Welcome to the micro-chaos mini site!

Here You can find papers, tools and other materials related to the phenomenon called micro chaos (small amplitude chaotic oscillations caused by digital effects - sampling, delay and round off - in control systems)

Cell Mapping Tools


For the open source cell mapping C++ library, please visit the following site: Cell mapping tools documentation (gyebro.com/cmdocs).

All papers and news with the micro-chaos tag:

Multi-baker map as a model of digital PD control

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.

Experimental investigation of micro-chaos

Micro-chaos is a phenomenon when small amplitude chaotic oscillations are inflicted by digital effects (sampling, roundoff and processing delay). In previous works, various digitally controlled unstable linear mechanical systems were analysed; the corresponding micro-chaos maps were derived and the coexistence of several disconnected chaotic attractors was proven. The distance of the farthest attractor from the desired state can be quite large, while the size of these attractors is usually negligible from practical point of view.

Methods for the Quick Analysis of Micro-chaos

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.

Cell mapping methods for investigating micro-chaos

Micro-chaos is a phenomenon when small amplitude chaotic oscillations are caused by digital effects (sampling, round-off and processing delay). In previous works, various digitally controlled unstable linear mechanical systems were analysed and the corresponding micro-chaos maps were derived. It was proven [1, 2], that several chaotic attractors coexist in the state space of these maps, and the distance of the farthest attractor from the desired state can be quite large. This is why the phenomenon could be the source of remarkable control error.

Numerical Exploration of Micro-chaotic Behaviour

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.