Determining the Spectrum of the Nonlinear Local Lyapunov Exponents in a Multidimensional Chaotic System

For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent （NLLE） from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.

Bright and dark small amplitude nonlinear localized modes in a quantum one-dimensional Klein-Gordon chain

By means of the Glauber＇s coherent state method combined with multiple-scale method, this paper investigates the localized modes in a quantum one-dimensional Klein-Gordon chain and finds that the equation of motion of annihilation operator is reduced to the nonlinear Schroedinger equation. Interestingly, the model can support both bright and dark small amplitude travelling and non-travelling nonlinear localized modes in different parameter spaces.

Localized waves in three-component coupled nonlinear Schrdinger equation

We study the generalized Darboux transformation to the three-component coupled nonlinear Schr ¨odinger equation.First-and second-order localized waves are obtained by this technique.In first-order localized wave,we get the interactional solutions between first-order rogue wave and one-dark,one-bright soliton respectively.Meanwhile,the interactional solutions between one-breather and first-order rogue wave are also given.In second-order localized wave,one-dark-one-bright soliton together with second-order rogue wave is presented in the first component,and two-bright soliton together with second-order rogue wave are gained respectively in the other two components.Besides,we observe second-order rogue wave together with one-breather in three components.Moreover,by increasing the absolute values of two free parameters,the nonlinear waves merge with each other distinctly.These results further reveal the interesting dynamic structures of localized waves in the three-component coupled system.

GLOBAL EXISTENCE FOR THE NONHOMOGENEOUS QUASILINEAR WAVE EQUATION WITH A LOCALIZED WEAKLY NONLINEAR DISSIPATION IN EXTERIOR DOMAINS

It is proved that the global existence for the nonhomogeneous quasilinear wave equation with a localized weakly nonlinear dissipation in exterior domains.

Comparison of Nonlinear Local Lyapunov Vectors with Bred Vectors, Random Perturbations and Ensemble Transform Kalman Filter Strategies in a Barotropic Model

The breeding method has been widely used to generate ensemble perturbations in ensemble forecasting due to its simple concept and low computational cost. This method produces the fastest growing perturbation modes to catch the growing components in analysis errors. However, the bred vectors （BVs） are evolved on the same dynamical flow, which may increase the dependence of perturbations. In contrast, the nonlinear local Lyapunov vector （NLLV） scheme generates flow-dependent perturbations as in the breeding method, but regularly conducts the Gram-Schmidt reorthonormalization processes on the perturbations. The resulting NLLVs span the fast-growing perturbation subspace efficiently, and thus may grasp more com- ponents in analysis errors than the BVs. In this paper, the NLLVs are employed to generate initial ensemble perturbations in a barotropic quasi-geostrophic model. The performances of the ensemble forecasts of the NLLV method are systematically compared to those of the random pertur- bation （RP） technique, and the BV method, as well as its improved version--the ensemble transform Kalman filter （ETKF） method. The results demonstrate that the RP technique has the worst performance in ensemble forecasts, which indicates the importance of a flow-dependent initialization scheme. The ensemble perturbation subspaces of the NLLV and ETKF methods are preliminarily shown to catch similar components of analysis errors, which exceed that of the BVs. However, the NLLV scheme demonstrates slightly higher ensemble forecast skill than the ETKF scheme. In addition, the NLLV scheme involves a significantly simpler algorithm and less computation time than the ETKF method, and both demonstrate better ensemble forecast skill than the BV scheme.

Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method

The backward nonlinear local Lyapunov exponent method(BNLLE)is applied to quantify the predictability of warm and cold events in the Lorenz model.Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space.The maximum prediction lead times of each warm(cold)event on individual circles concentric with the distribution of warm(cold)regime events are roughly the same,whereas the maximum prediction lead time of events on other circles are different.Statistical results show that warm events are more predictable than cold events.

APPROXIMATE POWER OF HETEROSCEDASTICITY TEST IN NONLINEAR MODELS WITH ARIMA（0,1,0） ERRORS

This paper presents an approach for estimating power of the score test, based on an asymptotic approximation to the power of the score test under contiguous alternatives. The method is applied to the problem of power calculations for the score test of heteroscedasticity in European rabbit data （Ratkowsky, 1983）. Simulation studies are presented which indicate that the asymptotic approximation to the finite-sample situation is good over a wide range of parameter configurations.

