard-PSpice results. In addition, fractional-order chaotic three-dimensional proposed autonomous VdPD type oscillator is studied. The lowest order of the commensurate form of this oscillator to exhibit chaotic behavior is found to be 2.979. The stability analysis of the controlled fractional-order-form of proposed three-dimensional autonomous VdPD type oscillator at its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Finally, an example of observer-based synchronization applied to unidirectional coupled identical proposed chaotic fractional-order oscillator is illustrated. It is shown that synchronization can be achieved for appropriate coupling strength.     

Spectral characterization of hydrogen-like atoms confined by oscillating systems

The spectral characterization of Coulomb systems confined by a homogeneous pseudo-Gaussian oscillator is investigated. This is done using the efficient computational method of generating functionals. Also, this method is used for the spectral characterization of homogeneous harmonic oscillator confinement, treated as a particular case of pseudo-Gaussian oscillator confinement. Finally, confinement by an impenetrable sphere of finite radius is considered by studying its conjugate effect along with a harmonic oscillator.     

An efficient approach to solving fractional Van der Pol–Duffing jerk oscillator

The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator, involving fractional-order by the simplest method. An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach. The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order. A detailed solving process is given for the obtained oscillator with the traditional system.     

The relativistic bound states for a new ring-shaped harmonic oscillator

In this paper a new ring-shaped harmonic oscillator for spin 1/2 particles is studied, and the corresponding eigenfunctions and eigenenergies are obtained by solving the Dirac equation with equal mixture of vector and scalar potentials. Several particular cases such as the ring-shaped non-spherical harmonic oscillator, the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.     

Pseudospin symmetry and the relativistic ring-shaped non-spherical harmonic oscillator

A generalized relativistic harmonic oscillator for spin 1/2 particles is studied. The eigenfunctions and eigenenergies are obtained for the ring-shaped non-spherical harmonic oscillator by solving Dirac equation with equal mixture of vector and scalar potentials in opposite signs, for which pseudospin symmetry is exact. Several particular cases such as the ring-shaped harmonic oscillator, non-spherical harmonic oscillator, and spherical harmonic oscillator are also discussed.     

Inversion in harmonic noise driven bistable oscillators

We consider the dynamics in a asymmetrical bistable oscillator driven externally by a noisy harmonic oscillator. The basic for our approach is the energy dynamics in each well of the oscillator in the low friction limit. We discuss the possibility of energy inversion states generated by harmonic noise and show this effect by simulations.     

新型高功率虚阴极径向反射速调管振荡器

 提出了一种新型的高功率虚阴极径向反射速调管振荡器,它结合了虚阴极振荡器容易起振和速调管微波产生效率较高的特点。利用虚阴极反射电子束对调制腔的正反馈,可以减小起振电流和起振时间,而且提高了微波产生效率。它是一种结构简单、紧凑的器件。用2.5维PIC程序对这种器件进行了数值模拟研究。得到的数值模拟结果表明,输入电压620 kV,电流25 kA,输出微波周期平均功率为2.5 GW。虚阴极振荡频率被锁定,频率为1.25 GHz。     

Power Series Expansion of Propagator for Path Integral and Its Applications

In this paper we obtain a propagator of path integral for a harmonic oscillator and a driven harmonic oscillator by using the power series expansion. It is shown that our result for the harmonic oscillator is more exact than the previous one obtained with other approximation methods. By using the same method, we obtain a propagator of path integral for the driven harmonic oscillator, which does not have any exact expansion. The more exact propagators may improve the path integral results for these systems.     

On asymptotics of the solution of the moving oscillator problem

Asymptotic behavior of the solution of the moving oscillator problem is examined for large and small values of the spring stiffness for the general case of non-zero beam initial conditions. In the limiting case of infinite spring stiffness, it is shown that the moving oscillator problem for a simply supported beam is not equivalent, in a strict sense, to the moving mass problem. The two problems are shown to be equivalent in terms of the beam displacements but are not equivalent in terms of stresses (the higher order derivatives of the two solutions differ). In the general case, the force acting on the beam from the oscillator is shown to contain a high-frequency component , which does not vanish and can even grow when the spring coefficient tends to infinity. The magnitude of this force and its dependence on the oscillator parameters can be estimated by considering the asymptotics of the solution for the initial stage of the oscillator motion. It is shown that, for the case of a simply supported beam, the magnitude of the high-frequency force depends linearly on the oscillator eigenfrequency and velocity. The deficiency of the moving mass model is principally that it fails to predict stresses in the supporting structure. For small values of the spring stiffness, the moving oscillator problem is shown to be equivalent to the moving force problem. The discussion is amply illustrated by results of numerical experiments.     

