Comparative Analysis of NES and TMD Performance via High-Dimensional Invariant Manifolds

Detection of outliers or anomalies is one of the vital issues in pattern-driven data mining. Outlier detection detects the inconsistent behavior of individual objects. It is an important sector in the data mining field with several different applications such as detecting credit card fraud, hacking discovery and discovering criminal activities. It is necessary to develop tools used to uncover the critical information established in the extensive data. This paper investigated a novel method for detecting cluster outliers in a multidimensional dataset, capable of identifying the clusters and outliers for datasets containing noise. The proposed method can detect the groups and outliers left by the clustering process, like instant irregular sets of clusters (C) and outliers (O), to boost the results. The results obtained after applying the algorithm to the dataset improved in terms of several parameters. For the comparative analysis, the accurate average value and the recall value parameters are computed. The accurate average value is 74.05% of the existing COID algorithm, and our proposed algorithm has 77.21%. The average recall value is 81.19% and 89.51% of the existing and proposed algorithm, which shows that the proposed work efficiency is better than the existing COID algorithm.

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How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models

Nonlinear Dynamics ◽

10.1007/s11071-021-06957-4 ◽

2021 ◽

Author(s):

Shobhit Jain ◽

George Haller

Keyword(s):

Finite Element ◽

Degrees Of Freedom ◽

Invariant Manifolds ◽

Mechanical Systems ◽

Forced Response ◽

High Dimensional ◽

Nonlinear Phenomena ◽

Finite Element Models ◽

Engineering Structures ◽

Backbone Curves

AbstractInvariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful tools for the computation of forced response curves, backbone curves, detached resonance curves (isolas) via exact reduced-order models. For conservative nonlinear mechanical systems, Lyapunov subcenter manifolds and their reduced dynamics provide a way to identify nonlinear amplitude–frequency relationships in the form of conservative backbone curves. Despite these powerful predictions offered by invariant manifolds, their use has largely been limited to low-dimensional academic examples. This is because several challenges render their computation unfeasible for realistic engineering structures described by finite element models. In this work, we address these computational challenges and develop methods for computing invariant manifolds and their reduced dynamics in very high-dimensional nonlinear systems arising from spatial discretization of the governing partial differential equations. We illustrate our computational algorithms on finite element models of mechanical structures that range from a simple beam containing tens of degrees of freedom to an aircraft wing containing more than a hundred–thousand degrees of freedom.

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Glyph sorting: Interactive visualization for multi-dimensional data

Information Visualization ◽

10.1177/1473871613511959 ◽

2013 ◽

Vol 14 (1) ◽

pp. 76-90 ◽

Cited By ~ 26

Author(s):

David HS Chung ◽

Philip A Legg ◽

Matthew L Parry ◽

Rhodri Bown ◽

Iwan W Griffiths ◽

...

Keyword(s):

Comparative Analysis ◽

Visual Search ◽

Conceptual Framework ◽

High Dimensional ◽

Event Analysis ◽

Match Analysis ◽

Technical Aspects ◽

Visualization Process ◽

Intuitive Manner

Glyph-based visualization is an effective tool for depicting multivariate information. Since sorting is one of the most common analytical tasks performed on individual attributes of a multi-dimensional dataset, this motivates the hypothesis that introducing glyph sorting would significantly enhance the usability of glyph-based visualization. In this article, we present a glyph-based conceptual framework as part of a visualization process for interactive sorting of multivariate data. We examine several technical aspects of glyph sorting and provide design principles for developing effective, visually sortable glyphs. Glyphs that are visually sortable provide two key benefits: (1) performing comparative analysis of multiple attributes between glyphs and (2) to support multi-dimensional visual search. We describe a system that incorporates focus and context glyphs to control sorting in a visually intuitive manner and for viewing sorted results in an interactive, multi-dimensional glyph plot that enables users to perform high-dimensional sorting, analyse and examine data trends in detail. To demonstrate the usability of glyph sorting, we present a case study in rugby event analysis for comparing and analysing trends within matches. This work is undertaken in conjunction with a national rugby team. From using glyph sorting, analysts have reported the discovery of new insight beyond traditional match analysis.

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High-Dimensional Chaos Control Algorithms for Improved Piezoelectric Energy Harvesting Using a Bistable Structure

Volume 8: 27th Conference on Mechanical Vibration and Noise ◽

10.1115/detc2015-46654 ◽

2015 ◽

Author(s):

Daniel Geiyer ◽

Jeffrey L. Kauffman

Keyword(s):

Energy Harvesting ◽

Chaotic Attractor ◽

Chaos Control ◽

Invariant Manifolds ◽

Basin Of Attraction ◽

High Energy ◽

High Dimensional ◽

Piezoelectric Energy Harvesting ◽

Piezoelectric Energy ◽

Large Period

Linear cantilevered piezoelectric energy harvesters typically rely on excitation around a resonance frequency for peak operation. Compounding the problem, typical ambient environments either vary dynamically in time or possess energy distributed across a wide spectrum of frequencies. Nonlinear broadband techniques have been implemented with success, but rely on chance that steady-state high energy orbits result as opposed to the low energy or chaotic trajectories that coexist in the basin of attraction. This work aims to implement two high dimensional chaotic controllers for large period orbits located within the chaotic attractor. The first control law is defined using traditional OGY, while the second uses the principles of invariant manifolds and is therefore independent of the system Jacobian. Comparison of the two control methods aims to show that invariant principles are less computationally intensive and result in equivalent stabilized orbits. Furthermore, the only necessary measurement for control design is a single time series representing a state of the system. This article compares two methods of chaos control and their ability to stabilize a large period orbit within the chaotic attractor for improved broadband piezoelectric energy harvesting.

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SLOW INVARIANT MANIFOLDS AS CURVATURE OF THE FLOW OF DYNAMICAL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408022457 ◽

2008 ◽

Vol 18 (11) ◽

pp. 3409-3430 ◽

Cited By ~ 18

Author(s):

JEAN-MARC GINOUX ◽

BRUNO ROSSETTO ◽

LEON O. CHUA

Keyword(s):

Dynamical Systems ◽

Invariant Manifolds ◽

Piecewise Linear ◽

Slow Manifold ◽

Analytical Equation ◽

Singular Perturbation Theory ◽

High Dimensional ◽

Geometric Singular Perturbation Theory ◽

Flow Curvature ◽

Dimensional Dynamical System

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

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