Showing papers on "Nonlinear system published in 2009"

Dimensionality Reduction: A Comparative Review

Abstract: In recent years, a variety of nonlinear dimensionality reduction techniques have been proposed that aim to address the limitations of traditional techniques such as PCA and classical scaling. The paper presents a review and systematic comparison of these techniques. The performances of the nonlinear techniques are investigated on artificial and natural tasks. The results of the experiments reveal that nonlinear techniques perform well on selected artificial tasks, but that this strong performance does not necessarily extend to real-world tasks. The paper explains these results by identifying weaknesses of current nonlinear techniques, and suggests how the performance of nonlinear dimensionality reduction techniques may be improved.

Spectral analysis of nonlinear flows

Abstract: We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

Spectral analysis of nonlinear flows

Abstract: We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

Abstract: In recent years two nonlinear dispersive partial differential equations have attracted a lot of attention due to their integrable structure. We prove that both equations arise in the modeling of the propagation of shallow water waves over a flat bed. The equations capture stronger nonlinear effects than the classical nonlinear dispersive Benjamin-Bona-Mahoney and Korteweg-de Vries equations. In particular, they accomodate wave breaking phenomena.

VORO++: a three-dimensional voronoi cell library in C++.

Abstract: Voro++ is a free software library for the computation of three dimensional Voronoi cells. It is primarily designed for applications in physics and materials science, where the Voronoi tessellation can be a useful tool in the analysis of densely-packed particle systems, such as granular materials or glasses. The software comprises of several C++ classes that can be modified and incorporated into other programs. A command-line utility is also provided that can use most features of the code. Voro++ makes use of a direct cell-by-cell construction, which is particularly suited to handling special boundary conditions and walls. It employs algorithms which are tolerant for numerical precision errors, and it has been successfully employed on very large particle systems.

Time Series Prediction Using Support Vector Machines: A Survey

Abstract: Time series prediction techniques have been used in many real-world applications such as financial market prediction, electric utility load forecasting , weather and environmental state prediction, and reliability forecasting. The underlying system models and time series data generating processes are generally complex for these applications and the models for these systems are usually not known a priori. Accurate and unbiased estimation of the time series data produced by these systems cannot always be achieved using well known linear techniques, and thus the estimation process requires more advanced time series prediction algorithms. This paper provides a survey of time series prediction applications using a novel machine learning approach: support vector machines (SVM). The underlying motivation for using SVMs is the ability of this methodology to accurately forecast time series data when the underlying system processes are typically nonlinear, non-stationary and not defined a-priori. SVMs have also been proven to outperform other non-linear techniques including neural-network based non-linear prediction techniques such as multi-layer perceptrons.The ultimate goal is to provide the reader with insight into the applications using SVM for time series prediction, to give a brief tutorial on SVMs for time series prediction, to outline some of the advantages and challenges in using SVMs for time series prediction, and to provide a source for the reader to locate books, technical journals, and other online SVM research resources.

Delay Compensation for Nonlinear, Adaptive, and PDE Systems

Abstract: Preface 1. Introduction Part I. Linear Delay-ODE Cascades 2. Basic Predictor Feedback 3. Predictor Observers 4. Inverse Optimal Redesign 5. Robustness to Delay Mismatch 6. Time-Varying Delay Part II. Adaptive Control 7. Delay-Adaptive Full-State Predictor Feedback 8. Delay-Adaptive Predictor with Estimation of Actuator State 9. Trajectory Tracking Under Unknown Delay and ODE Parameters Part III. Nonlinear Systems 10. Nonlinear Predictor Feedback 11. Forward-Complete Systems 12. Strict- Feedforward Systems 13. Linearizable Strict-Feedforward Systems Part IV. PDE-ODE Cascades 14. ODEs with General Transport-Like Actuator Dynamics 15. ODEs with Heat PDE Actuator Dynamics 16. ODEs with Wave PDE Actuator Dynamics 17. Observers for ODEs Involving PDE Sensor and Actuator Dynamics Part V. Delay-PDE and PDE-PDE Cascades 18. Unstable Reaction-Diffusion PDE with Input Delay 19. Antistable Wave PDE with Input Delay 20. Other PDE-PDE Cascades Appendices A. Poincare, Agmon, and Other Basic Inequalities B. Input-Output Lemmas for LTI and LTV Systems C. Lyapunov Stability and ISS for Nonlinear ODEs D. Bessel Functions E. Parameter Projection F. Strict-Feedforward Systems: A General Design G. Strict-Feedforward Systems: A Linearizable Class H. Strict-Feedforward Systems: Not Linearizable References Index

Nonlinear Learning using Local Coordinate Coding

Abstract: This paper introduces a new method for semi-supervised learning on high dimensional nonlinear manifolds, which includes a phase of unsupervised basis learning and a phase of supervised function learning. The learned bases provide a set of anchor points to form a local coordinate system, such that each data point x on the manifold can be locally approximated by a linear combination of its nearby anchor points, and the linear weights become its local coordinate coding. We show that a high dimensional nonlinear function can be approximated by a global linear function with respect to this coding scheme, and the approximation quality is ensured by the locality of such coding. The method turns a difficult nonlinear learning problem into a simple global linear learning problem, which overcomes some drawbacks of traditional local learning methods.

Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Pms-48)

Abstract: This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

Nonlinear Auto-Oscillator Theory of Microwave Generation by Spin-Polarized Current

Abstract: This paper formulates a general analytic approach to the theory of microwave generation in magnetic nano-structures driven by spin-polarized current and reviews analytic results obtained in this theory. The proposed approach is based on the universal model of an auto-oscillator with negative damping and nonlinear frequency shift. It is demonstrated that this universal model, when applied to the case of a spin-torque oscillator (STO) based on a current-driven magnetic nano-pillar or nano-contact, gives adequate description of most of the experimentally observed properties of STO. In particular, the model describes the power and frequency of the generated microwave signal as functions of the bias current and magnetic field, predicts the magnitude and properties of the generation linewidth, and explains the STO behavior under the influence of periodic and stochastic external signals: frequency modulation, phase-locking to external signals, mutual phase-locking in an array of STO, broadening of the generation linewidth near the generation threshold, etc. The proposed nonlinear auto-oscillator theory is rather general and can be used not only for the development of practical nano-sized STO, but, also, for the description of nonlinear auto-oscillating systems of any physical nature.

Entanglement, Nonlinear Dynamics, and the Heisenberg Limit

Abstract: We show that the quantum Fisher information provides a sufficient condition to recognize multi-particle entanglement in a N qubit state. The same criterion gives a necessary and sufficient condition for sub shot-noise phase sensitivity in the estimation of a collective rotation angle θ . The analysis therefore singles out the class of entangled states which are {\it useful} to overcome classical phase sensitivity in metrology and sensors. We finally study the creation of useful entangled states by the non-linear dynamical evolution of two decoupled Bose-Einstein condensates or trapped ions.

Regularity theory for fully nonlinear integro‐differential equations

Abstract: We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump Levy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C 1,� regularity for general fully nonlinear integro- differential equations. Our estimates remain uniform as the degree of the equation approaches two, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations.

Fundamentals of vehicle–track coupled dynamics

Abstract: This paper presents a framework to investigate the dynamics of overall vehicle-track systems with emphasis on theoretical modelling, numerical simulation and experimental validation. A three-dimensional vehicle-track coupled dynamics model is developed in which a typical railway passenger vehicle is modelled as a 35-degree-of-freedom multi-body system. A traditional ballasted track is modelled as two parallel continuous beams supported by a discrete-elastic foundation of three layers with sleepers and ballasts included. The non-ballasted slab track is modelled as two parallel continuous beams supported by a series of elastic rectangle plates on a viscoelastic foundation. The vehicle subsystem and the track subsystem are coupled through a wheel-rail spatial coupling model that considers rail vibrations in vertical, lateral and torsional directions. Random track irregularities expressed by track spectra are considered as system excitations by means of a time-frequency transformation technique. A fast explicit integration method is applied to solve the large nonlinear equations of motion of the system in the time domain. A computer program named TTISIM is developed to predict the vertical and lateral dynamic responses of the vehicle-track coupled system. The theoretical model is validated by full-scale field experiments, including the speed-up test on the Beijing-Qinhuangdao line and the high-speed running test on the Qinhuangdao-Shenyang line. Differences in the dynamic responses analysed by the vehicle-track coupled dynamics and by the classical vehicle dynamics are ascertained in the case of vehicles passing through curved tracks.

The Hysteresis Bouc-Wen Model, a Survey

Abstract: Structural systems often show nonlinear behavior under severe excitations generated by natural hazards. In that condition, the restoring force becomes highly nonlinear showing significant hysteresis. The hereditary nature of this nonlinear restoring force indicates that the force cannot be described as a function of the instantaneous displacement and velocity. Accordingly, many hysteretic restoring force models were developed to include the time dependent nature using a set of differential equations. This survey contains a review of the past, recent developments and implementations of the Bouc-Wen model which is used extensively in modeling the hysteresis phenomenon in the dynamically excited nonlinear structures.

Introduction to Nonlinear Dispersive Equations

Abstract: 1. The Fourier Transform.- 2. Interpolation of Operators.- 3. Sobolev Spaces and Pseudo-Differential Operators.- 4. The Linear Schrodinger Equation.- 5. The Non-Linear Schrodinger Equation.- 6. Asymptotic Behavior for NLS Equation.- 7. Korteweg-de Vries Equation.- 8. Asymptotic Behavior for k-gKdV Equations.- 9. Other Nonlinear Dispersive Models.- 10. General Quasilinear Schrodinger Equation.- Proof of Theorem 2.8.- Proof of Lemma 4.2.- References.- Index.

