Nonlinear resonances phenomena in a modified Josephson junction model

In this paper,the equivalent circuit of the non-autonomous Josephson junction(JJ)is presented and the effect of the proper frequency on the phaseφis studied.We also study nonlinear resonance phenomena in the oscillations of a modified Josephson junction(MJJ).These oscillations are probed through a system of nonlinear differential equations and the multiple time scale method is employed to investigate all different types of resonance that occur.The results of primary,superharmonic and subharmonic resonances are obtained analytically.We show that the system exhibits hardening and softening behaviors,as well as hysteresis and amplitude hopping phenomena in primary and superharmonic resonances,and only the hysteresis phenomenon in subharmonic resonance.In addition,the stabilities and the steady state solutions in each type of resonances are kindly evaluated.The number of equilibrium points that evolve with time and their stabilities are also studied.Finally,the equations of motion are numerically integrated to check the correctness of analytical calculations.We further show that the dynamics of the MJJ is strongly influenced by its parameters.

Nonlinear resonance characteristics of a dual-rotor system with a local defect on the inner ring of the inter-shaft bearing

In this paper,the nonlinear resonance characteristics of a dual-rotor system are investigated with the consideration of a local defect on the inter-shaft bearing of the system.A simplified model of the dual-rotor system is proposed by considering that there is a local defect on the inner ring of inter-shaft bearing.The local defect is modelled as an inverted isosceles trapezoidal groove,which can make great influence on the inter-shaft bearing force due to the change of radial clearance of the inter-shaft bearing.The motion equations of the dual-rotor system are formulated by using the Lagrange equation.The Runge-Kutta method is employed to solve the motion equation.The amplitude-frequency response curve of the dual-rotor system is obtained,the abnormal resonance characteristics are analyzed.In addition,the influence of defect parameters,rotors and support parameters and inter-shaft bearing parameters on the amplitude-frequency characteristics of the system are discussed.The results show that there are two main resonance peaks and four abnormal resonance peaks on the amplitude-frequency response curve of the dual-rotor with a local defect on the inner ring of the inter-shaft bearing.Through analyzing the vibration response of the abnormal resonance peaks,it is found that the first two abnormal resonances are caused by the combined resonance,which are related to the inner ring fault and the rotational speed of high or low pressure rotors,and the last two resonances are caused by the induced resonance of the inner ring fault.At the same time,when the parameters of defect,rotors and support and inter-shaft bearing change,the resonance of the system also shows the corresponding change law.

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.

Resonance Rayleigh Scattering and Resonance Nonlinear Scattering Methods for Determination of Methylene Blue with 12-Tungstophosphoric Acid

In a pH=0.65―1.5 NaAc-HCl medium, methylene blue（MB） reacts with 12-tungstophosphoric acid （TPA） by virtue of electrostatic attraction and hydrophobic force to form a 3：2 ion-association complex. As a result, the intensities of resonance Rayleigh scattering（RRS）, second-order scattering（SOS） and frequency doubling scatte- ring（FDS） are enhanced greatly. The maximum scattering wavelengths of RRS, SOS and FDS are located at 316, 647 and 311 nm. The increments of scattering intensity（△I） are directly proportional to the concentration of MB in a certain range. The methods exhibited high sensitivity, and the detection limits（3s） for MB are 2.3 ng/mL（RRS method）, 5.6 ng/mL（SOS method） and 6.4 ng/mL（FDS method）, respectively. The effects of coexisting substances have been examined, and the results indicate that the methods have good selectivity. Based on the above researches, a new spectral method for the determination of trace amounts of MB has been developed. It can be applied to the determination of MB in human serum, and the recoveries are 97.5%―105.0%. The results are in good agreement with those obtained by the pharmacopoeia method. In this work, the optimum conditions of the reaction and the influencing factors were investigated. In addition, the reaction mechanism and the reasons of the enhancement of resonance light scattering were discussed.

MULTIPLE SOLUTIONS FOR HAMILTONIAN SYSTEMS WITH PERIODIC NONLINEARITY AND STRONG RESONANCE

I. Introduction In this paper we are looking for solutions of the following Hamiltonian system of second order: where x= （x1, x2） and V satisfies （V. 1） V: R×R2→R is a C1-function, 1-periodic In t, （V.2） V is periodic in x1 with the period T>0, （V. 3） V→O, Vx→O as |x2|→∞, uniformly in （t, x1）.

