Robust control of nonlinear stochastic systems by modelling conditional distributions of control signals

We introduce a novel inversion-based neuro-controller for solving control problems involving uncertain nonlinear systems that could also compensate for multi-valued systems. The approach uses recent developments in neural networks, especially in the context of modelling statistical distributions, which are applied to forward and inverse plant models. Provided that certain conditions are met, an estimate of the intrinsic uncertainty for the outputs of neural networks can be obtained using the statistical properties of networks. More generally, multicomponent distributions can be modelled by the mixture density network. In this work a novel robust inverse control approach is obtained based on importance sampling from these distributions. This importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The performance of the new algorithm is illustrated through simulations with example systems.

Numerically optimized implementations, mitigation and simulations of polarisation mode dispersion in real-time transmission systems

The aim of this thesis is to present numerical investigations of the polarisation mode dispersion (PMD) effect. Outstanding issues on the side of the numerical implementations of PMD are resolved and the proposed methods are further optimized for computational efficiency and physical accuracy. Methods for the mitigation of the PMD effect are taken into account and simulations of transmission system with added PMD are presented. The basic outline of the work focusing on PMD can be divided as follows. At first the widely-used coarse-step method for simulating the PMD phenomenon as well as a method derived from the Manakov-PMD equation are implemented and investigated separately through the distribution of a state of polarisation on the Poincaré sphere, and the evolution of the dispersion of a signal. Next these two methods are statistically examined and compared to well-known analytical models of the probability distribution function (PDF) and the autocorrelation function (ACF) of the PMD phenomenon. Important optimisations are achieved, for each of the aforementioned implementations in the computational level. In addition the ACF of the coarse-step method is considered separately, based on the result which indicates that the numerically produced ACF, exaggerates the value of the correlation between different frequencies. Moreover the mitigation of the PMD phenomenon is considered, in the form of numerically implementing Low-PMD spun fibres. Finally, all the above are combined in simulations that demonstrate the impact of the PMD on the quality factor (Q=factor) of different transmission systems. For this a numerical solver based on the coupled nonlinear Schrödinger equation is created which is otherwise tested against the most important transmission impairments in the early chapters of this thesis.

Evolution of optical pulses in fiber lines with lumped nonlinear devices as a mapping problem

We analyze the steady-state propagation of optical pulses in fiber transmission systems with lumped nonlinear optical devices (NODs) placed periodically in the line. For the first time to our knowledge, a theoretical model is developed to describe the transmission regime with a quasilinear pulse evolution along the transmission line and the point action of NODs. We formulate the mapping problem for pulse propagation in a unit cell of the line and show that in the particular application to nonlinear optical loop mirrors, the steady-state pulse characteristics predicted by the theory accurately reproduce the results of direct numerical simulations.

Auto-solitons guided by in-line nonlinear optical loop mirrors in dispersion-managed communication systems

Summary form only given. Both dispersion management and the use of a nonlinear optical loop mirror (NOLM) as a saturable absorber can improve the performance of a soliton-based communication system. Dispersion management gives the benefits of low average dispersion while allowing pulses with higher powers to propagate, which helps to suppress Gordon-Haus timing jitter without sacrificing the signal-to-noise ratio. The NOLM suppresses the buildup of amplifier spontaneous emission noise and background dispersive radiation which, if allowed to interact with the soliton, can lead to its breakup. We examine optical pulse propagation in dispersion-managed (DM) transmission system with periodically inserted in-line NOLMs. To describe basic features of the signal transmission in such lines, we develop a simple theory based on a variational approach involving Gaussian trial functions. It, has already been proved that the variational method is an extremely effective tool for description of DM solitons. In the work we manage to include in the variational description the point action of the NOLM on pulse parameters, assuming that the Gaussian pulse shape is inherently preserved by propagation through the NOLM. The obtained results are verified by direct numerical simulations

Evolution of optical pulses in fiber lines with lumped nonlinear devices as a mapping problem

We analyze the steady-state propagation of optical pulses in fiber transmission systems with lumped nonlinear optical devices (NODs) placed periodically in the line. For the first time to our knowledge, a theoretical model is developed to describe the transmission regime with a quasilinear pulse evolution along the transmission line and the point action of NODs. We formulate the mapping problem for pulse propagation in a unit cell of the line and show that in the particular application to nonlinear optical loop mirrors, the steady-state pulse characteristics predicted by the theory accurately reproduce the results of direct numerical simulations. © 2005 Springer Science+Business Media, Inc.