Nonlinear characteristic investigation of magnetorheological damper-rotor system with local nonlinearity

Magnetorheological(MR)dampers show superior performance in reducing rotor vibration,but their high nonlinearity will cause nonsynchronous response,resulting in fatigue and instability of rotors.Herein,we are devoted to the investigation of the nonlinear characteristics of MR damper mounted on a flexible rotor.First,Reynolds equations with bilinear constitutive equations of MR fluid are employed to derive nonlinear oil film forces.Then,the Finite Element(FE)model of rotor system is developed,where the local nonlinear support forces produced by MR damper and its coupling effects with the rotor are considered.A hybrid numerical method is proposed to solve the nonlinear FE motion equations of the MR damper-rotor system.To validate the proposed model,a rotor test bench with two dual-coil MR dampers is constructed,upon which experimental studies on the dynamic characteristics of MR damper-rotor system are carried out.The effects of different system parameters,including rotational speed,excitation current and amount of unbalance,on nonlinear dynamic behaviors of MR damper-rotor system are evaluated.The results show that the system may appear chaos,jumping,and other complex nonlinear phenomena,and the level of the nonlinearity can be effectively alleviated by applying suitable excitation current and oil supply pressure.

Impact of Perturbation Schemes on the Ensemble Prediction in a Coupled Lorenz Model

Based on a simple coupled Lorenz model,we investigate how to assess a suitable initial perturbation scheme for ensemble forecasting in a multiscale system involving slow dynamics and fast dynamics.Four initial perturbation approaches are used in the ensemble forecasting experiments:the random perturbation(RP),the bred vector(BV),the ensemble transform Kalman filter(ETKF),and the nonlinear local Lyapunov vector(NLLV)methods.Results show that,regardless of the method used,the ensemble averages behave indistinguishably from the control forecasts during the first few time steps.Due to different error growth in different time-scale systems,the ensemble averages perform better than the control forecast after very short lead times in a fast subsystem but after a relatively long period of time in a slow subsystem.Due to the coupled dynamic processes,the addition of perturbations to fast variables or to slow variables can contribute to an improvement in the forecasting skill for fast variables and slow variables.Regarding the initial perturbation approaches,the NLLVs show higher forecasting skill than the BVs or RPs overall.The NLLVs and ETKFs had nearly equivalent prediction skill,but NLLVs performed best by a narrow margin.In particular,when adding perturbations to slow variables,the independent perturbations(NLLVs and ETKFs)perform much better in ensemble prediction.These results are simply implied in a real coupled air–sea model.For the prediction of oceanic variables,using independent perturbations(NLLVs)and adding perturbations to oceanic variables are expected to result in better performance in the ensemble prediction.

Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

In order to quantify the influence of external forcings on the predictability limit using observational data,the author introduced an algorithm of the conditional nonlinear local Lyapunov exponent(CNLLE)method.The effectiveness of this algorithm is validated and compared with the nonlinear local Lyapunov exponent(NLLE)and signal-to-noise ratio methods using a coupled Lorenz model.The results show that the CNLLE method is able to capture the slow error growth constrained by external forcings,therefore,it can quantify the predictability limit induced by the external forcings.On this basis,a preliminary attempt was made to apply this method to measure the influence of ENSO on the predictability limit for both atmospheric and oceanic variable fields.The spatial distribution of the predictability limit induced by ENSO is similar to that arising from the initial conditions calculated by the NLLE method.This similarity supports ENSO as the major predictable signal for weather and climate prediction.In addition,a ratio of predictability limit(RPL)calculated by the CNLLE method to that calculated by the NLLE method was proposed.The RPL larger than 1 indicates that the external forcings can significantly benefit the long-term predictability limit.For instance,ENSO can effectively extend the predictability limit arising from the initial conditions of sea surface temperature over the tropical Indian Ocean by approximately four months,as well as the predictability limit of sea level pressure over the eastern and western Pacific Ocean.Moreover,the impact of ENSO on the geopotential height predictability limit is primarily confined to the troposphere.

Model tests and numerical analyses on horizontal impedance functions of inclined single piles embedded in cohesionless soil

Horizontal impedance functions of inclined single piles are measured experimentally for model soil-pile systems with both the effects of local soil nonlinearity and resonant characteristics.Two practical pile inclinations of 5掳 and 10掳 in addition to a vertical pile embedded in cohesionless soil and subjected to lateral harmonic pile head loadings for a wide range of frequencies are considered.Results obtained with low-to-high amplitude of lateral loadings on model soil-pile systems encased in a laminar shear box show that the local nonlinearities have a profound impact on the horizontal impedance functions of piles.Horizontal impedance functions of inclined piles are found to be smaller than the vertical pile and the values decrease as the angle of pile inclination increases.Distinct values of horizontal impedance functions are obtained for the ＇positive＇ and ＇negative＇ cycles of harmonic loadings,leading to asymmetric force-displacement relationships for the inclined piles.Validation of these experimental results is carried out through three-dimensional nonlinear finite element analyses,and the results from the numerical models are in good agreement with the experimental data.Sensitivity analyses conducted on the numerical models suggest that the consideration of local nonlinearity at the vicinity of the soil-pile interface influence the response of the soil-pile systems.

