Linearization of Intrinsically Nonlinear Oscillators

We show that an intrinsically nonlinear oscillator can always be transformed into a linear or harmonic oscillator by addition of a constant force, which shifts the equilibrium position of the oscillator.

Parameter estimation method for a linear frequency modulation signal with a Duffing oscillator based on frequency periodicity

In view of the complexity of existing linear frequency modulation(LFM)signal parameter estimation methods and the poor antinoise performance and estimation accuracy under a low signal-to-noise ratio(SNR),a parameter estimation method for LFM signals with a Duffing oscillator based on frequency periodicity is proposed in this paper.This method utilizes the characteristic that the output signal of the Duffing oscillator excited by the LFM signal changes periodically with frequency,and the modulation period of the LFM signal is estimated by autocorrelation processing of the output signal of the Duffing oscillator.On this basis,the corresponding relationship between the reference frequency of the frequencyaligned Duffing oscillator and the frequency range of the LFM signal is analyzed by the periodic power spectrum method,and the frequency information of the LFM signal is determined.Simulation results show that this method can achieve high-accuracy parameter estimation for LFM signals at an SNR of-25 dB.

CALCULATING ACCURATE PERIODIC MOTIONS OF HARMONICALLY FORCED PIECEWISE LINEAR OSCILLATORS

This paper presents an efficient numerical scheme for calculating the periodic motion of a harmonically forced piecewise linear oscillator very accurately. The scheme is based on the shooting technique with the traditional numerical Poincare mapping and its Jacobian replaced by the piecewise analytic ones. Thus, the scheme gets rid of the requirement of the current schemes for an assumed order of the oscillator trajectory passing through different linear regions. The numerical examples in the paper demonstrate that the new scheme, compared with the current schemes, enables one to cope with more complicated dynamics of harmonically forced piecewise linear oscillators.

The construction of conserved quantities for linearly coupled oscillators and study of symmetries about the conserved quantities

In this paper, the conserved quantities are constructed using two methods. The first method is by making an ansatz of the conserved quantity and then using the definition of Poisson bracket to obtain the coefficients in the ansatz. The main procedure for the second method is given as follows. Firstly, the coupled terms in Lagrangian are eliminated by changing the coordinate scales and rotating the coordinate axes, secondly, the conserved quantities are obtain in new coordinate directly, and at last, the conserved quantities are expressed in the original coordinates by using the inverse transform of the coordinates. The Noether symmetry and Lie symmetry of the infinitesimal transformations about the conserved quantities are also studied in this paper.

Study of measurements of air velocity transducer deriving average air velocity of roadway based on one-dimensional linear regression

One-dimensional linear regression equation between measured value of air velocity transducer and the average air velocity was established by experimental data. The effect is to be evaluated. Through judging the parameters, one-dimensional linear equation established is valid. Regression equation can approximately put the measurements of air velocity transducer into the value of average air velocity. The distribution of air velocity field is simulated using Comsol in the conditions of the same length of roadway, the same air velocity and different sections.

Modelling of the Vortex-Induced Motion of A Single-Column Platform with Non-Linear Mooring Stiffness Using the Coupled Wake Oscillators

A phenomenological model for predicting the vortex-induced motion (VIM) of a single-column platform with non- linear stiffness has been proposed. The VIM model is based on the couple of the Duffing-van der Pol oscillators and the motion equations with non-linear terms. The model with liner stiffness is presented for comparison and their results are compared with the experiments in order to calibrate the model. The computed results show that the predicted VIM amplitudes and periods of oscillation are in qualitative agreements with the experimental data. Compared with the results with linear stiffness, it is found that the application of non-linear stiffness causes the significant reductions in the in-line and transverse motion amplitudes. Under the non-linear stiffness constraint, the lock-in behavior is still identified at 8<Ur<15, and the trajectories of the VIM on the xy plane with eight-figure patterns are maintained. The results with different non-linear geometrically parameters show that both in-line and transverse non-linear characteristics can significantly affect the predict in-line and transverse motion amplitudes. Furthermore, the computed results for different aspect ratios indicate that the in-line and transverse motion amplitudes increase with the growth of aspect ratio, and the range of lock-in region is enlarged for the large aspect ratio.

