Higher Order Harmonic Terms Research Articles

Geometric phase and non-adiabatic resonance of the Rabi model

We investigate the effects of counterrotating terms on geometric phase and its relation to the resonance of the Rabi model. We apply the unitary transformation with a single parameter to the Rabi model and obtain the transformed Hamiltonian involving multiple harmonic terms. By combining the counter-rotating-hybridized rotating-wave method with time-dependent perturbation theory, we solve systematically time evolution operator and then obtain the geometric phase of the two-level system. Our results are beyond adiabatic approximation and rotating-wave approximation (RWA). Higher-order harmonic resonance happens when driving frequency is equal to higher-order subharmonic of the Rabi frequency. In comparison with numerically exact results, our calculated results are accurate over a wide range of parameters space, especially in higher-order harmonic resonance regimes. In these regimes we demonstrate geometric phases change dramatically while those of the RWA are smooth. The RWA is thoroughly invalid even if the driving strength is extremely weak. We find it is the higher-order harmonic terms that play an important role on the cyclic state and demonstrate the characters of geometric phase in higher-order harmonic resonance regime. We also present analytical formalism of the change rate of geometric phase and quasienergies, which agree well with numerically exact ones even in the strong driving case. The developed method can be applied to explore the dynamics of strongly driven qubits and physical properties of higher-order harmonic processes.

Energy Content Characterization of Water Waves Using Linear and Nonlinear Spectral Analysis

Abstract The survivability, safe operation, and design of marine vehicles and wave energy converters are highly dependent on accurate characterization and estimation of the energy content of the ocean wave field. In this study, analytical solutions of the nonlinear Schrödinger equation (NLS) using periodic inverse scattering transformation (IST) and its associated Riemann spectrum are used to obtain the nonlinear wave modes (eigen functions of the nonlinear equation consisting of multiple phase-locked harmonic components). These nonlinear wave modes are used in two approaches to develop a more accurate definition of the energy content. First, in an ad hoc approach, the amplitudes of the nonlinear wave modes are used with a linear energy calculation resulting in a semi-linear energy estimate. Next, a novel, mathematically exact definition of the energy content taking into account the nonlinear effects up to fifth order is introduced in combination with the nonlinear wave modes, the exact energy content of the wave field is computed. Experimental results and numerical simulations were used to compute and analyze the linear, ad hoc, and exact energy contents of the wave field, using both linear and nonlinear spectra. The ratio of the ad hoc and exact energy estimates to the linear energy content was computed to examine the effect of nonlinearity on the energy content. In general, an increasing energy ratio was observed for increasing nonlinearity of the wave field, with larger contributions from higher-order harmonic terms. It was confirmed that the significant increase in nonlinear energy content with respect to its linear counterpart is due to the increase in the number of nonlinear phase-locked (bound wave) modes.2

Modified nonlocal theory for investigation the specific aspects of nonlinear behavior of carbon nanotube as a nano-resonator

Purpose In this paper, the important aspects of vibration analysis of carbon nanotubes (CNTs) as nano-resonators are studied. This study has covered the important nonlinear phenomena such as jump super-harmonic and chaotic behavior. CNT is modeled by using the modified nonlocal theory (MNT). Design/methodology/approach In previous research studies, the effects of CNT’s rotary inertia, stiffness and shear modulus of the medium were neglected. So by considering these terms in MNT, a comprehensive model of vibrational behavior of carbon nanotube as a nanosensor is presented. The nanotube is modeled as a nonlocal nonlinear beam. The first eigenmode of an undamped simply supported beam is used to extract the nonlinear equation of CNT. Harmonic balance method is used to solve the equation, while to study its super-harmonic behavior, higher-order harmonic terms were used. Findings In light of frequency response equation, jump phenomenon and chaotic behavior of the nanotube with respect to the amplitude of excitation are investigated. Also in each section of the study, the effects of elastic medium and nonlocal parameters on the vibration behavior of nanotube are investigated. Furthermore, parts of the results in linear and nonlinear cases were compared with results of other references. Originality/value The present modification of the nonlocal theory is so important and useful for accurate investigation of the vibrational behavior of nano structures such as a nano-resonator.

