Chaos in Vallis' asymmetric Lorenz model for El Nino

AbstractWe consider Vallis’ symmetric and asymmetric Lorenz models for El Niño—systems of autonomous ordinary differential equations in 3D—with the usual parameters and, in both cases, by using rigorous numerics, we locate topological horseshoes in iterates of Poincaré return maps. The computer-assisted proofs follow the standard Mischaikow–Mrozek–Zgliczynski approach. The novelty is a dimension reduction method, a direct exploitation of numerical Lorenz-like maps associated to the two components of the Poincaré section

Learning the Structure of High-Dimensional Manifolds with Self-Organizing Maps for Accurate Information Extraction

This paper was submitted by the author prior to final official version. For official version please see http://hdl.handle.net/1911/70515This work aims to improve the capability of accurate information extraction from high-dimensional data, with a specific neural learning paradigm, the Self-Organizing Map (SOM). The SOM is an unsupervised learning algorithm that can faithfully sense the manifold structure and support supervised learning of relevant information from the data. Yet open problems regarding SOM learning exist. We focus on the following two issues. 1. Evaluation of topology preservation. Topology preservation is essential for SOMs in faithful representation of manifold structure. However, in reality, topology violations are not unusual, especially when the data have complicated structure. Measures capable of accurately quantifying and informatively expressing topology violations are lacking. One contribution of this work is a new measure, the Weighted Differential Topographic Function (WDTF), which differentiates an existing measure, the Topographic Function (TF), and incorporates detailed data distribution as an importance weighting of violations to distinguish severe violations from insignificant ones. Another contribution is an interactive visual tool, TopoView, which facilitates the visual inspection of violations on the SOM lattice. We show the effectiveness of the combined use of the WDTF and TopoView through a simple two-dimensional data set and two hyperspectral images. 2. Learning multiple latent variables from high-dimensional data. We use an existing two-layer SOM-hybrid supervised architecture, which captures the manifold structure in its SOM hidden layer, and then, uses its output layer to perform the supervised learning of latent variables. In the customary way, the output layer only uses the strongest output of the SOM neurons. This severely limits the learning capability. We allow multiple, k, strongest responses of the SOM neurons for the supervised learning. Moreover, the fact that different latent variables can be best learned with different values of k motivates a new neural architecture, the Conjoined Twins, which extends the existing architecture with additional copies of the output layer, for preferential use of different values of k in the learning of different latent variables. We also automate the customization of k for different variables with the statistics derived from the SOM. The Conjoined Twins shows its effectiveness in the inference of two physical parameters from Near-Infrared spectra of planetary ices

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

Doctor of Philosophy

dissertationWith modern computational resources rapidly advancing towards exascale, large-scale simulations useful for understanding natural and man-made phenomena are becoming in- creasingly accessible. As a result, the size and complexity of data representing such phenom- ena are also increasing, making the role of data analysis to propel science even more integral. This dissertation presents research on addressing some of the contemporary challenges in the analysis of vector fields--an important type of scientific data useful for representing a multitude of physical phenomena, such as wind flow and ocean currents. In particular, new theories and computational frameworks to enable consistent feature extraction from vector fields are presented. One of the most fundamental challenges in the analysis of vector fields is that their features are defined with respect to reference frames. Unfortunately, there is no single ""correct"" reference frame for analysis, and an unsuitable frame may cause features of interest to remain undetected, thus creating serious physical consequences. This work develops new reference frames that enable extraction of localized features that other techniques and frames fail to detect. As a result, these reference frames objectify the notion of ""correctness"" of features for certain goals by revealing the phenomena of importance from the underlying data. An important consequence of using these local frames is that the analysis of unsteady (time-varying) vector fields can be reduced to the analysis of sequences of steady (time- independent) vector fields, which can be performed using simpler and scalable techniques that allow better data management by accessing the data on a per-time-step basis. Nevertheless, the state-of-the-art analysis of steady vector fields is not robust, as most techniques are numerical in nature. The residing numerical errors can violate consistency with the underlying theory by breaching important fundamental laws, which may lead to serious physical consequences. This dissertation considers consistency as the most fundamental characteristic of computational analysis that must always be preserved, and presents a new discrete theory that uses combinatorial representations and algorithms to provide consistency guarantees during vector field analysis along with the uncertainty visualization of unavoidable discretization errors. Together, the two main contributions of this dissertation address two important concerns regarding feature extraction from scientific data: correctness and precision. The work presented here also opens new avenues for further research by exploring more-general reference frames and more-sophisticated domain discretizations

