Transition of planar Couette flow at infinite Reynolds numbers

An outline of the state space of planar Couette flow at high Reynolds numbers (Re<105) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of the hairpin vortex state (HVS) asymptotically approaches the primary (laminar) state with increasing Re. It is also predicted that the lower branch of the HVS at high Re belongs to the stability boundary that initiates a transition to turbulence, and that one of the unstable manifolds of the lower branch of HVS lies on the boundary. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite Re limit. © 2013 American Physical Society.

Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point

Even simple hybrid automata like the classic bouncing ball can exhibit Zeno behavior. The existence of this type of behavior has so far forced a large class of simulators to either ignore some events or risk looping indefinitely. This in turn forces modelers to either insert ad-hoc restrictions to circumvent Zeno behavior or to abandon hybrid automata. To address this problem, we take a fresh look at event detection and localization. A key insight that emerges from this investigation is that an enclosure for a given time interval can be valid independent of the occurrence of a given event. Such an event can then even occur an unbounded number of times. This insight makes it possible to handle some types of Zeno behavior. If the post-Zeno state is defined explicitly in the given model of the hybrid automaton, the computed enclosure covers the corresponding trajectory that starts from the Zeno point through a restarted evolution.

Analysis and optimisation of the performance of nonlinear optical communication systems

We investigate the feasibility of simultaneous suppressing of the amplification noise and nonlinearity, representing the most fundamental limiting factors in modern optical communication. To accomplish this task we developed a general design optimisation technique, based on concepts of noise and nonlinearity management. We demonstrate the immense efficiency of the novel approach by applying it to a design optimisation of transmission lines with periodic dispersion compensation using Raman and hybrid Raman-EDFA amplification. Moreover, we showed, using nonlinearity management considerations, that the optimal performance in high bit-rate dispersion managed fibre systems with hybrid amplification is achieved for a certain amplifier spacing – which is different from commonly known optimal noise performance corresponding to fully distributed amplification. Required for an accurate estimation of the bit error rate, the complete knowledge of signal statistics is crucial for modern transmission links with strong inherent nonlinearity. Therefore, we implemented the advanced multicanonical Monte Carlo (MMC) method, acknowledged for its efficiency in estimating distribution tails. We have accurately computed acknowledged for its efficiency in estimating distribution tails. We have accurately computed marginal probability density functions for soliton parameters, by numerical modelling of Fokker-Plank equation applying the MMC simulation technique. Moreover, applying a powerful MMC method we have studied the BER penalty caused by deviations from the optimal decision level in systems employing in-line 2R optical regeneration. We have demonstrated that in such systems the analytical linear approximation that makes a better fit in the central part of the regenerator nonlinear transfer function produces more accurate approximation of the BER and BER penalty. We present a statistical analysis of RZ-DPSK optical signal at direct detection receiver with Mach-Zehnder interferometer demodulation

