Experiment based determination of fractal dimensionality of Belousov-Zhabotinsky reaction

Tomas Nahlik, Anna Zhyrova and Dalibor Stys

Faculty of Fisheries and Protection of Waters, School of Complex Systems, University of South Bohemia, Zámek 136, 373 33 Nové Hrady, Czech Republic

Belousov-Zhabotinsky (BZ) reaction represents a simple model of citric acid cycle. The BZ reaction is one of oscillating chemical reactions also called as chemical clock. The non-equilibrium thermodynamics behind BZ reaction creates oscillating structures between colors. Easy setup of the experiments allows us to capture the oscillations by ordinary RGB digital camera. The captured images are transformed by Rényi entropy using different alpha parameters. The Rényi entropy is generalization of Shannon’s information entropy. The alpha parameter is related to weights of probability events. In other words, the alpha parameter describes the inner dimension of observable structures. The evaluations of different Rényi entropies in each color channel and therefore events of different probabilities are our state variables. The aim of the research is to develop dynamical model of BZ reaction with respect to its fractal dimension. We define two approaches how to use the Rényi entropy for image transformation: (a) computation from the whole image; (b) computation from the cross neighborhood. Our 39 state variables (13 from each color channel) obtained from image transformation are not independent to each other. To obtain uncorrelated state variables, we carry out the Principal Component Analysis (PCA). The first five principal components (PC) from transformation (a) are able to describe 94.15% of the original state space. Via transformation (b), only the first three PCs are necessary to describe 97.25% of the original state space. Uncorrelated state space (PC, (b)) is used for description of the BZ reaction trajectory. In this new state space, the trajectory is observably segmented into distinguishable clusters. According to the Rényi and Theiler definitions, the alpha parameters which contributed most to the PCs should be used for determination of the multifractal dimensionality of BZ reaction. Back to the original experimental data, we can pick up a typical image related to every single cluster. Therefore, we conclude that linear combination of several Rényi entropies is enough to describe properly and sufficiently the state space as well as the dimensionality of BZ reaction trajectory. Moreover, this uncorrelated state space is also directly suitable for the decomposition of BZ reaction state trajectory. The property of clusters distinguishability is caused by weighting of different probability events by alpha parameter in Rényi entropy equation. The theoretical presumption was also confirmed by several simulations of different common probabilistic distributions.