Three invariants of strange attractors derived through hypergeometric entropy

A new description of strange attractor systems through three geometrical and dynamical invariants is provided. They are the correlation dimension (D) and the correlation entropy (K), both having attracted attention over the past decades, and a new invariant called the correlation concentration (A) introduced in the present study. The correlation concentration is defined as the normalised mean distance between the reconstruction vectors, evaluated by the underlying probability measure on the infinite-dimensional embedding space. These three invariants determine the scaling behaviour of the system’s Rényi-type extended entropy, modelled by Kummer’s confluent hypergeometric function, with respect to the gauge parameter (ρ) coupled to the distance between the reconstruction vectors. The entropy function reproduces the known scaling behaviours of D and K in the ‘microscopic’ limit ρ→∞ while exhibiting a new scaling behaviour of A in the other, ‘macroscopic’ limit ρ→0. The three invariants are estimated simultaneously via nonlinear regression analysis without needing separate estimations for each invariant. The proposed method is verified through simulations in both discrete and continuous systems.