Multi-baker map as a model of digital PD control

Gábor Csernák, Gergely Gyebrószki, Gábor Stépán

Digital stabilization of unstable equilibria of linear systems may lead to small amplitude stochastic-like oscillations. We show that these vibrations can be related to a deterministic chaotic dynamics induced by sampling and quantization. A detailed analytical proof of chaos is presented for the case of a PD controlled oscillator: it is shown that there exists a finite attracting domain in the phase-space, the largest Lyapunov exponent is positive and the existence of a Smale horseshoe is also pointed out. The corresponding two-dimensional micro-chaos map is a multi-baker map, i.e., it consists of a finite series of baker's maps.

Published in: 
International Journal of Bifurcation and Chaos
Published at: 
Saturday, September 5, 2015