Analyzing the Multistability of Multilevel Systems Based on Vector Entropy Modeling

In dissipative systems, multistability refers to the coexistence of several possible stable states for a given set of parameters. Except for fairly simple cases, systems are multileveled in nature. The study of stable states in multilevel systems is difficult for several reasons. First and foremost among these are multidimensionality and the need to analyze states at different stages within the system. We believe that at the base level, the system is represented as a set of initial stochastic indicators characterizing the functioning of the system. Each subsequent level combines several indicators (subsets) from the previous level. In order to investigate the relationships not only between objects at the same level, but also between different levels, we use differential entropy modeling of a random vector. The model consists of two separate components: first, the entropy measuring chaos and secondly the entropy measuring self-organization. In general, a system can be considered as stable whenever a certain balance among the interrelationships and the level of chaos both within each level and between different levels is present. In dynamics, the entropy production is considered instead of the absolute value of the entropy leading to the cancellation/unification of units allowing comparisons of different objects’ features. Hence, this significantly increases the accuracy of estimating the entropy measuring chaos and expands the possibilities for studying the multistability of multilevel systems. The entropy measure of self-organization allows us to quantitatively assess the degree of linear and nonlinear correlation between different objects, especially correlations between components of a random vector or correlation between two or more sets of random variables. The presentation will present not only the methodology itself but also examples for its application using model examples and real data.