Numerical Exploration of Micro-chaotic Behaviour

Gergely Gyebrószki, Péter Tamás Nagy, Gábor Csernák
Finno-Ugric International Conference on Mechanics – 2013, book of abstracts
Published at: 
Monday, August 12, 2013
conference presentation

In [1] and [2] the micro-chaotic behaviour of digitally controlled systems was investigated. Micro-chaos is a phenomenon, when digital effects (sampling, round-off and processing delay) lead to small-amplitude chaotic oscillations. The so-called micro-chaos maps of various digitally controlled unstable linear mechanical systems were derived, and it was shown, that several disconnected attractors may exist in the phase-space of these maps. Although the size of these attractors is usually negligible, the distance of the farthest attractor from the desired state can be rather large. It has also been proved mathematically that in a couple of simple cases the evolving vibrations are indeed chaotic. Our current goal is to compile a set of numerical methods which can be used to quickly examine the micro-chaotic behaviour of realistic digitally controlled systems. For example, cell mapping techniques can be applied for the exploration of the phase-space or the duration of transients can be estimated, based on Lyapunov exponent calculation algorithms. In the present paper, these numerical methods are used on PD- and PID-controlled unstable linear systems to qualify the chaotic behaviour in a broad range of control parameters.

[1] G. Csernák, G. Stépán: Digital Control as Source of Chaotic Behavior, International Journal of Bifurcation and Chaos, 20(5), pp. 1365-1378, 2010.
[2] G. Csernák, G. Stépán: Sampling and Round-off, as Sources of Chaos in PD-controlled Systems, Proceedings of the 19th Mediterranean Conference on Control and Automation, June 20-23, 2011, Corfu.