Kaplan–Yorke conjecture: equality of Lyapunov and fractal dimensions of attractors

Establish that, for a dynamical system’s attractor, the Lyapunov dimension computed via the Kaplan–Yorke formula from the ordered Lyapunov exponents equals the attractor’s true fractal dimension. This resolves whether the Kaplan–Yorke dimension coincides with the actual fractal dimension for the class of chaotic attractors considered in multidimensional discrete-time systems.

Background

The paper analyzes chaotic attractors arising in 60-dimensional ring lattices of electrically coupled non-chaotic Rulkov neurons. Because direct geometric characterization of such high-dimensional attractors is infeasible, the authors estimate their fractal dimensions using the Kaplan–Yorke formula derived from the full Lyapunov spectrum.

They explicitly invoke the Kaplan–Yorke conjecture, which posits that the Lyapunov (Kaplan–Yorke) dimension equals the true fractal dimension. Validating this conjecture, especially for the type of high-dimensional coupled map lattices studied here, would justify the dimension estimates used throughout the work and solidify links between dynamical instability measures and geometric size of attractors.