The Hybrid Micro-chaos Map: Digitally Controlled Inverted Pendulum with Dry Friction

Digital effects (quantization, sampling and delay) can lead to small amplitude chaotic oscillations, called micro-chaos [1, 2]. Often, these vibrations are neglected due to their small amplitude or replaced by random noise, but doing so one might be unable to capture some important behavior of the digitally controlled system.One notable example is the PD-controlled inverted pendulum with quantization at the calculated control effort. Taking digital effects into account leads to separated chaotic attractors in the state space. While the amplitude of the chaotic oscillations is indeed small, these attractors are situated rather far from the desired position, introducing considerable control error. Micro-chaos is undoubtedly present in ideal models of computer-controlled mechanical systems, however an important question is still open: does it persist if a more complex model of reality is used? For instance, does it survive in the presence of dry friction?This paper answers the latter question analyzing the micro-chaos in a system with Coulomb friction. We introduce the so-called hybrid micro-chaos map that describes the behavior of a digitally controlled system with dry friction. Then, the theoretical analysis of this map is presented and numerical results are provided that were acquired using a new mathematical tool, the Clustered Simple Cell Mapping method.Lastly, we conclude, that the phenomena of micro-chaos can withstand the presence of Coulomb friction and chaotic attractors can coexist with sticking zones in the state space.

Jumps in Scale

One of cinema’s inventions is the possibility, thanks to montage, of giving equivalent onscreen size to objects across the scale, from the smallest to the largest – just as, in poetic discourse, a human army might be compared to ants walking in line. In a number of recent films, it has even become possible to imply, with the help of digital effects, the presence of an imaginary camera, capable of ‘moving through solid matter’, of circulating between atoms: a camera that has nothing in common any more with the human eye.