Analysis of magnetoencephaoographic data as a nonlinear dynamical system

This thesis presents the results from an investigation into the merits of analysing Magnetoencephalographic (MEG) data in the context of dynamical systems theory. MEG is the study of both the methods for the measurement of minute magnetic flux variations at the scalp, resulting from neuro-electric activity in the neocortex, as well as the techniques required to process and extract useful information from these measurements. As a result of its unique mode of action - by directly measuring neuronal activity via the resulting magnetic field fluctuations - MEG possesses a number of useful qualities which could potentially make it a powerful addition to any brain researcher's arsenal. Unfortunately, MEG research has so far failed to fulfil its early promise, being hindered in its progress by a variety of factors. Conventionally, the analysis of MEG has been dominated by the search for activity in certain spectral bands - the so-called alpha, delta, beta, etc that are commonly referred to in both academic and lay publications. Other efforts have centred upon generating optimal fits of "equivalent current dipoles" that best explain the observed field distribution. Many of these approaches carry the implicit assumption that the dynamics which result in the observed time series are linear. This is despite a variety of reasons which suggest that nonlinearity might be present in MEG recordings. By using methods that allow for nonlinear dynamics, the research described in this thesis avoids these restrictive linearity assumptions. A crucial concept underpinning this project is the belief that MEG recordings are mere observations of the evolution of the true underlying state, which is unobservable and is assumed to reflect some abstract brain cognitive state. Further, we maintain that it is unreasonable to expect these processes to be adequately described in the traditional way: as a linear sum of a large number of frequency generators. One of the main objectives of this thesis will be to prove that much more effective and powerful analysis of MEG can be achieved if one were to assume the presence of both linear and nonlinear characteristics from the outset. Our position is that the combined action of a relatively small number of these generators, coupled with external and dynamic noise sources, is more than sufficient to account for the complexity observed in the MEG recordings. Another problem that has plagued MEG researchers is the extremely low signal to noise ratios that are obtained. As the magnetic flux variations resulting from actual cortical processes can be extremely minute, the measuring devices used in MEG are, necessarily, extremely sensitive. The unfortunate side-effect of this is that even commonplace phenomena such as the earth's geomagnetic field can easily swamp signals of interest. This problem is commonly addressed by averaging over a large number of recordings. However, this has a number of notable drawbacks. In particular, it is difficult to synchronise high frequency activity which might be of interest, and often these signals will be cancelled out by the averaging process. Other problems that have been encountered are high costs and low portability of state-of-the- art multichannel machines. The result of this is that the use of MEG has, hitherto, been restricted to large institutions which are able to afford the high costs associated with the procurement and maintenance of these machines. In this project, we seek to address these issues by working almost exclusively with single channel, unaveraged MEG data. We demonstrate the applicability of a variety of methods originating from the fields of signal processing, dynamical systems, information theory and neural networks, to the analysis of MEG data. It is noteworthy that while modern signal processing tools such as independent component analysis, topographic maps and latent variable modelling have enjoyed extensive success in a variety of research areas from financial time series modelling to the analysis of sun spot activity, their use in MEG analysis has thus far been extremely limited. It is hoped that this work will help to remedy this oversight.

The development and stability analysis of a nonlinear growth model for microorganisms

A nonlinear dynamic model of microbial growth is established based on the theories of the diffusion response of thermodynamics and the chemotactic response of biology. Except for the two traditional variables, i.e. the density of bacteria and the concentration of attractant, the pH value, a crucial influencing factor to the microbial growth, is also considered in this model. The pH effect on the microbial growth is taken as a Gaussian function G0e-(f- fc)2/G1, where G0, G1 and fc are constants, f represents the pH value and fc represents the critical pH value that best fits for microbial growth. To study the effects of the reproduction rate of the bacteria and the pH value on the stability of the system, three parameters a, G0 and G1 are studied in detail, where a denotes the reproduction rate of the bacteria, G0 denotes the impacting intensity of the pH value to microbial growth and G1 denotes the bacterial adaptability to the pH value. When the effect of the pH value of the solution which microorganisms live in is ignored in the governing equations of the model, the microbial system is more stable with larger a. When the effect of the bacterial chemotaxis is ignored, the microbial system is more stable with the larger G1 and more unstable with the larger G0 for f0 > fc. However, the stability of the microbial system is almost unaffected by the variation G0 and G1 and it is always stable for f0 < fc under the assumed conditions in this paper. In the whole system model, it is more unstable with larger G1 and more stable with larger G0 for f0 < fc. The system is more stable with larger G1 and more unstable with larger G0 for f0 > fc. However, the system is more unstable with larger a for f0 < fc and the stability of the system is almost unaffected by a for f0 > fc. The results obtained in this study provide a biophysical insight into the understanding of the growth and stability behavior of microorganisms.

Nonlinear speech analysis algorithms mapped to a standard metric achieve clinically useful quantification of average Parkinson's disease symptom severity

The standard reference clinical score quantifying average Parkinson's disease (PD) symptom severity is the Unified Parkinson's Disease Rating Scale (UPDRS). At present, UPDRS is determined by the subjective clinical evaluation of the patient's ability to adequately cope with a range of tasks. In this study, we extend recent findings that UPDRS can be objectively assessed to clinically useful accuracy using simple, self-administered speech tests, without requiring the patient's physical presence in the clinic. We apply a wide range of known speech signal processing algorithms to a large database (approx. 6000 recordings from 42 PD patients, recruited to a six-month, multi-centre trial) and propose a number of novel, nonlinear signal processing algorithms which reveal pathological characteristics in PD more accurately than existing approaches. Robust feature selection algorithms select the optimal subset of these algorithms, which is fed into non-parametric regression and classification algorithms, mapping the signal processing algorithm outputs to UPDRS. We demonstrate rapid, accurate replication of the UPDRS assessment with clinically useful accuracy (about 2 UPDRS points difference from the clinicians' estimates, p < 0.001). This study supports the viability of frequent, remote, cost-effective, objective, accurate UPDRS telemonitoring based on self-administered speech tests. This technology could facilitate large-scale clinical trials into novel PD treatments.