SOME EXTENDED RESULTS OF“SUBHARMONIC RESONANCE BIFURCATION THEORY OF NONLINEAR MATHIEU EQUATION”

The authors of [1] discussed the subharmonic resonance bifurcation theory of nonlinear Mathieu equation and obtained six bifurcation diagrams in -plane. In this paper, we extended the results of[1] and pointed out that there may exist as many as fourteen bifurcation diagrams which are not topologically equivalent to each other.

Simulation of nonlinear behavior in an electron cyclotron resonance plasma

Some nonlinear behavior in electron cyclotron resonance plasma was investigated using a two-dimension hybrid-mode with self-consistent microwave absorption. The saturation,oscillations of plasma parameters (plasma density, potential, electron temperature) versus operating conditions (pressure, power) are discussed. Our simulation results are consistent qualitatively with many experimental measurements.

Third-Order Nonlinear Optical Response near the Plasmon Resonance Band of Cu2-xSe Nanocrystals

The third-order nonlinear optical properties of water-soluble Cu Se nanocrystals are studied in the near infrared range of 700-980 nm using a femtosecond pulsed laser by the Z-scan technique. It is observed that the nonlinear optical response of Cu Se nanocrystals is sensitively dependent on the excitation wavelength and exhibits the enhanced nonlinearity compared with other selenides such as ZnSe and CdSe. The W-shaped Z-scan trace, a mixture of the reversed saturated absorption and saturated absorption, is observed near the plasmon resonance band of Cu Se nanocrystals, which is attributed to the state-filling of free carriers generated by copper vacancies (self-doping effect) of Cu Se nanocrystals as well as the hot carrier thermal effect upon intense femtosecond laser excitation. The large nonlinear optical response and tunable plasmonic band make Cu Se nanocrystals promising materials for applications in ultra-fast all-optical switching devices as well as nonlinear nanosensors.

Couple stress-based nonlinear primary resonant dynamics of FGM composite truncated conical microshells integrated with magnetostrictive layers

The size-dependent geometrically nonlinear harmonically soft excited oscillation of composite truncated conical microshells(CTCMs)made of functionally graded materials(FGMs)integrated with magnetostrictive layers is analyzed.It is supposed that the FGM CTCMs are subjected to mechanical soft excitations together with external magnetic fields.An analytical framework is created by a microstructuredependent shell model having the 3rd-order distribution of shear deformation based on the modified couple stress(MCS)continuum elasticity.With the aid of the discretized form of differential operators developed via the generalized differential quadrature technique,a numerical solution methodology is introduced for obtaining the couple stress-based amplitude and frequency responses related to the primary resonant dynamics of the FGM CTCMs.Jump phenomena due to the loss of the first stability branch and falling down to the lower stable branch can be seen in the nonlinear primary resonance of the FGM CTCMs.It is demonstrated that the hardening type of nonlinearity results in bending the frequency response to the right side,and the MCS type of size effect weakens this pattern.Moreover,for higher material gradient indexes,the hardening type of nonlinearity is enhanced,and the MCS-based frequency response bends more considerably to the right side.

ON THE RESONANT GENERATION OF WEAKLY NONLINEAR STOKES WAVES IN REGIONS WITH FAST VARYING TOPOGRAPHY AND FREE SURFACE CURRENT

The effect of nonlinearity on the free surface wave resonated by an incident flow over rippled beds, which consist of fast varying topography superimposed on an otherwise slowly varying mean depth, is studied using a WKBJ-type perturbation approach. Synchronous, superharmonic and in particular subharmonic resonance were selectively excited over the fast varying topography with corresponding wavelengths. For a steady current the dynamical system is autonomous and the possible nonlinear steady states and their stability were investigated. When the current has a small oscillatory component the dynamical system becomes non-autonomous, chaos is now possible.