A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure, and complex two-dimensional (2D) integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude–frequency response surfaces of quasi-periodic responses and avoid nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge–Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

An analytical technique for solving a class of strongly nonlinear conservative systems

Recently, an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms. Usually, a set of nonlinear algebraic equations is solved in this method. However, analytical solutions of these algebraic equations are not always possible, especially in the case of a large oscillation. Previously such algebraic equations for the Duffing equation were solved in powers of a small parameter; but the solutions measure desired results when the amplitude is an order of 1. In this article different parameters of the same nonlinear problems are found, for which the power series produces desired results even for the large oscillation. Moreover, two or three terms of this power series solution measure a good result when the amplitude is an order of 1. Besides these, a suitable truncation formula is found in which the solution measures better results than existing solutions. The method is mainly illustrated by the Duffing oscillator but it is also useful for many other nonlinear problems.

Analytic theory for betatron radiation from relativistic electrons in ion plasma channels with magnetic field

We analytically solve the relativistic equation of motion for an electron in ion plasma channels and calculate the corresponding trajectory as well as the synchrotron radiation. The relativistic effect on a trajectory is strong, i.e., many high-order harmonic terms in the trajectory, when the ratio of the initial transverse velocity (vx0) to the longitudinal velocity (vz0) of the electron injected to ion plasma channels is high. Interestingly, these high-order harmonic terms result in a quite broad and intense radiation spectrum, especially at an oblique angle, in contrast to an earlier understanding. As the initial velocity ratio (vx0:vz0) decreases, the relativistic effect becomes weak; only the first and second harmonic terms remain in the transverse and longitudinal trajectories, respectively, which coincides with the result of Esarey et al. [Phys. Rev. E 65, 056505 (2002)]. Our formalism also allows the description of electron’s trajectory in the presence of an applied magnetic field. Critical magnetic fields for cyclotron motions are figured out and compared with semiclassical results. The cyclotron motion leads to more high-order harmonic terms than the trajectory without magnetic fields and causes an immensely broad spectrum with vastly large radiation amplitude for high initial velocity ratios (vx0:vz0). The radiation from hard x-ray to gamma-ray regions can be generated with a broad radiation angle, thus available for applications.

Relay feedback identification for processes under drift and noisy environments

Relay feedback identification methods are widely used to find the process ultimate information and tune proportional-integral-derivative controllers. The conventional relay feedback method has several disadvantages, which include poor estimates of the process ultimate information for low-order processes, chattering of relay for noisy environments, and asymmetric relay responses for constant biases or slow drifts in the process outputs. Methods to mitigate each of the above disadvantages are available. However, a systematic method to treat all of them has not been studied yet. Here, simple relay feedback methods that resolve these problems by introducing band-pass filters in the feedback loop are proposed. The high-pass filter part in band-pass filter removes a constant bias or low frequency drift, and the low-pass filter part removes high frequency noise and high-order harmonic terms in the relay feedback oscillation, resulting better estimates of the process ultimate information. Because filters used for the proposed methods are able to reject constant biases, the process steady state gains can be estimated without disturbing the relay feedback oscillations and first order plus time delay (FOPTD) models can be obtained by combining the process steady state gains with the relay oscillation information. © 2010 American Institute of Chemical Engineers AIChE J, 2011

Area Method for a Biased Relay Feedback System

The relay feedback system often shows an asymmetric response because of initial transient states, disturbances, and process nonlinearity. To restore a symmetric response of the relay feedback system, an iterative adjustment of input or output bias is required. Instead of trying to obtain a symmetric response, the asymmetric response can be analyzed to estimate the ultimate data or a parametric model of process. However, the asymmetric response causes additional errors in estimating the ultimate properties of the process. An area method is proposed to reduce these errors. Because integrals (areas) of the relay responses reduce the effects of the high-order harmonic terms significantly, the proposed method shows better accuracy in obtaining frequency response data and parametric models compared with previous approaches.