Visuelle Analyse großer Partikeldaten

Partikelsimulationen sind eine bewährte und weit verbreitete numerische Methode in der Forschung und Technik. Beispielsweise werden Partikelsimulationen zur Erforschung der Kraftstoffzerstäubung in Flugzeugturbinen eingesetzt. Auch die Entstehung des Universums wird durch die Simulation von dunkler Materiepartikeln untersucht. Die hierbei produzierten Datenmengen sind immens. So enthalten aktuelle Simulationen Billionen von Partikeln, die sich über die Zeit bewegen und miteinander interagieren. Die Visualisierung bietet ein großes Potenzial zur Exploration, Validation und Analyse wissenschaftlicher Datensätze sowie der zugrundeliegenden Modelle. Allerdings liegt der Fokus meist auf strukturierten Daten mit einer regulären Topologie. Im Gegensatz hierzu bewegen sich Partikel frei durch Raum und Zeit. Diese Betrachtungsweise ist aus der Physik als das lagrange Bezugssystem bekannt. Zwar können Partikel aus dem lagrangen in ein reguläres eulersches Bezugssystem, wie beispielsweise in ein uniformes Gitter, konvertiert werden. Dies ist bei einer großen Menge an Partikeln jedoch mit einem erheblichen Aufwand verbunden. Darüber hinaus führt diese Konversion meist zu einem Verlust der Präzision bei gleichzeitig erhöhtem Speicherverbrauch. Im Rahmen dieser Dissertation werde ich neue Visualisierungstechniken erforschen, welche speziell auf der lagrangen Sichtweise basieren. Diese ermöglichen eine effiziente und effektive visuelle Analyse großer Partikeldaten

Numerical solution of differential equations through empirical eigenfunction expansions

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 1995.Includes bibliographical references (leaves 184-190).by Peter S. Wyckoff.Ph.D

Computational dynamical systems analysis : Bogdanov-Takens points and an economic model

The subject of this thesis is the bifurcation analysis of dynamical systems (ordinary differential equations and iterated maps). A primary aim is to study the branch of homoclinic solutions that emerges from a Bogdanov-Takens point. The problem of approximating such branch has been studied intensively but neither an exact solution was ever found nor a higher-order approximation has been obtained. We use the classical ``blow-up'' technique to reduce an appropriate normal form near a Bogdanov-Takens bifurcation in a generic smooth autonomous ordinary differential equations to a perturbed Hamiltonian system. With a regular perturbation method and a generalization of the Lindstedt-Poincare' perturbation method, we derive two explicit third-order corrections of the unperturbed homoclinic orbit and parameter value. We prove that both methods lead to the same homoclinic parameter value as the classical Melnikov technique and the branching method. We show that the regular perturbation method leads to a ``parasitic turn'' near the saddle point while the Lindstedt-Poincare' solution does not have this turn, making it more suitable for numerical implementation. To obtain the normal form on the center manifold, we apply the standard parameter dependent center manifold reduction combined with the normalization, using the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we correct the parameter transformation existing in the literature. The generic homoclinic predictors are applied to explicitly compute the homoclinic solutions in the Gray-Scott kinetic model. The actual implementation of both predictors in the MATLAB continuation package MatCont and five numerical examples illustrating its efficiency are discussed. Besides, the thesis discusses the possibility to use the derived homoclinic predictor of generic ordinary differential equations to continue the branches of homoclinic tangencies in the Bogdanov-Takens map. The second part of this thesis is devoted to the application of bifurcation theory to analyze the dynamic and chaotic behaviors of a nonlinear economic model. The thesis studies the monopoly model with cubic price and quadratic marginal cost functions. We present fundamental corrections to the earlier studies of the model and a complete discussion of the existence of cycles of period 4. A numerical continuation method is used to compute branches of solutions of period 5, 10, 13 and 17 and to determine the stability regions of these solutions. General formulas for solutions of period 4 are derived analytically. We show that the solutions of period 4 are never linearly asymptotically stable. A nonlinear stability criterion is combined with basin of attraction analysis and simulation to determine the stability region of the 4-cycles. This corrects the erroneous linear stability analysis in previous studies of the model. The chaotic and periodic behavior of the monopoly model are further analyzed by computing the largest Lyapunov exponents, and this confirms the above mentioned results

Capture and generalisation of close interaction with objects

Robust manipulation capture and retargeting has been a longstanding goal in both the fields of animation and robotics. In this thesis I describe a new approach to capture both the geometry and motion of interactions with objects, dealing with the problems of occlusion by the use of magnetic systems, and performing the reconstruction of the geometry by an RGB-D sensor alongside visual markers. This ‘interaction capture’ allows the scene to be described in terms of the spatial relationships between the character and the object using novel topological representations such as the Electric Parameters, which parametrise the outer space of an object using properties of the surface of the object. I describe the properties of these representations for motion generalisation and discuss how they can be applied to the problems of human-like motion generation and programming by demonstration. These generalised interactions are shown to be valid by demonstration of retargeting grasping and manipulation to robots with dissimilar kinematics and morphology using only local, gradient-based planning