Framework Of Nonlinear Dynamics Research Articles

Prediction of Extreme Ship Motions in Irregular Waves

AbstractThe occurrence of indirectly excited large amplitude roll motions of ships is investigated under the influence of a time‐varying righting lever curve in waves. In order to account for the irregular character of ocean waves appropriately, the ship motions are described within the framework of nonlinear dynamics and stochastic process theory. The analysis of roll motions in regular waves provides a qualitative understanding of the occurrence of unpredictable large amplitude roll motions as observed in ocean waves. However, quantitative information on the rolling behavior in irregular waves may be retrieved only from a probabilistic analysis. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Configurational forces and their application in solid mechanics

The theory of configurational forces is presented within the non-linear dynamic framework. The configurational force balance is recast in a form which resembles the equation of motion. Specializations to statics and the small strain case are given. The general use of configurational forces is discussed by help of numerous examples. Additionally the occurrence of configurational forces in the Finite Element Method is illustrated. The numerically induced configurational forces can be used to improve the numerical resolution.

On characterization of turbulent stellar convection

An attempt is made to characterize convective fluid motion in a stellar-type convective envelope within the framework of non-linear dynamics. For this analysis, we have simulated threedimensional turbulent compressible convection by solving numerically the full set of Navier‐ Stokes equations in a rectangular box with a two-layer stable‐unstable stratification. The fluid motions are evolved with time until the fluid reaches a statistically steady state. A time series of the horizontally-averaged vertical velocity of the convective fluid is collected from the middle of the unstable zone. The correlation dimension and the largest Lyapunov exponent computed for this time series characterize the fluid motion as that of a low-dimensional strange attractor. The application of the tools of non-linear dynamics in astrophysics is relatively recent, except in the case of celestial mechanics, in ∂

A nonlinear dynamical systems framework for structural diagnosis and prognosis

This work aims to establish a nonlinear dynamics framework for diagnosis and prognosis in structural dynamic systems. The objective is to develop an analytically sound means for extracting features, which can be used to characterize damage, from modal-based input–output data in complex hybrid structures with heterogeneous materials and many components. Although systems like this are complex in nature, the premise of the work here is that damage initiates and evolves in the same phenomenological way regardless of the physical system according to nonlinear dynamic processes. That is, bifurcations occur in healthy systems as a result of damage. By projecting a priori the equations of motion of high-dimensional structural dynamic systems onto lower dimensional center, or so-called ‘damage’, manifolds, it is demonstrated that model reduction near bifurcations might be a useful way to identify certain features in the input–output data that are helpful in identifying damage. Normal forms describing local co-dimension one and two bifurcations (e.g. transcritical, subcritical pitchfork, and asymmetric pitchfork bifurcations) are assumed to govern the initiation and evolution of damage in a low-order model. Real-world complications in damage prognosis involving spatial bifurcations, global bifurcation phenomena, and the sensitivity of damage to small changes in initial conditions are also briefly discussed.

Reproducibility of the Structure and Properties of Parts and Their Description Within the Framework of Nonlinear Dynamics

The problem of the reproducibility of complex structures is discussed from the standpoint of the applied theory of dynamic systems. Some fundamental concepts of the theory of nonlinear fluctuations (bifurcations, fractal sets, space and time networks of interacting subsystems) that can be helpful in applications are described on a qualitative level.

Sustained oscillations in living cells.

Glycolytic oscillations in yeast have been studied for many years simply by adding a glucose pulse to a suspension of cells and measuring the resulting transient oscillations of NADH. Here we show, using a suspension of yeast cells, that living cells can be kept in a well defined oscillating state indefinitely when starved cells, glucose and cyanide are pumped into a cuvette with outflow of surplus liquid. Our results show that the transitions between stationary and oscillatory behaviour are uniquely described mathematically by the Hopf bifurcation. This result characterizes the dynamical properties close to the transition point. Our perturbation experiments show that the cells remain strongly coupled very close to the transition. Therefore, the transition takes place in each of the cells and is not a desynchronization phenomenon. With these two observations, a study of the kinetic details of glycolysis, as it actually takes place in a living cell, is possible using experiments designed in the framework of nonlinear dynamics. Acetaldehyde is known to synchronize the oscillations. Our results show that glucose is another messenger substance, as long as the glucose transporter is not saturated.