Complex networks in climate dynamics. Comparing linear and nonlinear network construction methods

Abstract: Complex network theory provides a powerful framework to statistically investigate the topology of local and non-local statistical interrelationships, i.e. teleconnections, in the climate system. Climate networks constructed from the same global climatological data set using the linear Pearson correlation coefficient or the nonlinear mutual information as a measure of dynamical similarity between regions, are compared systematically on local, mesoscopic and global topological scales. A high degree of similarity is observed on the local and mesoscopic topological scales for surface air temperature fields taken from AOGCM and reanalysis data sets. We find larger differences on the global scale, particularly in the betweenness centrality field. The global scale view on climate networks obtained using mutual information offers promising new perspectives for detecting network structures based on nonlinear physical processes in the climate system.

Reversible hysteresis for broadband magnetopiezoelastic energy harvesting

Abstract: We model and experimentally validate a nonlinear energy harvester capable of bidirectional hysteresis. In particular, both hardening and softening response within the quadratic potential field of a power generating piezoelectric beam (with a permanent magnet end mass) is invoked by tuning nonlinear magnetic interactions. Not only is this technique shown to increase the bandwidth of the device but experimental results additionally verify the capability to outperform linear resonance. Engaging this nonlinear phenomenon is ideally suited to efficiently harvest energy from ambient excitations with slowly varying frequencies.

Positive solutions of nonlinear problems involving the square root of the Laplacian

Abstract: We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

Dispersion Properties of Optical Polymers

Abstract: In this report dispersion properties of dierent types of optical polymers are discussed on the base of measured refractive indices and the Cauchy‐Schott approximation. A number of dispersion curves are presented in the visible and near infrared spectral regions between 400 and 1060 nm. A comparison with some optical glasses with similar refraction is performed. The nonlinear dependence of dn=d‚ of polymer materials and test glasses on the wavelength is calculated and analyzed. Normalized dispersion curves at 550 nm and 880 nm are presented to illustrate better the dispersion of the polymers in the considered spectral regions, separately. Abbe numbers are calculated to exhibit the mean and partial dispersion.

Neural network approach to continuous-time direct adaptive optimal control for partially unknown nonlinear systems

Abstract: In this paper we present in a continuous-time framework an online approach to direct adaptive optimal control with infinite horizon cost for nonlinear systems. The algorithm converges online to the optimal control solution without knowledge of the internal system dynamics. Closed-loop dynamic stability is guaranteed throughout. The algorithm is based on a reinforcement learning scheme, namely Policy Iterations, and makes use of neural networks, in an Actor/Critic structure, to parametrically represent the control policy and the performance of the control system. The two neural networks are trained to express the optimal controller and optimal cost function which describes the infinite horizon control performance. Convergence of the algorithm is proven under the realistic assumption that the two neural networks do not provide perfect representations for the nonlinear control and cost functions. The result is a hybrid control structure which involves a continuous-time controller and a supervisory adaptation structure which operates based on data sampled from the plant and from the continuous-time performance dynamics. Such control structure is unlike any standard form of controllers previously seen in the literature. Simulation results, obtained considering two second-order nonlinear systems, are provided.

Semi-empirical dissipation source functions for ocean waves: Part I, definition, calibration and validation

Abstract: New parameterizations for the spectra dissipation of wind-generated waves are proposed. The rates of dissipation have no predetermined spectral shapes and are functions of the wave spectrum and wind speed and direction, in a way consistent with observation of wave breaking and swell dissipation properties. Namely, the swell dissipation is nonlinear and proportional to the swell steepness, and dissipation due to wave breaking is non-zero only when a non-dimensional spectrum exceeds the threshold at which waves are observed to start breaking. An additional source of short wave dissipation due to long wave breaking is introduced to represent the dissipation of short waves due to longer breaking waves. Several degrees of freedom are introduced in the wave breaking and the wind-wave generation term of Janssen (J. Phys. Oceanogr. 1991). These parameterizations are combined and calibrated with the Discrete Interaction Approximation of Hasselmann et al. (J. Phys. Oceangr. 1985) for the nonlinear interactions. Parameters are adjusted to reproduce observed shapes of directional wave spectra, and the variability of spectral moments with wind speed and wave height. The wave energy balance is verified in a wide range of conditions and scales, from gentle swells to major hurricanes, from the global ocean to coastal settings. Wave height, peak and mean periods, and spectral data are validated using in situ and remote sensing data. Some systematic defects are still present, but the parameterizations yield the best overall results to date. Perspectives for further improvement are also given.