Analytical solutions to nonlinear conservative oscillator with fifth-order nonlinearity

This paper describes analytical and numerical methods to analyze the steady state periodic response of an oscillator with symmetric elastic and inertia nonlinearity. A new implementation of the homotopy perturbation method （HPM） and an ancient Chinese method called the max-rain approach are presented to obtain an approximate solution. The major concern is to assess the accuracy of these approximate methods in predicting the system response within a certain range of system parameters by examining their ability to establish an actual （numerical） solution. Therefore, the analytical results are compared with the numerical results to illustrate the effectiveness and convenience of the proposed methods.

Nonstandard Numerical Schemes for a Linear Stochastic Oscillator with Additive Noise

In order to simulate a linear stochastic oscillator with additive noise,improved nonstandard optimal(INSOPT) schemes are derived utilizing the nonstandard finite difference(NSFD)technique and the improvement technique.These proposed schemes reproduce long time features of the oscillator solution exactly.Their abilities in preserving the symplecticity,the linear growth property of the second moment and the oscillation property of the solution of the stochastic oscillator system on long time interval are studied.It can be shown that the component { x_n}_(n≥1) of the INSOPT schemes switch signs infinitely many times as n →∞,almost surely.Further,the mean-square convergence order of 1 is obtained for these INSOPT schemes.Finally,numerical experiments illustrate intuitively the results obtained in this paper.

Decoherence of a Quantum Nonlinear Oscillator Under a Non-zero Temperature Thermal Bath

The characteristic time τD for decoherence process of a quantum nonlinear oscillator system under a nonzero temperature thermal bath is studied by expanding the linear entropy. By numerical analysis, it is shown that at a non-zero temperature, the quantum coherence decays much faster than at zero temperature. Moreover, the non-zero temperature thermal bath will bring a crucial suppression to the quantum effects of the observables, which causes these quantum effects to become unable to persist up to the Ehrenfest time but is insufticient to destroy the quantum-classical transition.

Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator

The SU（1,1） coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU（1,1） coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr Sommerfeld quantization rule yields the exact expression for the energy spectrum.

Time Dependent Gaussian Equivalent Linearization of Duffing Oscillator Using Continuous Wavelet Transform

Evolutionary response analysis of Duffing oscillator using Gaussian equivalent linearization in wavelet based time-frequency frame work is presented here. Cubic （i.e., odd type） non-linearity associated with stiffness and damping is modeled. The goal of this research is to develop the mathematical model of an equivalent linear system which is applicable for different non-stationary input processes （i.e., either summation of amplitude modulated stationary orthogonal processes or digitally simulated non-stationary processes）. The instantaneous parameters of the ELTVS （equivalent linear time varying system） are evaluated by minimizing the error between the displacements of non-linear and equivalent linear systems in wavelet domain. For this purpose, three different basis functions （i.e., Mexican Hat, Morlet and a modified form of Littlewood-Paley） are used. The unknown parameters （i.e., natural frequency and damping） of the ELTVS are optimized in stochastic least square sense. Numerical results are presented for different types of input to show the applicability and accuracy of the proposed wavelet based linearization technique.

Classical and Quantum Behavior of Generalized Oscillators in Terms of Linear Canonical Transformations

The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.

Admissible Periodic Bouncing Solutions for A Class of Semi-linear and Non-conservative Impact Oscillators

In this paper, by using the topological degree method and some limiting arguments, the existence of admissible periodic bouncing solutions for a class of non-conservative semi-linear impact equations is proved.

STABILITY OF VISCOUS SHOCK WAVES FOR THE ONE-DIMENSIONAL COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density- dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel~ and the continuation argument.