Numerical analysis of artificial neural network and volterra-based nonlinear equalizers for coherent optical OFDM

One major drawback of coherent optical orthogonal frequency-division multiplexing (CO-OFDM) that hitherto remains unsolved is its vulnerability to nonlinear fiber effects due to its high peak-to-average power ratio. Several digital signal processing techniques have been investigated for the compensation of fiber nonlinearities, e.g., digital back-propagation, nonlinear pre- and post-compensation and nonlinear equalizers (NLEs) based on the inverse Volterra-series transfer function (IVSTF). Alternatively, nonlinearities can be mitigated using nonlinear decision classifiers such as artificial neural networks (ANNs) based on a multilayer perceptron. In this paper, ANN-NLE is presented for a 16QAM CO-OFDM system. The capability of the proposed approach to compensate the fiber nonlinearities is numerically demonstrated for up to 100-Gb/s and over 1000km and compared to the benchmark IVSTF-NLE. Results show that in terms of Q-factor, for 100-Gb/s at 1000km of transmission, ANN-NLE outperforms linear equalization and IVSTF-NLE by 3.2dB and 1dB, respectively.

Amplitude response of a unilaterally constrained nonlinear micromechanical resonator

Dynamical systems that involve impacts frequently arise in engineering. This Letter reports a study of such a system at microscale that consists of a nonlinear resonator operating with an unilateral impact. The microresonators were fabricated on silicon-on-insulator wafers by using a one-mask process and then characterised by using the capacitively driving and sensing method. Numerical results concerning the dynamics of this vibro-impact system were verified by the experiments. Bifurcation analysis was used to provide a qualitative scenario of the system steady-state solutions as a function of both the amplitude and the frequency of the external driving sinusoidal voltage. The results show that the amplitude of resonant peak is levelled off owing to the impact effect and that the bandwidth of impacting is dependent upon the nonlinearity and the operating conditions.

From filters to features:Scale-space analysis of edge and blur coding in human vision

To make vision possible, the visual nervous system must represent the most informative features in the light pattern captured by the eye. Here we use Gaussian scale-space theory to derive a multiscale model for edge analysis and we test it in perceptual experiments. At all scales there are two stages of spatial filtering. An odd-symmetric, Gaussian first derivative filter provides the input to a Gaussian second derivative filter. Crucially, the output at each stage is half-wave rectified before feeding forward to the next. This creates nonlinear channels selectively responsive to one edge polarity while suppressing spurious or "phantom" edges. The two stages have properties analogous to simple and complex cells in the visual cortex. Edges are found as peaks in a scale-space response map that is the output of the second stage. The position and scale of the peak response identify the location and blur of the edge. The model predicts remarkably accurately our results on human perception of edge location and blur for a wide range of luminance profiles, including the surprising finding that blurred edges look sharper when their length is made shorter. The model enhances our understanding of early vision by integrating computational, physiological, and psychophysical approaches. © ARVO.

Assessing interactions of linear and nonlinear neuronal sources using MEG beamformers:a proof of concept

