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ON COMPUTATIONAL MODELING OF CYBER-PHYSICAL SYSTEMS ON THE BASIS OF HEXAGONAL LATTICE
Abstract
This work initiates a study toward developing and applying computation models of cyber-physical systems on the basis of hexagonal lattice. The focus is on mathematical description of continuous population dynamics combined with dynamic logic used for discrete events. First, we introduce a class of lattice differential equations with time delay simulating interactions within hexagonal pixels. Spatial operator is modeling diffusion-like interaction between hexagonal pixels. We then use the syntax of dynamic logic to describe discrete states of the hexagonal pixel. Electrical signal which is simulated as a number of pixels with the certain state is important from viewpoint of CPS design. Stability research is focused on a notion of practical stability. For this purpose we constructed specific randomized multivariate algorithm which provides a probabilistic estimate of the practical stability of the system. In particular, we use Monte Carlo technique. It analyses both initial conditions and time delay and rate parameters. The experimental results obtained provide a complete analysis of immunosensor model stability with respect to changes of time delay. Namely, as the time delay was increased, the stable endemic solution changed at a critical value to a stable limit cycle. Further, when increasing the time delay, the behavior changed from convergence to simple limit cycle to convergence to complicated limit cycles with an increasing number of local maxima and minima per cycle until at sufficiently high time delay the behavior became chaotic. Such behavior can be seen using both phase portraits, tile plots and simulated electrical signal. Mathematical models and algorithms, which are developed in the work, may be considered as additional skills of cyber-physical system. There is shown their software implementation as methods in language R.
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https://www.redblobgames.com/grids/hexagons/
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