Nonlinear phenomena in contemporary vocal music.

Complex and multiphonic voice signals of vocal improvisors are analyzed within the framework of nonlinear dynamics. Evidence is given that nonlinear phenomena are extensively used by performers associated with contemporary music. Narrow-band spectrograms of complex vocalizations are used to visualize the appearance of nonlinear phenomena (spectral bifurcation diagrams). Possible production mechanisms are discussed in connection with previous research, personal performance and pedagogical experience. Examples for period doubling, biphonation and irregular aperiodic phonation in vocal sonorities of contemporary vocal improvisors are given, and glottal whistle production encompassed with biphonation and triphonation is shown. Furthermore, coincidences of harmonics-formant matching associated with abrupt transitions to subharmonics and biphonation in the vocal output are provided. This also shows the recurrent use of nonlinear phenomena by performers. It is argued that mechanisms such as source-tract coupling or vocal fold desynchronization due to asymmetry are used in a reproducible way for musical tasks.     

Low-dimensional nonlinearity of ENSO and its impact on predictability

Using a hybrid coupled model, we perform a bred vector (BV) analysis and retrospective ENSO (El Niño and the Southern Oscillation) forecast for the period from 1881 to 2000. The BV local dimension and BV-skewness inherent to the intensity of nonlinearity are analyzed. Emphasis is placed on exploring the nature of the low-dimensional nonlinearity of the ENSO system and the relationship between BV-skewness and model prediction skills. The results show that ENSO is a low-dimensional nonlinear system, and the BV-skewness is a good measure of its predictability at the decadal/interdecadal time scales. As the low-dimensional nonlinearity of ENSO is weakened, high predictability is attained, and vice versa. The low-dimensional nonlinearity of ENSO is also investigated and verified using observations.Another finding in this study is the relationship between the error growth rate (BV-rate) and actual prediction skill. While there is a good positive correlation between them in some decades, the BV-rate demonstrates a strong inverse correlation with the prediction skill in other decades. The BV-rate components contributed by the nonlinear process play a dominant role in quantifying ENSO predictability. The possible mechanism for the link between BV-rate, BV-skewness and ENSO predictability is discussed.     

Stochastic climate dynamics: Random attractors and time-dependent invariant measures

This article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s: the nonlinear deterministic and the linear stochastic one. This unification, via the theory of random dynamical systems (RDS), allows one to consider the detailed geometric structure of the random attractors associated with nonlinear, stochastically perturbed systems. We report on high-resolution numerical studies of two idealized models of fundamental interest for climate dynamics. The first of the two is a stochastically forced version of the classical Lorenz model. The second one is a low-dimensional, nonlinear stochastic model of the El Niño-Southern Oscillation (ENSO). These studies provide a good approximation of the two models’ global random attractors, as well as of the time-dependent invariant measures supported by these attractors; the latter are shown to have an intuitive physical interpretation as random versions of Sinaï-Ruelle-Bowen (SRB) measures.     

Stochastic Perturbation of Power Law Optical Solitons

The soliton perturbation theory is used to study and analyze the stochastic perturbation of optical solitons, with power law nonlinearity, in addition to deterministic perturbations, that is governed by the nonlinear Schrödinger’s equation. The Langevin equations are derived and analysed. The deterministic perturbations that are considered here are due to filters and nonlinear damping.     