Measures of complexity for cancellous bone

The problem of quantifying the structure of cancellous bone has been addressed in the past by histomorphometry and more recently by imaging techniques using X-ray attenuation. The current approaches compute and describe parts of the construction of the trabecular net. We developed a new technique which quantifies cancellous bone of human lumbar vertebrae as a whole. The interactions, transactions, and interrelationships of all parts of the structural composition of the trabeculae are accounted for and quantified. The method is based on the concept of structural complexity within the framework of nonlinear dynamics. The methodology was developed by using axial high resolution computed tomography images. The technique was transferred to quantitative computed tomography images and is based on the non-invasive assessment of 50 human L3 specimens. The value of Houndsfield units per pixel representing trabecular bone of the vertebrae was transformed into color-encoded and alphabet-encoded symbols. The procedure of transformation of the X-ray attenuation pixels into symbols was necessary as a basis on which measures of complexity were introduced to assess the composition of symbols within the images. The development of a generalization of symbolic dynamics, a mathematical method, to work with two-dimensional images was a prerequisite. The results of this study demonstrate that the structural composition of cancellous bone declines more rapidly than bone mineral density during the loss of bone. This outcome strongly suggests an exponential relationship between bone mineral density and the architectural composition of cancellous bone. Normal trabecular bone has a complex ordered structure. The structural composition during the osteopenic phase of bone loss is characterized by lower structural complexity and a significantly higher level of architectural disorder. A high grade of osteoporosis leads again to an ordered structure, although its structural complexity is minimal.

SUBHARMONICS, BIPHONATION, AND DETERMINISTIC CHAOS IN MAMMAL VOCALIZATION

ABSTRACT To establish a framework for discussing mammalian vocalizations, relevant terminology and concepts from the theory of nonlinear dynamics are introduced. It is suggested that a variety of nonlinear phenomena including subharmonics, biphonation, and deterministic chaos are normally occurring phonatory events. The whole spectrum of these phenomena can be found in the repertoire of the African wild dog Lycaon pictus. In addition, examples of nonlinear phenomena in a wide range of other mammalian taxa will be presented. Moreover, some artifacts in sound spectrographic analysis are listed which may be misinterpreted as nonlinear phenomena. Within the framework of nonlinear dynamics, a consistent terminology is proposed and our observations are related to laryngeal sound production mechanisms. Finally, some hypotheses concerning the communicative potential of the described phenomena are discussed.

Dynamics of the neural discharge in snail neurons

Spike trains recorded under weak sinusoidal driving from central neurons of Lymnaea stagnalis appear quite irregular and envisage the possibility of an underlying chaotic process. Therefore, the sequences of interspike intervals are analyzed in the framework of non-linear dynamics. Since, for several reasons, these sequences are rather short, the analysis is performed by using methods of non-linear forecasting. To reject the null hypothesis that the original time series is a realization of a linear stochastic process with the same autocorrelation function, the results obtained on the original data are compared with those from surrogate data sets. Some ‘non-linear’ predictability occurs only in narrow regions of the space of stimulus parameters and the frequency of perturbation is critical in determining it. Moreover, it is shown that such behavior can be qualitatively mimicked by the FitzHugh-Nagumo model driven by a weak sinusoidal signal plus noise. It is argued that the narrowness of the non-linear predictability regions renders quite unlikely the detection of deterministic dynamics in the activity of these neurons.

Spatiotemporal pattern formation in reactions at surfaces

The recent development of novel experimental methods and progress in theoretical modelling within the framework of nonlinear dynamics provide detailed insight into the rich variety of spatiotemporal concentration patterns associated with catalytic and electrochemical reactions at solid—gas and solid—liquid interfaces, respectively.

Analysis of Vocal Disorders With Methods From Nonlinear Dynamics

Several authors have recently demonstrated the intimate relationship between nonlinear dynamics and observations in vocal fold vibration (Herzel, 1993; Mende, Herzel, & Wermke, 1990; Titze, Baken, & Herzel, 1993). The aim of this paper is to analyze vocal disorders from a nonlinear dynamics point of view. Basic concepts and analysis techniques from nonlinear dynamics are reviewed and related to voice. The voices of several patients with vocal disorders are analyzed using traditional voice analysis techniques and methods from nonlinear dynamics. The two methods are shown to complement each other in many ways. Likely physiological mechanisms of the observed nonlinear phenomena are presented, and it is shown how much of the terminology in the literature describing rough voice can be unified within the framework of nonlinear dynamics.

