Sensitivity Resonance and Attractor Morphing Quantified by Sensitivity Vector Fields for Parameter Reconstruction

Abstract

A novel method of parameter variation reconstruction for systems exhibiting chaotic dynamics is presented. The algorithm reconstructs variations of system parameters without the need for explicit system equations of motion, or knowledge of the nominal parameter values. The concept of a sensitivity vector field (SVF) is developed. This construct captures geometrical deformations to the dynamical attractor of the system in state space. These fields are collected by means of a proposed unique approach referred to as point cloud averaging (PCA). PCA is applied to discrete time series data from the system with nominal parameter values (healthy) and the system with changed parameters. Test variations are reconstructed from an optimal basis of SVF snapshots which is generated by means of proper orthogonal decomposition. The method is applied to two system models, a magneto-elastic oscillator (MEO) and an atomic force microscope (AFM). The method is shown to be highly accurate, and capable of identifying multiple simultaneous variations. The success of the method as applied to an AFM and a MEO indicates a potential for highly accurate readings by exploiting the geometric features of observed chaotic vibrations. An exciting new phenomenon referred to as sensitivity resonance was also observed, and some implications regarding its use in further improving algorithm performance are discussed.

Sensitivity vector fields in time-delay coordinate embeddings: Theory and experiment

“…Thus, our focus is constructing eSVs using local models exclusively, specifically using embedded point cloud averaging (ePCA) (similar to the original PCA [7]). …”

“…X t k holds the n embedded states from the varied data set in neighborhood k, and m is the embedding dimension. This is the original PCA formulation [7]. For ePCA, we enforce the additional condition that…”

“…For example, when a system is significantly nonlinear, when it has a low quality factor, or when multiple simultaneous parametric variations need to be distinguished from one another, other methods may be preferable. A specific alternative championed here is based on sensitivity vector fields (SVFs) [5][6][7], which quantify how dynamical system attractors deform under parametric variations. Because a SVF consists of a field of vectors distributed in state space, it can be successful even when a frequency-shift method would fail.…”

Sensitivity vector fields for atomic force microscopes

“…The proposed approach can be used in many areas, such as system identification, sensing, damage detection, and others [9,26]. The approach allows the detection of simultaneous variations of multiple parameters by exploiting the morphology of chaotic attractors.…”

“…Hashmi and Epureanu [9] have shown the basic concept for the application of SVFs to AFM. Herein, that approach is expanded upon for a multi-mode system where mode shapes vary due to perturbation in system parameters.…”