A generalized high-order correlation dimension for strange attractors

Correlation dimension is a well-known measurement for characterizing strange attractors but only exploring low-order correlation in complex orbit structure. We propose a generalized correlation dimension of strange attractors based on algebraic topology. By mapping a strange attractor into consecutive graphs under different similarity scales, we identify a power law relation between the number of a specific order of clique and the similarity scale, defining a finite algebraic exponent. Interestingly, we find that the algebraic exponent follows a linear growth pattern as a function of the order of clique. We demonstrate that this is a universal principle governing topological structure of strange attractors.