高功率微波振荡器的相位控制

基于相对论返波管振荡器,提出了一种小信号牵引相位控制的方法,通过外加小信号对振荡器起振过程的引导,实现对输出微波的相位控制。相比于传统的方法,该方法可以利用ns量级的脉冲在较宽的带宽和较低的注入功率下实现对高功率微波振荡器的相位控制。在X波段相对论返波管振荡器相位控制实验中,利用百kW级的小信号,实现了对GW级高功率微波的输出相位控制,相位抖动小于±15°。     

Generation of pulsed squeezed light in a mode-locked optical parametric oscillator

The generation of pulsed squeezed light using an optical parametric oscillator (OPO) is discussed. This mode-locked optical parametric oscillator consists of a nonlinear crystal in a cavity which is resonant for both signal and idler waves and which is synchronously pumped by the second-harmonic of an acousto-optically mode-locked cw Nd: YAG laser. The fundamental wavelength of the pump laser provides local oscillator pulses for balanced homodyne detection of squeezed vacuum pulses emitted by the oscillator when operated below oscillation threshold. Photocurrent noise reduction to 30% below the classical shot-noise limit is observed, corresponding to squeezing of the field to a level approximately a factor of two below the mean square vacuum noise.     

Injection locking of a diode laser locked to a Zeeman frequency stabilized laser oscillator

We describe the design and performance of an injection-locked diode laser locked to a stabilized, single frequency, unmodulated diode laser. The master oscillator is a grating-tuned, external cavity diode laser which is stabilized on a Doppler free alkali metal resonance transition frequency via Zeeman locking. The master oscillator frequency is shifted by an acousto-optic modulator, which provides optical isolation of the master oscillator laser while tuning of the acousto-optic modulation frequency can also provide frequency offset tuning. The slave laser is a free running diode which is injection-locked by a small fraction of the frequency shifted master oscillator light. Good long- and short-time frequency stability are observed for both the Zeeman-locked master oscillator and the injection-locked slave laser.     

Asymmetry of Coordinate and Velocity in Noisy Oscillator

The velocity variable of a noise oscillator as an internal or external noise is proposed, the spectrum of which is quite different from that of the coordinate variable of the same noise oscillator. The former leads to ballistic diffusion for a free particle in long time limit and the asymptotical results of the system are sensitive to the initial condition. However, diffusion process induced by the coordinate of noise oscillator is a normal one and depends on the initial condition only in the transient time. This allows us to classify two kinds of non-Markovian processes: normal one and strong one, just like the processes induced by the coordinate and the velocity of noise oscillator, respectively. Applying to a correlation ratchet, we have found that the steady current of a particle subjected to the velocity of noise oscillator is opposite to that subjected to its coordinate, thus the former shows greenness and the latter redness.     

Prescriptions for gravitational initial data as judged by a simple radiating model

Several methods of prescribing initial data for gravitational and matter fields, which are intended to eliminate extraneous radiation that is not produced by the matter source, are analysed in a simple exactly soluble radiating model. The model consists of an harmonic oscillator coupled to a scalar field along future light cones of Minkowski space time. In particular we analyze the asymptotic regime of the oscillator and find it is characterized essentially by two distinct decay modes. They differ in the way they behave both in the limit of small coupling constant and in a certain Newtonian limit. As a criterion to select initial data for the field with no extra radiation, we require that these initial data sets should put the oscillator from the start into the asymptotic regime. The underlying hypothesis here is that initial transients result from excitation of the oscillator by incoming radiation. We then see that the requirement of a uniform Newtonian limit leads to unique data for the scalar field for each arbitrary data set for the oscillator. We further find that this unique data set indeed satisfies our criterion.     

