Ideal Quantification of Chaos at Finite Resolution

In this paper, we demonstrate the current limitations of quantifying chaos using floating-point numbers at finite time. Alternatively, we introduce a stable method for quantifying chaos of a given dynamical system using rational numbers at finite resolution. The proposed method is based on constructing a combinatorial representation of a given dynamical system, expressing it as a graph, performing some graph algorithms, and finally deriving conclusions about the original dynamical system. By adaptively partitioning the phase space of a given dynamical system, it is possible to approximate its topology at the desired finite resolution efficiently and independently of the initial conditions. Independence on initial conditions allows deriving ubiquitous conclusions about a given dynamical system. Partition elements with disjoint interiors and dyadic boundaries are employed to build a transparent partition that does not interfere with the map behaviour. Such dyadic partition allows constructing an ideal combinatorial representation for verifying qualitative properties more accurately and calculating quantitative properties more precisely. As an application, combinatorial Lyapunov exponent of the logistic map is computed.