CNCS Graduate Certificate Recipient

Jonathan M. Nichols

Thesis Title: Applications of Nonlinear Time-Series Analysis

Ph.D. Final Defense Date:  June 6, 2002

Ph.D. Dissertation Committee:

Lawrence N. Virgin (Chair)
Henri P. Gavin
Henry S. Greenside
Laurens E. Howle
Michael D. Todd

Abstract:

In this work, new applications in chaos theory and nonlinear time-series analysis are explored. Tools for attractor-based analysis are developed along with a complete description of invariant measures. The focus is on the computation of dimension and Lyapunov spectra from a single time-history for the purposes of system identification. The need for accurate attractor reconstruction is stressed as it may have severe effects on the quality of estimated invariants and of attractor based predictions.

These tools are then placed in the context of several different problems of importance to the engineering community. Dimension and Lyaponuv spectra are used to indicate the operating regime of a nonlinear mechanical oscillator. Subtle changes to the way in which the oscillator is forced may give rise to a response with different state space characteristics. These differences are clearly discernible using invariant measures yet are undetectable using linear-based techniques. A state space approach is also used to extract damping estimates from the oscillator by means of the complete Lyapunov spectrum. The sum of the exponents may be thought of as the average divergence of the system which will, for a viscous damping model, provide quantitative information about the coefficient of viscous damping.

The notion of chaotic excitation of a linear system is also explored. A linear structure subject to chaotic excitation will effectively act as a filter. The resulting dynamical interaction gives rise to response (filtered) attractors which possess information about the linear system. Differences in the geometric properties of the filtered attractors are used to detect damage in structures. These attractor-based statistics are shown to be more robust indicators of damage than linear-based statistics (e.g. mode shapes, frequencies, etc.). The same procedure is also used to estimate the coefficient of viscous damping for a multi-degree-of-freedom linear structure.