Bubble nonlinear dynamics and stimulated scattering process

A complete understanding of the bubble dynamics is deemed necessary in order to achieve their full potential applications in industry and medicine. For this purpose it is first needed to expand our knowledge of a single bubble behavior under different possible conditions including the frequency and pressure variations of the sound field. In addition, stimulated scattering of sound on a bubble is a special effect in sound field, and its characteristics are associated with bubble oscillation mode. A bubble in liquid can be considered as a representative example of nonlinear dynamical system theory with its resonance, and its dynamics characteristics can be described by the Keller–Miksis equation. The nonlinear dynamics of an acoustically excited gas bubble in water is investigated by using theoretical and numerical analysis methods. Our results show its strongly nonlinear behavior with respect to the pressure amplitude and excitation frequency as the control parameters, and give an intuitive insight into stimulated sound scattering on a bubble. It is seen that the stimulated sound scattering is different from common dynamical behaviors, such as bifurcation and chaos, which is the result of the nonlinear resonance of a bubble under the excitation of a high amplitude acoustic sound wave essentially. The numerical analysis results show that the threshold of stimulated sound scattering is smaller than those of bifurcation and chaos in the common condition.

Nonlinear energy harvesting from vibratory disc-shaped piezoelectric laminates

Implementing resonators with geometrical nonlinearities in vibrational energy harvesting systems leads to considerable enhancement of their operational bandwidths. This advantage of nonlinear devices in comparison to their linear counterparts is much more obvious especially at small-scale where transition to nonlinear regime of vibration occurs at moderately small amplitudes of the base excitation. In this paper the nonlinear behavior of a disc-shaped piezoelectric laminated harvester considering midplane-stretching effect is investigated. Extended Hamilton’s principle is exploited to extract electromechanically coupled governing partial differential equations of the system. The equations are firstly order-reduced and then analytically solved implementing perturbation method of multiple scales. A nonlinear finite element method(FEM) simulation of the system is performed additionally for the purpose of verification which shows agreement with the analytical solution to a large extent. The frequency response of the output power at primary resonance of the harvester is calculated to investigate the effect of nonlinearity on the system performance. Effect of various parameters including mechanical quality factor, external load impedance and base excitation amplitude on the behavior of the system are studied. Findings indicate that in the nonlinear regime both output power and operational bandwidth of the harvester will be enhanced by increasing the mechanical quality factor which can be considered as a significant advantage in comparison to linear harvesters in which these two factors vary in opposite ways as quality factor is changed.

Influence of surface micro-beams with large deflection on the resonance frequency of a quartz crystal resonator in thickness-shear mode vibrations

We study the dynamic behavior of a quartz crystal resonator （QCR） in thickness-shear vibrations with the upper surface covered by an array of micro-beams （MBs） under large deflection. Through taking into account the continuous conditions of shear force and bending moment at the interface of MBs/resonator, dependences of frequency shift of the compound QCR system versus material parameter and geometrical parameter are illustrated in detail for nonlinear and linear vibrations. It is found that the frequency shift produces a little right （left） translation for increasing elastic modulus （length/radius ratio） of MBs. Moreover, the frequency right （left） translation distance caused by nonlinear deformation becomes more serious in the second-order mode than in the first-order one,

Nonlinear dynamic response of beam and its application in nanomechanical resonator

Nonlinear dynamic response of nanomechanical resonator is of very important characteristics in its application. Two categories of the tension-dominant and curvaturedominant nonlinearities are analyzed. The dynamic nonlinearity of four beam structures of nanomechanical resonator is quantitatively studied via a dimensional analysis approach. The dimensional analysis shows that for the nanomechanical resonator of tension-dominant nonlinearity, its dynamic nonlinearity decreases monotonically with increasing axial loading and increases monotonically with the increasing aspect ratio of length to thickness; the dynamic nonlinearity can only result in the hardening effects. However, for the nanomechanical resonator of the curvature-dominant nonlinearity, its dynamic nonlinearity is only dependent on axial loading. Compared with the tension-dominant nonlinearity, the curvature-dominant nonlinearity increases monotonically with increasing axial loading; its dynamic nonlinearity

