BMe Research Grant
Nowadays, digitally (or computer) controlled devices are widespread (from assembly line robots to 3D printers), but designing and operating them introduces new challenges. Digital control typically occurs periodically, at specific sampling times. At these moments, the computer examines the current state and calculates the necessary amount of intervention, then drives the control components. Usually time delay is also present, since some amount of computation time is needed between the measurement of the state and the control. Since digital signals are represented by finite number of bits, round-off occurs during analog to digital conversions.
These digital effects (sampling, time delay and round-off) can cause small amplitude chaotic oscillations; the so-called micro-chaos.
The discovery of the phenomenon of micro-chaos is corresponding to Dr. Gábor Stépán, Dr. György Haller and Dr. Enikov Eniko [1, 2] during the second half of the 90’s. In the past two decades, Dr. Gábor Csernák has actively contributed to the corresponding research areas.
Figure 1.: Strange attractor in the phase space of a micro-chaos map .
Due to its small amplitude, micro-chaos is usually neglected in practice, or considered as random noise. This step, although makes impossible to examine the underlying patterns of micro-chaotic behaviour or the possibility to exploit its presence in some cases.
In 2019 I have finished the open source package containing my implementation of cell-mapping algorithms (cell-mapping C++ library) which is capable of using multiple processors effectively. I have generated larger cell mapping simulations than ever before (using 4 billion cells), which show the global state space structure and fractal like attractors and basin boundaries at the same time.
Figure 8: (Click on the figure) 4 GigaPixel cell mapping result showing the state space of the micro-chaos map. Coloured regions indicate basin of attraction of fractal like, chaotic attractors situated at the x-axis.
I have analysed the so-called twofold-quantization case (when both the measured signals and the control output is quantized), which fits the reality very well [G9].
Experiments have been carried out with direct-driven inverted pendulums utilizing microcontrollers and encoders (see Fig 11) [G5]. The newest generation of experimental device (along with the latest mathematical models) enabled me to record irregular motions which can be used to prove chaos experimentally as well. One of these measurements is shown on Fig 12 and Video 2.
Figure 12: Experimentally obtained picture of a chaotic attractor. An unstable saddle point is situated in the middle of the chaotic attractor and its sawtooth-like fractal structure is clearly visible.
Video 2: Micro-chaotic oscillation in case of twofold-quantization (in case of artificially increased rounding resolutions). The measured position-velocity plot of the motion is shown on Fig. 12.
The formalism of twofold-quantization in micro-chaos and taking friction into account enabled us to experimentally show the micro-chaotic behaviour. These results should provide foundation to further publications which will help popularize the field of micro-chaos.
[G8.] Gyebrószki Gergely, Dr. Csernák Gábor: The Hybrid Micro-chaos Map: Digitally Controlled Inverted Pendulum with Dry Friction, Periodica Polytechnica Mechanical Engineering, 63(2), pp. 148–155.
[G9.] Gyebrószki Gergely, Dr. Csernák Gábor: Twofold quantization in digital control: deadzone crisis and switching line collision, Nonlinear Dynamics, 2019. (submitted, under review)
C.S. Hsu: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems, Applied Mathematical Sciences 64, 1987, Springer, Singapore