Influence of Sea Surface Temperature on the Predictability of Idealized Tropical Cyclone Intensity

The role of sea surface temperature(SST)forcing in the development and predictability of tropical cyclone(TC)intensity is examined using a large set of idealized numerical experiments in the Weather Research and Forecasting(WRF)model.The results indicate that the onset time of rapid intensification of TC gradually decreases,and the peak intensity of TC gradually increases,with the increased magnitude of SST.The predictability limits of the maximum 10 m wind speed(MWS)and minimum sea level pressure(MSLP)are~72 and~84 hours,respectively.Comparisons of the analyses of variance for different simulation time confirm that the MWS and MSLP have strong signal-to-noise ratios(SNR)from 0-72 hours and a marked decrease beyond 72 hours.For the horizontal and vertical structures of wind speed,noticeable decreases in the magnitude of SNR can be seen as the simulation time increases,similar to that of the SLP or perturbation pressure.These results indicate that the SST as an external forcing signal plays an important role in TC intensity for up to 72 hours,and it is significantly weakened if the simulation time exceeds the predictability limits of TC intensity.

Estimation of the Monthly Precipitation Predictability Limit in China Using the Nonlinear Local Lyapunov Exponent

By using the nonlinear local Lyapunov exponent and nonlinear error growth dynamics, the predictability limit of monthly precipitation is quantitatively estimated based on daily observations collected from approx- imately 500 stations in China for the period 1960-2012. As daily precipitation data are not continuous in space and time, a transformation is first applied and a monthly standardized precipitation index （SPI） with Gaussian distribution is constructed. The monthly SPI predictability limit （MSPL） is quantitatively calcu- lated for SPI dry, wet, and neutral phases. The results show that the annual mean MSPL varies regionally for both wet and dry phases： the MSPL in the wet （dry） phase is relatively higher （lower） in southern China than in other regions. Further, the pattern of the MSPL for the wet phase is almost opposite to that for the dry phase in both autumn and winter. The MSPL in the dry phase is higher in winter and lower in spring and autumn in southern China, while the MSPL values in the wet phase are higher in summer and winter than those in spring and autumn in southern China. The spatial distribution of the MSPL resembles that of the prediction skill of monthly precipitation from a dynamic extended-range forecast system.

The Predictability Limit of Oceanic Mesoscale Eddy Tracks in the South China Sea

Employing the nonlinear local Lyapunov exponent (NLLE) technique, this study assesses the quantitative predictability limit of oceanic mesoscale eddy (OME) tracks utilizing three eddy datasets for both annual and seasonal means. Our findings reveal a discernible predictability limit of approximately 39 days for cyclonic eddies (CEs) and 44 days for anticyclonic eddies (AEs) within the South China Sea (SCS). The predictability limit is related to the OME properties and seasons. The long-lived, large-amplitude, and large-radius OMEs tend to have a higher predictability limit. The predictability limit of AE (CE) tracks is highest in autumn (winter) with 52 (53) days and lowest in spring (summer) with 40 (30) days. The spatial distribution of the predictability limit of OME tracks also has seasonal variations, further finding that the area of higher predictability limits often overlaps with periodic OMEs. Additionally, the predictability limit of periodic OME tracks is about 49 days for both CEs and AEs, which is 5-10 days higher than the mean values. Usually, in the SCS, OMEs characterized by high predictability limit values exhibit more extended and smoother trajectories and often move along the northern slope of the SCS.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

The nonlinear transverse vibrations of ordered and disordered twodimensional(2D)two-span composite laminated plates are studied.Based on the von Karman’s large deformation theory,the equations of motion of each-span composite laminated plate are formulated using Hamilton’s principle,and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin’s method.The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales.The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out.The effects of the disorder ratio and ply angle on the two different resonances are analyzed.From the numerical results,it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon,and with the increase of the disorder ratio,the vibration localization phenomenon will become more obvious.Moreover,the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration,and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.