Integrals of relay feedback responses for extracting process information

AbstractRelay feedback methods have been widely used in tuning proportional‐integral‐derivative controllers automatically. Previous approaches usually use specific point data such as the oscillation amplitude and period of the relay response. In this article, we propose new identification methods, which use integrals of the relay response instead of the point data. The proposed methods guarantee better accuracy and advantages in obtaining the ultimate information of process as well as parametric models compared with previous approaches, because effects of the high order harmonic terms are suppressed significantly using the integrals of the relay responses. © 2007 American Institute of Chemical Engineers AIChE J, 2007

Determination of a Coupling Function in Multicoupled Oscillators

A new method to determine a coupling function in a phase model is theoretically derived for coupled self-sustained oscillators and applied to Belousov-Zhabotinsky (BZ) oscillators. The synchronous behavior of two coupled BZ reactors is explained extremely well in terms of the coupling function thus obtained. This method is expected to be applicable to weakly coupled multioscillator systems, in which mutual coupling among nearly identical oscillators occurs in a similar manner. The importance of higher-order harmonic terms involved in the coupling function is also discussed.

A bar signature and central disc in the gaseous and stellar velocity fields of NGC 5448

We analyse SAURON kinematic maps of the inner kpc of the early-type (Sa) barred spiral galaxy NGC5448. The observed morphology and kinematics of the emission-line gas is patchy and perturbed, indicating clear departures from circular motion. The kinematics of the stars is more regular, and display a small inner disk-like system embedded in a large-scale rotating structure. We focus on the [OIII] gas, and use a harmonic decomposition formalism to analyse the gas velocity field. The higher-order harmonic terms and the main kinematic features of the observed data are consistent with an analytically constructed simple bar model. The bar model is derived using linear theory, considering an m=2 perturbation mode, and with bar parameters which are consistent with the large-scale bar detected via imaging. We also study optical and near infra-red images to reveal the asymmetric extinction in NGC5448, and we recognise that some of the deviations between the data and the analytical bar model may be due to these complex dust features. Our study illustrates how the harmonic decomposition formalism can be used as a powerful tool to quantify non-circular motions in observed gas velocity fields.

The Dwarf Galaxy DDO 47 as a Dark Matter Laboratory: Testing Cusps Hiding in Triaxial Halos

We present an analysis of high-resolution H I data of the dwarf galaxy DDO 47, aimed at testing the hypothesis that dark halo triaxiality might induce noncircular motions resulting in rotation curves best fitted by cored halos, even if the dark matter halo is intrinsically cuspy. This hypothesis could be invoked in order to reconcile the predictions of the standard ΛCDM theory with the rotation curves of disk galaxies. DDO 47 is an ideal case to test this hypothesis because it has a very regular velocity field, its rotation curve is best fitted by a cored halo, and an NFW halo is inconsistent with the data. We analyze the velocity field through the higher order harmonic terms in order to search for kinematical signatures of alleged noncircular motions needed to hide a cusp: the result is that globally noncircular motions are at a level of 2-3 km s-1, and they are more likely to be associated with the presence of some spiral structure than with a global elongated potential (e.g., a triaxial halo). These noncircular motions are far from being sufficient to account for the discrepancy with the ΛCDM predictions. We therefore conclude that the dark matter halo around the dwarf galaxy DDO 47 is truly cored and that a cusp cannot be hidden by noncircular motions.

CEMS Evidence of the Presurface Zone Enhancement of the Spin-Density in Chromium

Spin-density enhancement in the presurface zone of a single crystal (110) Cr was observed by using the 119Sn Mössbauer effect in the conversion electron mode (CEMS). The observed enhancement is by a factor of two smaller and by a factor of 30–50 deeper than theoretical predictions. The spin structure is different from the one in bulk chromium and can be explained in terms of higher-order harmonics.

Angular dependence of characteristic fields in spin-valves

Spin-valves are commonly characterised by a series of representative field values such as the coercivity of the magnetic layers, the exchange bias field and the interlayer coupling field. As the angle of an applied field is varied, these values have been observed to vary as periodic functions containing many high order harmonic terms. We analyse the magnetic behaviour of a spin-valve in terms of an extended Stoner–Wohlfarth model, and show that only the fundamental terms are necessary in the free energy of the system for these complex angular dependences to arise. We present illustrative data for spin-valve multilayer films. The complex angular dependences are seen to arise when the unidirectional and uniaxial energy terms are of comparable size, as in the free layer of a weakly coupled spin-valve.