Auto-solitons guided by in-line nonlinear optical loop mirrors in dispersion-managed communication systems

Summary form only given. Both dispersion management and the use of a nonlinear optical loop mirror (NOLM) as a saturable absorber can improve the performance of a soliton-based communication system. Dispersion management gives the benefits of low average dispersion while allowing pulses with higher powers to propagate, which helps to suppress Gordon-Haus timing jitter without sacrificing the signal-to-noise ratio. The NOLM suppresses the buildup of amplifier spontaneous emission noise and background dispersive radiation which, if allowed to interact with the soliton, can lead to its breakup. We examine optical pulse propagation in dispersion-managed (DM) transmission system with periodically inserted in-line NOLMs. To describe basic features of the signal transmission in such lines, we develop a simple theory based on a variational approach involving Gaussian trial functions. It, has already been proved that the variational method is an extremely effective tool for description of DM solitons. In the work we manage to include in the variational description the point action of the NOLM on pulse parameters, assuming that the Gaussian pulse shape is inherently preserved by propagation through the NOLM. The obtained results are verified by direct numerical simulations

Local Bifurcations in a Nonlinear Model of a Bioreactor

This paper is partially supported by the Bulgarian Science Fund under grant Nr. DO 02– 359/2008.

Nonlinear semi-analytical model for simulation of few-mode fiber transmission

In this letter, a nonlinear semi-analytical model (NSAM) for simulation of few-mode fiber transmission is proposed. The NSAM considers the mode mixing arising from the Kerr effect and waveguide imperfections. An analytical explanation of the model is presented, as well as simulation results for the transmission over a two mode fiber (TMF) of 112 Gb/s using coherently detected polarization multiplexed quadrature phase-shift-keying modulation. The simulations show that by transmitting over only one of the two modes on TMFs, long-haul transmission can be realized without increase of receiver complexity. For a 6000-km transmission link, a small modal dispersion penalty is observed in the linear domain, while a significant increase of the nonlinear threshold is observed due to the large core of TMF. © 2006 IEEE.

Design rules for nonlinear spectral compression in optical fibers

We present comprehensive design rules to optimize the process of spectral compression arising from nonlinear pulse propagation in an optical fiber. Extensive numerical simulations are used to predict the performance characteristics of the process as well as to identify the optimal operational conditions within the space of system parameters. It is shown that the group velocity dispersion of the fiber is not detrimental and, in fact, helps achieve optimum compression. We also demonstrate that near-transform-limited rectangular and parabolic pulses can be generated in the region of optimum compression.

Interaction dynamics in small networks of nonlinear elements

We study a small circuit of coupled nonlinear elements to investigate general features of signal transmission through networks. The small circuit itself is perceived as building block for larger networks. Individual dynamics and coupling are motivated by neuronal systems: We consider two types of dynamical modes for an individual element, regular spiking and chattering and each individual element can receive excitatory and/or inhibitory inputs and is subjected to different feedback types (excitatory and inhibitory; forward and recurrent). Both, deterministic and stochastic simulations are carried out to study the input-output relationships of these networks. Major results for regular spiking elements include frequency locking, spike rate amplification for strong synaptic coupling, and inhibition-induced spike rate control which can be interpreted as a output frequency rectification. For chattering elements, spike rate amplification for low frequencies and silencing for large frequencies is characteristic