Nonlinear phenomena in the natural howling of a dog-wolf mix

It was reported to the first author that a female dog-wolf mix showed anomalously rough-sounding vocalization. Spectral analysis of recordings of the vocalization revealed frequency occurrences of subharmonics, biphonation (two independent pitches) and chaos. Since these nonlinear phenomena are currently widely discussed as integral to mammalian vocalization [Wilden et al., Bioacoustics 9, 171-196 (1988)] or as indicators of vocal pathologies [Herzel et al., J. Speech Hearing Res. 37, 1008-1019 (1994); Riede et al., Z. Sgtkde 62 Suppl: 198-203 (1997)], we sought to understand the production mechanism of the observed vocal instabilities. First the frequency of nonlinear phenomena in the calls was determined for the female and four additional individuals. It turned out that these phenomena appear, but much less frequently in the repertoire of the four other animals. The larynges of the female and two other individuals were dissected post mortem. There was no apparent asymmetry of the vocal folds but a slight asymmetry of the arytenoid cartilages. The most pronounced difference, however, was an upward extension of both vocal folds of the female. This feature is reminiscent of "vocal lips" (syn. "vocal membranes") in some primates and bats. Spectral analysis of the female's voice showed clear similarities with an intensively studied voice of a human who produces biphonation intentionally. Finally, the possible communicative relevance of nonlinear phenomena is discussed.     

Analysis of pain-related vocalization in young pigs

The assessment of pain constitutes a major issue for animal welfare research. The objective of this study was to classify vocalizations during castration pain and to assess alterations in vocalizations under local anaesthesia. The alterations in vocalization were measured by multiparametric call analysis. A total of 4537 calls of 70 young pigs were evaluated. With the data of this study three call types are distinguishable (grunt, squeal, scream). A high percentage (94.64%) of calls that could be classified in one of the three call types during the castration process within the confidence level of 95% was found. The comparison of the occurrence of the call types during treatments gives evidence for pain-related use of screams. The piglets castrated without local anaesthesia produced almost double the number of screams as piglets castrated with anaesthesia. The comparison of the recorded sound parameters reveals the particular position of screams in the call repertoire of young pigs. Screams are significantly different in their sound parameters than grunts or squeals. Castration in comparison to mere restraint produced a comprehensive change in sound parameters, with castration calls becoming more extended and more powerful. The findings in this study also show differences in the effectiveness of the parameters which indicate pain. Parameters that describe a single event in a call, such as peak level or peak frequency give better results than parameters that describe an average, such as weighted frequency and main frequency. The research indicated that pain-related changes of calls in piglets can be identified. On the basis of the results, automatic classification of call types during management operations may be developed. This could contribute to objective animal welfare assessment.     

基于Chebyshev多项式逼近的随机 van der Pol系统的倍周期分岔分析

应用 Chebyshev 多项式逼近法研究了谐和激励作用下具有随机参数的随机van der Pol系统 的倍周期分岔现象.随机系统首先被转化成等价的确定性系统，然后通过数值方法求得响应 ，借此探索了随机van der Pol系统丰富的随机倍周期分岔现象.数值模拟显示随机van der Pol 系统存在与确定性系统极为相似的倍周期分岔行为，但受随机因素的影响，又有与之不 同之处.数值结果表明，Chebyshev 多项式逼近是研究非线性系统动力学问题的一种新的有 效方法. 关键词： Chebyshev 多项式 随机van der Pol 系统 倍周期分岔     

基于Laguerre多项式逼近法的随机双势阱Duffing系统的分岔和混沌研究

应用Laguerre正交多项式逼近法研究了含有随机参数的双势阱Duffing系统的分岔和混沌行为.系统参数为指数分布随机变量的非线性动力系统首先被转化为等价的确定性扩阶系统，然后通过数值方法求得其响应.数值模拟结果的比较表明，含有随机参数的双势阱Duffing系统保持着与确定性系统相类似的倍周期分岔和混沌行为，但是由于随机因素的影响，在局部小区域内随机参数系统的动力学行为会发生突变. 关键词： 双势阱Duffing系统 指数分布概率密度函数 Laguerre多项式逼近 随机分岔     

Generalized spectral decomposition for stochastic nonlinear problems

We present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multi-wavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low-dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional nonlinear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space.     