Temperature modulation effects on a material’s properties: Thermodynamics and dielectric relaxation during polymerization

Because of the nonlinear change in the properties of a material with change in temperature, sinusoidal temperature modulation will alter the magnitude of a measured property and the rate of an irreversible transformation. These effects are mathematically described and elaborated. From real-time dielectric spectroscopy and calorimetric measurements performed on a model liquid undergoing stepwise network structure polymerization until its vitrification, it is shown that the effects of sinusoidal temperature modulation are substantial, and need to be considered in the analysis of the data. Features that appear in the plots against time vanish in plots against the extent of polymerization, an intrinsic property. Fourier transform of the heat flow in response to a sinusoidal temperature modulation may help determine the magnitude of this effect in the form of second or higher order harmonic terms. This in turn may be used to calculate the temperature and time dependence of the nonlinear effects. These fundamental aspects will be useful for the current efforts in incorporating sinusoidal temperature modulation in calorimetry.

Hexamethylenetetramine Sebacate, an Incommensurate Structure with Large Nonsinusoidal Modulations: Comparison of Two Refinement Strategies

The organic salt hexamethylenetetramine sebacate presents a 1D modulation with strong satellite reflections. Single-crystal X-ray intensities of main reflections and satellites up to sixth order have been measured at room temperature and used to refine this incommensurate structure. Two incommensurate structure models of hexamethylenetetramine (hexamine) sebacate are proposed and compared. The first is based on displacive atomic modulation functions and the second one uses displacive modulation and occupational step functions. Both models require high-order harmonic terms to describe the strongly nonsinusoidal modulation components. The structural characteristics of hexamine sebacate are similar in the two models. The structure is a layer structure consisting of alternating sheets of hexamine and sebacic acid connected by H bonds. Each layer of sebacic acid molecules consists of close-packed aliphatic chains with their axis slightly tilted with respect to the layer normal. The chains form alternating areas, each of which is characterized by a common orientation of the chains. These characteristics have also been observed in hexamethylenetetramine suberate.

Harmonic analysis of Mössbauer spectra and its applications

A new iterative procedure for a numerical analysis of Mossbauer spectra in terms of higher-order harmonics is used to study spin-and charge-density waves in single-crystal and polycrystalline samples of chromium and chromium-vanadium alloys.

Toeplitz Jacobian Matrix for Nonlinear Periodic Vibration

The main difference between a linear system and a nonlinear system is in the non-uniqueness of solutions manifested by the singular Jacobian matrix. It is important to be able to express the Jacobian accurately, completely, and efficiently in an algorithm to analyze a nonlinear system. For periodic response, the incremental harmonic balance (IHB) method is widely used. The existing IHB methods, however, requiring double summations to form the Jacobian matrix, are often extremely time-consuming when higher order harmonic terms are retained to fulfill the completeness requirement. A new algorithm to compute the Jacobian is to be introduced with the application of fast Fourier transforms (FFT) and Toeplitz formulation. The resulting Jacobian matrix is constructed explicitly by three vectors in terms of the current Fourier coefficients of response, depending respectively on the synchronizing mass, damping, and stiffness functions. The part of the Jacobian matrix depending on the nonlinear stiffness is actually a Toeplitz matrix. A Toeplitz matrix is a matrix whose k, r position depends only on their difference k-r. The other parts of the Jacobian matrix depending on the nonlinear mass and damping are Toeplitz matrices modified by diagonal matrices. If the synchronizing mass is normalized in the beginning, we need only two real vectors to construct the Toeplitz Jacobian matrix (TJM), which can be treated in one complex fast Fourier transforms. The present method of TJM is found to be superior in both computation time and storage than all existing IHB methods due to the simplified explicit analytical form and the use of FFT.

Influence of crystalline boundaries on spin- and charge-density waves in chromium.

Using the $^{119}\mathrm{Sn}$ M\"ossbauer effect we show that the crystalline boundaries have a profound influence on the shape of M\"ossbauer spectra. Based on the newly developed method of the harmonic analysis of the M\"ossbauer spectra, we analyze the measured spectra in terms of higher-order harmonics and reveal a relationship between the harmonics and the size of grains: the smaller the size, the greater the contribution of the harmonics, with the third-order one being negative and dominating. We also demonstrate that the present results support the up-to-date picture of the spin-density waves in the form of single Q domains.

Harmonic analysis of Mössbauer spectra

A new iterative procedure for a numerical analysis of Mössbauer spectra in terms of higher-order harmonics is outlined and tested for 119Sn simulated and measured Mössbauer spectra of chromium. Its usefulness to study spin- and charge-density waves of chromium is demonstrated.