Nonresonant and Resonant Nonlinear Absorption of CdSe-Based Nanoplatelets

We present a comprehensive understanding of the nonHneer absorption characteristics of CdSe- based nanoplatelets （NPLs） synthesized by the solution-phase method and the colloidal atomic layer deposition approach through Z- scan techniques at 532nm with picosecond pulses. The CdSe NPLs exhibit strong two-photon induced free carrier absorption （effective three-photon absorption） upon the nonresonant excitation, resulting in a remarkable optical limiting behavior with the limiting threshold of approximately 75 GW/cm2. A nonlinear optical switching from saturable absorption （SA） to reverse saturable absorption （RSA） with increasing the laser intensity is observed when coating CdSe NPLs with a monolayer of CdS shell to realize the resonant absorption. The SA behavior originates from the ground state bleaching and the RSA behavior is attributed to the free carrier absorption. These findings explicitly demonstrate the potential applications of CdSe-based NPLs in nonlinear optoelectronics such as optical limiting devices, optical pulse compressors and optical switching devices.

Relating Some Nonlinear Systems to a Cold Plasma Magnetoacoustic System

Using a Gurevich-Krylov solution that describes the propagation of nonlinear magnetoacoustic waves in a cold plasma, we construct solutions of various other nonlinear systems. These include, for example, Madelung fluid, reaction diffusion, Broer-Kaup, Boussinesq, and Hamilton-Jacobi-Bellman systems. We also construct dilaton field solutions for a Jackiw-Teitelboim black hole with a negative cosmological constant. The black hole metric corresponds to a cold plasma metric by way of a change of variables, and the plasma dilatons and cosmological constant also have an expression in terms of parameters occurring in the Gurevich-Krylov solution. A dispersion relation, moreover, links the magnetoacoustic system and a resonance nonlinear Schrödinger equation.

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and transverse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic.This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a primary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.

Synthesis of Poly[(3-octanoylpyrrole-2,5-diyl)-p-(N,N-dimethylamino)benzylidene] and Its Properties by Nitrogen Ion Implantation

A novel soluble π-conjugated polymer, poly[(3-octanoylpyrrole-2,5-diyl)-p-(N,N-dimethylamino)benzylidene](POPDMABE), was synthesized firstly by the condensation of 3-octanoylpyrrole with para-dimethylaminobenzaldehyde. The chemical structure of the polymer was characterized by FTIR and 1H NMR spectrometries. The polymer is a potential nonlinear optical(NLO) material. According to the function of optical forbidden band gap(E_g) and photon energy(hν), the optical forbidden band gaps of the polymer before and after ion implantation were calculated. The resonant third-order nonlinear optical properties of POPDMABE before and after ion implantation were also studied by using the degenerate four-wave mixing(DFWM) technique at 532 nm. When the energy is 25 keV and the dose is 2.2×10 17 ions/cm 2, the {polymer′s} optical forbidden band gap is about 1.63 eV which is smaller than that of the non-implanted sample(1.98 eV) and the resonant third-order NLO susceptibility of POPDMABE is about 4.3×10 -7 esu, 1 order of magnitude higher than that of the non-implanted sample(4.1×10 -8 esu). The results show that nitrogen ion implantation is an effective method to improve the resonant third-order NLO property of the polymer.

Complete crossing of Fano resonances in an optical microcavity via nonlinear tuning

We report on the modeling, simulation, and experimental demonstration of complete mode crossings of Fano resonances within chip-integrated microresonators. The continuous reshaping of resonant line shapes is achieved via nonlinear thermo-optical tuning when the cavity-coupled optical pump is partially absorbed by the material.The locally generated heat then produces a thermal field, which influences the spatially overlapping optical modes, allowing us to alter the relative spectral separation of resonances. Furthermore, we exploit such tunability to continuously probe the coupling between different families of quasi-degenerate modes that exhibit asymmetric Fano interactions. As a particular case, we demonstrate a complete disappearance of one of the modal features in the transmission spectrum as predicted by Fano [Phys. Rev. 124, 1866(1961)]. The phenomenon is modeled as a third-order nonlinearity with a spatial distribution that depends on the stored optical field and thermal diffusion within the resonator. The performed nonlinear numerical simulations are in excellent agreement with the experimental results, which confirm the validity of the developed theory.

Semilinear Elliptic Resonant Problems at Higher Eigenvalue with Unbounded Nonlinear Terms

In this paper we study the existence of nontrivial solutions of a class of asymptotically linear elliptic resonant problems at higher eigenvalues with the nonlinear term which may be un- bounded by making use of the Morse theory for a C^2-function at both isolated critical point and infinity.