On the applicability of low-dimensional models for convective flow reversals at extreme Prandtl numbers

Constructing simpler models, either stochastic or deterministic, for exploring the phenomenon of flow reversals in fluid systems is in vogue across disciplines. Using direct numerical simulations and nonlinear time series analysis, we illustrate that the basic nature of flow reversals in convecting fluids can depend on the dimensionless parameters describing the system. Specifically, we find evidence of low-dimensional behavior in flow reversals occurring at zero Prandtl number, whereas we fail to find such signatures for reversals at infinite Prandtl number. Thus, even in a single system, as one varies the system parameters, one can encounter reversals that are fundamentally different in nature. Consequently, we conclude that a single general low-dimensional deterministic model cannot faithfully characterize flow reversals for every set of parameter values.     

Analysis of stochastic bifurcation and chaos in stochastic Duffing--van der Pol system via Chebyshev polynomial approximation

The Chebyshev polynomial approximation is applied to investigate the stochastic period-doubling bifurcation and chaos problems of a stochastic Duffing--van der Pol system with bounded random parameter of exponential probability density function subjected to a harmonic excitation. Firstly the stochastic system is reduced into its equivalent deterministic one, and then the responses of stochastic system can be obtained by numerical methods. Nonlinear dynamical behaviour related to stochastic period-doubling bifurcation and chaos in the stochastic system is explored. Numerical simulations show that similar to its counterpart in deterministic nonlinear system of stochastic period-doubling bifurcation and chaos may occur in the stochastic Duffing--van der Pol system even for weak intensity of random parameter. Simply increasing the intensity of the random parameter may result in the period-doubling bifurcation which is absent from the deterministic system.     

The harmonic-to-noise ratio applied to dog barks.

Dog barks are typically a mixture of regular components and irregular (noisy) components. The regular part of the signal is given by a series of harmonics and is most probably due to regular vibrations of the vocal folds, whereas noise refers to any nonharmonic (irregular) energy in the spectrum of the bark signal. The noise components might be due to chaotic vibrations of the vocal-fold tissue or due to turbulence of the air. The ratio of harmonic to nonharmonic energy in dog barks is quantified by applying the harmonics-to-noise ratio (HNR). Barks of a single dog breed were recorded in the same behavioral context. Two groups of dogs were considered: a group of ten healthy dogs (the normal sample), and a group of ten unhealthy dogs, i.e., dogs treated in a veterinary clinic (the clinic sample). Although the unhealthy dogs had no voice disease, differences in emotion or pain or impacts of surgery might have influenced their barks. The barks of the dogs were recorded for a period of 6 months. The HNR computation is based on the Fourier spectrum of a 50-ms section from the middle of the bark. A 10-point moving average curve of the spectrum on a logarithmic scale is considered as estimator of the noise level in the bark, and the maximum difference of the original spectrum and the moving average is defined as the HNR measure. It is shown that a reasonable ranking of the voices is achievable based on the measurement of the HNR. The HNR-based classification is found to be consistent with perceptual evaluation of the barks. In addition, a multiparametric approach confirms the classification based on the HNR. Hence, it may be concluded that the HNR might be useful as a novel parameter in bioacoustics for quantifying the noise within a signal.     

Small Open Chemical Systems Theory：Its Implications to Darwinian Evolution Dynamics,Complex Self-Organization and Beyond

The study of biological cells in terms of mesoscopic, nonequilibrium, nonlinear, stochastic dynamics of open chemical systems provides a paradigm for other complex, self-organizing systems with ultra-fast stochastic fluctuations, short-time deterministic nonlinear dynamics, and long-time evolutionary behavior with exponentially distributed rare events, discrete jumps among punctuated equilibria, and catastrophe.     

Characterizing noise in nonhuman vocalizations: Acoustic analysis and human perception of barks by coyotes and dogs

Measuring noise as a component of mammalian vocalizations is of interest because of its potential relevance to the communicative function. However, methods for characterizing and quantifying noise are less well established than methods applicable to harmonically structured aspects of signals. Using barks of coyotes and domestic dogs, we compared six acoustic measures and studied how they are related to human perception of noisiness. Measures of harmonic-to-noise-ratio (HNR), percent voicing, and shimmer were found to be the best predictors of perceptual rating by human listeners. Both acoustics and perception indicated that noisiness was similar across coyote and dog barks, but within each species there was significant variation among the individual vocalizers. The advantages and disadvantages of the various measures are discussed.     