Optical wave turbulence and wave condensation in a nonlinear optical experiment

We present theory, numerical simulations and experimental observations of a 1D optical wave system. We show that this system is of a dual cascade type, namely, the energy cascading directly to small scales, and the photons or wave action cascading to large scales. In the optical context the inverse cascade is particularly interesting because it means the condensation of photons. We show that the cascades are induced by a six-wave resonant interaction process described by weak turbulence theory. We show that by starting with weakly nonlinear randomized waves as an initial condition, there exists an inverse cascade of photons towards the lowest wavenumbers. During the cascade nonlinearity becomes strong at low wavenumbers and, due to the focusing nature of the nonlinearity, it leads to modulational instability resulting in the formation of solitons. Further interaction of the solitons among themselves and with incoherent waves leads to the final condensate state dominated by a single strong soliton. In addition, we show the existence of the direct energy cascade numerically and that it agrees with the wave turbulence prediction.

Analysis of Molecular Dynamics Simulations of Protein Folding

Microsecond long Molecular Dynamics (MD) trajectories of biomolecular processes are now possible due to advances in computer technology. Soon, trajectories long enough to probe dynamics over many milliseconds will become available. Since these timescales match the physiological timescales over which many small proteins fold, all atom MD simulations of protein folding are now becoming popular. To distill features of such large folding trajectories, we must develop methods that can both compress trajectory data to enable visualization, and that can yield themselves to further analysis, such as the finding of collective coordinates and reduction of the dynamics. Conventionally, clustering has been the most popular MD trajectory analysis technique, followed by principal component analysis (PCA). Simple clustering used in MD trajectory analysis suffers from various serious drawbacks, namely, (i) it is not data driven, (ii) it is unstable to noise and change in cutoff parameters, and (iii) since it does not take into account interrelationships amongst data points, the separation of data into clusters can often be artificial. Usually, partitions generated by clustering techniques are validated visually, but such validation is not possible for MD trajectories of protein folding, as the underlying structural transitions are not well understood. Rigorous cluster validation techniques may be adapted, but it is more crucial to reduce the dimensions in which MD trajectories reside, while still preserving their salient features. PCA has often been used for dimension reduction and while it is computationally inexpensive, being a linear method, it does not achieve good data compression. In this thesis, I propose a different method, a nonmetric multidimensional scaling (nMDS) technique, which achieves superior data compression by virtue of being nonlinear, and also provides a clear insight into the structural processes underlying MD trajectories. I illustrate the capabilities of nMDS by analyzing three complete villin headpiece folding and six norleucine mutant (NLE) folding trajectories simulated by Freddolino and Schulten [1]. Using these trajectories, I make comparisons between nMDS, PCA and clustering to demonstrate the superiority of nMDS. The three villin headpiece trajectories showed great structural heterogeneity. Apart from a few trivial features like early formation of secondary structure, no commonalities between trajectories were found. There were no units of residues or atoms found moving in concert across the trajectories. A flipping transition, corresponding to the flipping of helix 1 relative to the plane formed by helices 2 and 3 was observed towards the end of the folding process in all trajectories, when nearly all native contacts had been formed. However, the transition occurred through a different series of steps in all trajectories, indicating that it may not be a common transition in villin folding. The trajectories showed competition between local structure formation/hydrophobic collapse and global structure formation in all trajectories. Our analysis on the NLE trajectories confirms the notion that a tight hydrophobic core inhibits correct 3-D rearrangement. Only one of the six NLE trajectories folded, and it showed no flipping transition. All the other trajectories get trapped in hydrophobically collapsed states. The NLE residues were found to be buried deeply into the core, compared to the corresponding lysines in the villin headpiece, thereby making the core tighter and harder to undo for 3-D rearrangement. Our results suggest that the NLE may not be a fast folder as experiments suggest. The tightness of the hydrophobic core may be a very important factor in the folding of larger proteins. It is likely that chaperones like GroEL act to undo the tight hydrophobic core of proteins, after most secondary structure elements have been formed, so that global rearrangement is easier. I conclude by presenting facts about chaperone-protein complexes and propose further directions for the study of protein folding.