Control of friction oscillator by Lyapunov redesign based on delayed state feedback

The stability and boundedness of mechanical system have been one of important research topics. In this paper ultimate boundedness of a dry friction oscillator, belonging to nonsmooth mechanical system, is investigated by proposing a controller design method. Firstly a sufficient condition of the stability for the nominal system with delayed state feedback is derived by constructing a Lyapunov-Krasovskii function. The delayed feedback gain matrix is calculated by applying linear matrix inequality method. Secondly on the basis of the delayed state feedback, a continuous function is designed by Lyapunov redesign to ensure that the solutions of the friction oscillator system are ultimately bounded under the overall control. Moreover, the ultimate bound can be adjusted in practice by choosing appropriate parameter. Accordingly friction-induced vibration or instability can be controlled effectively. Numerical results show that the pro- posed method is valid.

A memristor oscillator based on a twin-T network

A novel inductance-free nonlinear oscillator circuit with a single bifurcation parameter is presented in this paper. This circuit is composed of a twin-T oscillator, a passive RC network, and a flux-controlled memristor. With an increase in the control parameter, the circuit exhibits complicated chaotic behaviors from double periodicity. The dynamic properties of the circuit are demonstrated by means of equilibrium stability, Lyapunov exponent spectra, and bifurcation diagrams. In order to confirm the occurrence of chaotic behavior in the circuit, an analog realization of the piecewise-linear flux-controlled memristor is proposed, and Pspice simulation is conducted on the resulting circuit.

Design of Efficient Tunable Terahertz-Wave Parametric Oscillator

An efficient widely tunable terahertz（THz）-wave parametric oscillator （TPO） has been designed based on the basic principle of optical parametric oscillator. The design of TPO comprises the theoretical calculations for wavelength dependent refractive index dependent wavelength when the frequency is in the far infrared region, the low-loss parametric gain has been discussed as a same form as the parametric gain in the optical region, realized the non-collinear phase-matching conditions of the terahertz optical parametric oscillator and structure of lithium niobate （LiNbO3） for TPO. The tunable spectrum range of terahertz-wave has been achieved. To increase THz output, a cut exit was made at the comer of the LiNbO3 crystal.

ON DOUBLE PEAK PROBABILITY DENSITY FUNCTIONS OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND RANDOM EXCITATIONS

The principal resonance of Duffing random external excitation was investigated. oscillator to combined deterministic and The random excitation was taken to be white noise or harmonic with separable random amplitude and phase. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The one peak probability density function of each of the two stable stationary solutions was calculated by the linearization method. These two one-peak-density functions were combined using the probability of realization of the two stable stationary solutions to obtain the double peak probability density function. The theoretical analysis are verified by numerical results.

Chaotic oscillations in a piecewise linear spring-mass system

Chaotic oscillations are useful in assessing the health of a structure. Hence, simple chaotic systems which can easily be realized mechanically or electro-mechanically are highly desired. We study a new pieeewise linear spring-tnass system. The chaotic behaviour in this system is characterized using bifurcation diagrams and the invariant parameters of the dynamics. We also show that there exists a stochastic analogue of this system, which mimics the dynamical features of its deterministic counterpart. This allows a greater flexibility in practical designs as the chaotic oscillations are obtained either deterministically or stochastically. Also, the oscillations are low dimensional, which reduces the computational resources needed for obtaining the invariant parameters of this system.

One-dimensional Chain Topology in [Ag(C_6H_6NCl)_2](ClO_4), Generated through Weak Interaction between Molecules

The mononuclear complex [Ag(C6H6NCl)2](ClO() has been prepared and structurally analyzed by single-crystal X-ray diffraction. The complex crystallizes in the monoclinic system, space group C2/c with unit cell parameters: a=15.5314(2), b=8.0247(8), c=15.3701(2)?.β=118.832(2)°, V=1678.2(3)?3, Z=4, Mr=462.46, Dc=1.830Mg/m3, F(000)=912, μ(MoKα) = 1.694cm-1. The final R and wR are 0.0472 and 0.1272 for 1484 observed reflections with I≥3σ(I). The Ag atom is coordinated by two nitrogen atoms of 4-chloromethyl-pyridine in a linear coordination geometry. Each molecule is further linked by the weak interaction between the Cl and Ag atoms to form a one-dimensional chain structure with Ag－Cl distance of 3.240?.