Coupling of the relaxation and resonant elements in the autonomous chaotic relaxation oscillator (ACRO)

If a harmonic oscillator is embedded in a relaxation oscillator, the resulting system may behave like an autonomous chaotic relaxation oscillator (ACRO). The discharge transient of the relaxation oscillator excites sinusoidal oscillations in the harmonic oscillator and these sinusoids affect when the next discharge occurs. This can lead to chaotic intervals in the oscillator periods. A simple electronic model of the ACRO is studied over a wide range of parameters using numerical, analytic, and experimental techniques. The dynamics of the ACRO is found to be determined by three parameters: (1) tuning, (2) coupling, and (3) damping. Complex, intermittent outputs can always be inhibited by increasing the damping of the harmonic oscillator. For weak damping, strong coupling yields chaotic periods. With weak damping and weak coupling, complex behavior only occurs if the relaxation oscillator is tuned near a resonance of the harmonic oscillator. A new path to chaos, called a disruption bifurcation, is the source for intermittency in the ACRO. This bifurcation occurs when the amplitude of internal resonances is excited to the degree that existing limit cycles are disrupted.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

Parametric frequency transformation in a superconducting waveguide line with an integrated Josephson oscillator

The parametric frequency division in a coplanar waveguide line with an integrated single-contact rf SQUID (Josephson oscillator) is discussed. It is assumed that the oscillator is excited by pump pulses whose carrier frequency can be a multiple of the plasma frequency of the oscillator. It is shown that the Josephson oscillator excited at the pump frequency can induce frequency division by emitting subharmonics that are multiples of the fundamental frequency (fractional resonances). Parameters for which parametric frequency transformation occurs are determined. The possible generalization of this effect to the quantum case in which correlated microwave photons (entangled photon states) can be generated is discussed.     

Widely tunable photonic crystal fiber Fabry-Perot optical parametric oscillator

We present a widely tunable low-threshold chi(3) optical parametric oscillator. The oscillator cavity is formed by butt coupling dichroic mirrors to either end of a highly nonlinear index-guiding photonic crystal fiber. This yields a singly resonant Fabry-Perot oscillator with a high feedback fraction for the resonant parametric sideband. The tuning range of the output parametric sideband stretches from 23 to 164 THz above the pump frequency. The threshold power of the oscillator is only 15 W.     

过阻尼谐振子的随机共振

研究了由交叉相关高斯白噪声驱动的过阻尼谐振子的随机共振,其中加法噪声被周期信号所调制,运用平稳关联函数的傅里叶变换,导出了过阻尼谐振子随机模型信噪比的精确表达式.结果揭示:在过阻尼谐振子的随机模型中存在二类随机共振.一类随机共振表现为信噪比随乘法噪声强度Q变化的曲线存在共振峰,另一类随机共振表现为信噪比随振子频率ω变化的曲线存在共振峰.大幅度改变信号频率Ω值的大小,信噪比随乘法噪声强度Q变化的曲线有单峰,一峰一谷和单调变化三种不同的形式.     

Simulation Investigation of Millimeter Wave Varactor-tuned Transmission Cavity-stabilized Oscillator

A new circuit configuration for millimeter wave varactor-tuned transmission cavity-stabilized oscillator has been proposed in this paper. Compared to conventional varactor-tuned reflection cavity-stabilized oscillator, in this configuration, a high quality factor transmission cavity directly coupled to varactor diode is employed to improve the performances of the oscillator. The operation frequency of this oscillator can be tuned by varying the resonant frequency of the transmission cavity through changing bias voltage of the varactor diode. An equivalent circuit model for the oscillator has been presented in order to theoretically investigate the performance characteristics of the oscillator. On the basis of this model, electrical tuning characteristics have been studied. Mode jumping phenomena during electrical tuning process have been analyzed for obtaining stable operations of the oscillator. The analytical formulae of quality factor and efficiency have been derived in terms of relevant circuit parameters. Particular emphasis has been paid on several circuit parameters which have a substantial impact on circuit performance. Some design considerations have been pointed out according to the simulation results, which are useful to the design and fabrication of this type of oscillators.