Objective: This study aimed to explore methods of assessing interactions between neuronal sources using MEG beamformers. However, beamformer methodology is based on the assumption of no linear long-term source interdependencies [VanVeen BD, vanDrongelen W, Yuchtman M, Suzuki A. Localization of brain electrical activity via linearly constrained minimum variance spatial filtering. IEEE Trans Biomed Eng 1997;44:867-80; Robinson SE, Vrba J. Functional neuroimaging by synthetic aperture magnetometry (SAM). In: Recent advances in Biomagnetism. Sendai: Tohoku University Press; 1999. p. 302-5]. Although such long-term correlations are not efficient and should not be anticipated in a healthy brain [Friston KJ. The labile brain. I. Neuronal transients and nonlinear coupling. Philos Trans R Soc Lond B Biol Sci 2000;355:215-36], transient correlations seem to underlie functional cortical coordination [Singer W. Neuronal synchrony: a versatile code for the definition of relations? Neuron 1999;49-65; Rodriguez E, George N, Lachaux J, Martinerie J, Renault B, Varela F. Perception's shadow: long-distance synchronization of human brain activity. Nature 1999;397:430-3; Bressler SL, Kelso J. Cortical coordination dynamics and cognition. Trends Cogn Sci 2001;5:26-36]. Methods: Two periodic sources were simulated and the effects of transient source correlation on the spatial and temporal performance of the MEG beamformer were examined. Subsequently, the interdependencies of the reconstructed sources were investigated using coherence and phase synchronization analysis based on Mutual Information. Finally, two interacting nonlinear systems served as neuronal sources and their phase interdependencies were studied under realistic measurement conditions. Results: Both the spatial and the temporal beamformer source reconstructions were accurate as long as the transient source correlation did not exceed 30-40 percent of the duration of beamformer analysis. In addition, the interdependencies of periodic sources were preserved by the beamformer and phase synchronization of interacting nonlinear sources could be detected. Conclusions: MEG beamformer methods in conjunction with analysis of source interdependencies could provide accurate spatial and temporal descriptions of interactions between linear and nonlinear neuronal sources. Significance: The proposed methods can be used for the study of interactions between neuronal sources. © 2005 International Federation of Clinical Neurophysiology. Published by Elsevier Ireland Ltd. All rights reserved.

Nonlinear dynamics of a periodically driven Duffing resonator coupled to a Van der Pol oscillator

We explore the dynamics of a periodically driven Duffing resonator coupled elastically to a van der Pol oscillator in the case of 1?:?1 internal resonance in the cases of weak and strong coupling. Whilst strong coupling leads to dominating synchronization, the weak coupling case leads to a multitude of complex behaviours. A two-time scales method is used to obtain the frequency-amplitude modulation. The internal resonance leads to an antiresonance response of the Duffing resonator and a stagnant response (a small shoulder in the curve) of the van der Pol oscillator. The stability of the dynamic motions is also analyzed. The coupled system shows a hysteretic response pattern and symmetry-breaking facets. Chaotic behaviour of the coupled system is also observed and the dependence of the system dynamics on the parameters are also studied using bifurcation analysis.

A preliminary study into emergent behaviours in a lattice of interacting nonlinear resonators and oscillators

Future sensor arrays will be composed of interacting nonlinear components with complex behaviours with no known analytic solutions. This paper provides a preliminary insight into the expected behaviour through numerical and analytical analysis. Specically, the complex behaviour of a periodically driven nonlinear Duffing resonator coupled elastically to a van der Pol oscillator is investigated as a building block in a 2D lattice of such units with local connectivity. An analytic treatment of the 2-device unit is provided through a two-time-scales approach and the stability of the complex dynamic motion is analysed. The pattern formation characteristics of a 2D lattice composed of these units coupled together through nearest neighbour interactions is analysed numerically for parameters appropriate to a physical realisation through MEMS devices. The emergent patterns of global and cluster synchronisation are investigated with respect to system parameters and lattice size.

A study of cluster analysis techniques and their applications

This thesis seeks to describe the development of an inexpensive and efficient clustering technique for multivariate data analysis. The technique starts from a multivariate data matrix and ends with graphical representation of the data and pattern recognition discriminant function. The technique also results in distances frequency distribution that might be useful in detecting clustering in the data or for the estimation of parameters useful in the discrimination between the different populations in the data. The technique can also be used in feature selection. The technique is essentially for the discovery of data structure by revealing the component parts of the data. lhe thesis offers three distinct contributions for cluster analysis and pattern recognition techniques. The first contribution is the introduction of transformation function in the technique of nonlinear mapping. The second contribution is the us~ of distances frequency distribution instead of distances time-sequence in nonlinear mapping, The third contribution is the formulation of a new generalised and normalised error function together with its optimal step size formula for gradient method minimisation. The thesis consists of five chapters. The first chapter is the introduction. The second chapter describes multidimensional scaling as an origin of nonlinear mapping technique. The third chapter describes the first developing step in the technique of nonlinear mapping that is the introduction of "transformation function". The fourth chapter describes the second developing step of the nonlinear mapping technique. This is the use of distances frequency distribution instead of distances time-sequence. The chapter also includes the new generalised and normalised error function formulation. Finally, the fifth chapter, the conclusion, evaluates all developments and proposes a new program. for cluster analysis and pattern recognition by integrating all the new features.