The equation of motion for the trajectory of a circling particle with an overdamped circle center

Inspired by biological microorganisms swimming in circles in liquid with low Reynolds number, I developed the dynamic theory for computing the helical trajectory of a circling particle with an overdamped circle center. The equation of motion for the circling particle is a hybrid equation of deterministic terms and stochastic terms. Observing the motion of a swimming microorganism, I found the strength of stochastic fluctuations should be much smaller than that governs deterministic dynamics. This dynamic theory predicts a nonlinear transverse motion perpendicular to the direction of external force. Both the living microorganism and artificial circling particle are applicable for an experimental check of this prediction. For the convenience of easy theoretical research, I further derived the probability conservation equations based on this dynamic theory both in two-dimensional and three-dimensional space.     

The Diffusion Kernel Filter

A particle filter method is presented for the discrete-time filtering problem with nonlinear Itô stochastic ordinary differential equations (SODE) with additive noise supposed to be analytically integrable as a function of the underlying vector-Wiener process and time. The Diffusion Kernel Filter is arrived at by a parametrization of small noise-driven state fluctuations within branches of prediction and a local use of this parametrization in the Bootstrap Filter. The method applies for small noise and short prediction steps. With explicit numerical integrators, the operations count in the Diffusion Kernel Filter is shown to be smaller than in the Bootstrap Filter whenever the initial state for the prediction step has sufficiently few moments. The established parametrization is a dual-formula for the analysis of sensitivity to gaussian-initial perturbations and the analysis of sensitivity to noise-perturbations, in deterministic models, showing in particular how the stability of a deterministic dynamics is modeled by noise on short times and how the diffusion matrix of an SODE should be modeled (i.e. defined) for a gaussian-initial deterministic problem to be cast into an SODE problem. From it, a novel definition of prediction may be proposed that coincides with the deterministic path within the branch of prediction whose information entropy at the end of the prediction step is closest to the average information entropy over all branches. Tests are made with the Lorenz-63 equations, showing good results both for the filter and the definition of prediction.     

Pathological tremors: Deterministic chaos or nonlinear stochastic oscillators?

Pathological tremors exhibit a nonlinear oscillation that is not strictly periodic. We investigate whether the deviation from periodicity is due to nonlinear deterministic chaotic dynamics or due to nonlinear stochastic dynamics. To do so, we apply various methods from linear and nonlinear time series analysis to tremor time series. The results of the different methods suggest that the considered types of pathological tremors represent nonlinear stochastic second order processes. Finally, we evaluate whether two earlier proposed features capturing nonlinear effects in the time series allow for a discrimination between two pathological forms of tremor for a much larger sample of time series than previously investigated. (c) 2000 American Institute of Physics.     

一类位于加周期分岔中的貌似混沌的随机神经放电节律的识别

识别非周期神经放电节律是混沌还是随机一直是一个重要的科学问题. 在神经起步点实验中发现了一类介于周期k和周期k+1(k=1,2)节律之间非周期自发放电节律, 其行为是长串的周期k簇和周期k+1簇的交替. 确定性理论模型Chay模型展示出了周期k和周期k+1节律的共存行为. 噪声在共存区诱发出了与实验结果类似的非周期节律, 说明该类节律是噪声引起的两类簇的跃迁. 非线性预报及其回归映射揭示该节律具有确定性机理; 将两类簇分别转换为0和1得到一个二进制序列, 对该序列进行概率分析获得了两类簇跃迁的随机机理. 这不仅说明该节律是具有确定性结构的随机节律而不是混沌, 还为深入识别现实神经系统的混沌和随机节律提供了典型示例和有效方法.