NOISE-INFLUENCED DYNAMICS OF NONLINEAR OSCILLATORS

Abstract

Noise is usually considered detrimental to the performance of a system and the effects of noise are usually mitigated through design and/or control. In this dissertation, noise-influenced phenomena and qualitative changes in responses of nonlinear systems with noise are explored. Here, the author considers a range of nonlinear dynamical systems, including an array of nonlinear, coupled oscillators, a vertically excited pendulum, the Duffing oscillator, and a Rayleigh-Duffing mixed type oscillator. These systems are studied analytically and numerically via stochastic direct numerical integration, and analytically via the Fokker-Planck equation. The array of nonlinear, coupled oscillators is also experimentally studied. The topics covered in this dissertation are as follows: i) the destruction and formation of energy localizations in an array of oscillators, ii) a technique to stabilize an inverted pendulum by using noise, iii) a noise-utilizing control scheme, iv) the effects of noise on the response of a nonlinear system that exhibits chaotic behavior, v) and the effects of phase lag on the information rate of a Duffing oscillator. The understanding gained through this dissertation efforts can be of benefit to a variety of nonlinear systems, including structural systems at the macro-scale, micro-scale, and nano-scale.