Effects of an asymmetric friction on the nonlinear equilibration of a baroclinic system

Following some recent linear and nonlinear studies the authors examine, using numerical simulations of a classical two-layer model, the effect of an asymmetric friction on the nonlinear equilibrium of moderately unstable baroclinic systems, The results show that the presence of an asymmetric friction leads to a significant wave scale selection: ''long'' waves (in terms of their zonal wavelengths) emerge with a traditional asymmetric friction (with the upper layer less viscous than the lower layer), while only ''short'' waves dominate with a nontraditional asymmetric friction (with the lower layer less viscous than the upper layer). The role of the nonlinear interactions and. more precisely, the effects of an asymmetric friction on the wave-mean flow and wave-wave interactions; and their consequences on the wave scale selection are examined.

Nonlinear breathing modes at a defect

Recent molecular dynamics (MD) simulations of Cubero et al (1999) of a DNA duplex containing the 'rogue' base difluorotoluene (F) in place of a thymine (T) base show that breathing events can occur on the nanosecond timescale, whereas breathing events in a normal DNA duplex take place on the microsecond timescale. The main aim of this paper is to analyse a nonlinear Klein-Gordon lattice model of the DNA duplex including both nonlinear interactions between opposing bases and a defect in the interaction at one lattice site; each of which can cause localisation of energy. Solutions for a breather mode either side of the defect are derived using multiple-scales asymptotics and are pieced together across the defect to form a solution which includes the effects of the nonlinearity and the defect. We consider defects in the inter-chain interactions and in the along chain interactions. In most cases we find in-phase breather modes and/or out-of-phase breather modes, with one case displaying a shifted mode.

INVESTIGATIONS INTO MECHANISMS UNDERLYING EXTREME WAVE FORMATIONS AND COMPUTATIONALLY INTENSIVE SIMULATIONS

Various mechanisms have been proposed to explain extreme waves or rogue waves in an oceanic environment including directional focusing, dispersive focusing, wave-current interaction, and nonlinear modulational instability. The Benjamin-Feir instability (nonlinear modulational instability), however, is considered to be one of the primary mechanisms for rogue-wave occurrence. The nonlinear Schrodinger equation is a well-established approximate model based on the same assumptions as required for the derivation of the Benjamin-Feir theory. Solutions of the nonlinear Schrodinger equation, including new rogue-wave type solutions are presented in the author's dissertation work. The solutions are obtained by using a predictive eigenvalue map based predictor-corrector procedure developed by the author. Features of the predictive map are explored and the influences of certain parameter variations are investigated. The solutions are rescaled to match the length scales of waves generated in a wave tank. Based on the information provided by the map and the details of physical scaling, a framework is developed that can serve as a basis for experimental investigations into a variety of extreme waves as well localizations in wave fields. To derive further fundamental insights into the complexity of extreme wave conditions, Smoothed Particle Hydrodynamics (SPH) simulations are carried out on an advanced Graphic Processing Unit (GPU) based parallel computational platform. Free surface gravity wave simulations have successfully characterized water-wave dispersion in the SPH model while demonstrating extreme energy focusing and wave growth in both linear and nonlinear regimes. A virtual wave tank is simulated wherein wave motions can be excited from either side. Focusing of several wave trains and isolated waves has been simulated. With properly chosen parameters, dispersion effects are observed causing a chirped wave train to focus and exhibit growth. By using the insights derived from the study of the nonlinear Schrodinger equation, modulational instability or self-focusing has been induced in a numerical wave tank and studied through several numerical simulations. Due to the inherent dissipative nature of SPH models, simulating persistent progressive waves can be problematic. This issue has been addressed and an observation-based solution has been provided. The efficacy of SPH in modeling wave focusing can be critical to further our understanding and predicting extreme wave phenomena through simulations. A deeper understanding of the mechanisms underlying extreme energy localization phenomena can help facilitate energy harnessing and serve as a basis to predict and mitigate the impact of energy focusing.