Methods for the Quick Analysis of Micro-chaos

Gergely Gyebrószki, Gábor Csernák
Applied Non-Linear Dynamical Systems Volume 93 of the series Springer Proceedings in Mathematics & Statistics (pp 383-395)
Published at: 
Monday, September 29, 2014
book chapter

Micro-chaos is a phenomenon when sampling, round-off and processing delay (shortly, digital effects) lead to chaotic oscillations with small amplitude. In previous works [1], the so-called micro-chaos maps of various digitally controlled unstable linear mechanical systems were derived and the possibility of the coexistence of several disconnected attractors was highlighted. The typical size of these attractors is usually negligible from the practical point of view, but the distance of the farthest attractor from the desired state can be rather large. This is why the phenomenon of micro-chaos can be the source of significant control error. In this paper, a set of numerical methods (e.g. cell mapping techniques for the exploration of the phase-space structure) is assembled in order to create a toolkit for the quick analysis of micro-chaotic behaviour. The elaborated methods are tested on models of PD-controlled unstable systems and the practically important characteristics of chaotic behaviour are determined.

1. Csernák, G., Stépán, G.: Digital control as source of chaotic behavior. Int. J. Bifur. Chaos 20(5), 1365–1378 (2010)