Psychiatry and the new dynamics

Self-organizing and self-tuning neural networks are conceptualized as nonlinear dynamical systems. It is shown that biological psychiatry can be encompassed within this nonlinear dynamical framework, where it becomes coherent with trends in cognitive science. A state space representation that is a hyperspace with an energy topology and a fractal dimension is utilized. The state space representation provides a unifying framework for the classification of mental disorders. Clinical phenomena in schizophrenia can be explained in terms of nonlinear dynamics. The framework is also productive of psychiatric research. These results motivate the development of a nonlinear dynamical psychiatry.

Time dependencies in the occurrences of epileptic seizures

A new method of analysis, developed within the framework of nonlinear dynamics, is applied to patient recorded time series of the occurrence of epileptic seizures. These data exhibit broad band spectra and generally have no obvious structure. The goal is to detect hidden internal dependencies in the data without making any restrictive assumptions, such as linearity, about the structure of the underlying system. The basis of our approach is a conditional probabilistic analysis in a phase space reconstructed from the original data. The data, recorded from patients with intractable epilepsy over a period of 1–3 years, consist of the times of occurrences of hundreds of partial complex seizures. Although the epileptic events appear to occur independently, we show that the epileptic process is not consistent with the rules of a homogeneous Poisson process or generally with a random (IID) process. More specifically, our analysis reveals dependencies of the occurrence of seizures on the occurrence of preceding seizures. These dependencies can be detected in the interseizure interval data sets as well as in the rate of seizures per time period. We modeled patient's inaccuracy in recording seizure events by the addition of uniform white noise and found that the detected dependencies are persistent after addition of noise with standard deviation as great as 1 3 of the standard deviation of the original data set. A linear autoregressive analysis fails to capture these dependencies or produces spurious ones in most of the cases.

NONLINEAR PHENOMENA IN CIRCULAR PARTICLE ACCELERATORS I: SEXTUPOLAR NONLINEARITIES

In this paper we relate the methods of the theory of dynamical systems to problems of accelerator design and optimization. It is written in review article style and any prior knowledge of accelerator physics is not required. We derive the Poincaré map for the equation of horizontal transverse motion of a test particle in a circular accelerator in the presence of any number of sextupolar nonlinearities. This map is studied within the framework of nonlinear dynamics and we identify several optimization strategies for the design of an accelerator magnet lattice. The dynamic aperture problem is analysed and we obtain, for a beam line with two sextupoles, the dependence of the dynamic aperture on the line parameters. The mechanisms of aperture reduction near rational values of the tune, ν0 = p/q ≠ 1/3, are due to the creation of heteroclinic tangles and do not affect the stability of the design orbit. For a machine with an arbitrary number of localized sextupolar fields, we obtain the dependence of the nonlinear tune on the amplitude of the particles and the parameters of the accelerator. The predictions for the nonlinear tune agree with the numerical simulations within the maximum relative error of 6% at large amplitudes (~ dynamic aperture) and within the relative error range 1%–2% at intermediate amplitudes (~ half dynamic aperture).

Analysis of rough voice from a nonlinear dynamics point of view

Because the vocal folds are nonlinear systems with many degrees of freedom, bifurcations and chaos should be expected for certain parameter configurations. Indeed, bifurcations and chaos appear in some of the simplest models of vocal fold vibration. They are also seen in more complex models, in models that incorporate left–right asymmetries, and in biomechanical simulations of vocal fold oscillations. Furthermore, an acoustical analysis of many types of rough voice (e.g., creaky voice, vocal fry, and newborn infant cries) reveals an intimate relationship between voice mechanics and bifurcations and chaos. Likely physiological mechanisms of the observed nonlinear phenomena are presented. Basic concepts and analysis techniques from nonlinear dynamics are reviewed and compared with traditional voice analysis schemes. Finally, it is shown how much of the terminology in the literature describing rough voice can be unified within the framework of nonlinear dynamics. [This research was supported by Grant. No. P60 DC00976 from the National Institute on Deafness and Other Communication Disorders.]

Reaction-transport dynamics and dislocation patterns in deformed materials

Dislocation patterns arise in a large variety of situations where crystalline materials are submitted to mechanical constraints such as cyclic or monotonous loadings. It may be shown that, under the combined effect of defect motion and interactions, uniform dislocation distribution may become unstable versus spatial modulations leading to the formation of various kinds of patterns going from cellular to wall or labyrinth structures. The formation of the wall structure of persistent slip bands in fatigued metals is discussed and analyzed in the framework of nonlinear dynamics of the reaction-transport type. The important of the concepts of instability and self-organization in the fundamental understanding of materials instabilities and defect microstructures is emphasized.