Eden Conjecture on Maximizers of Global Lyapunov Dimension

Determine whether, for autonomous ordinary differential equation systems, the global Lyapunov dimension defined by the Kaplan–Yorke formula attains its maximum on an equilibrium or on a periodic orbit rather than on a chaotic trajectory; specifically, ascertain if the supremum over initial conditions in the Kaplan–Yorke expression for Lyapunov dimension is achieved by equilibria or periodic orbits (the Eden conjecture).

Background

The paper defines the global Lyapunov dimension d_L via the Kaplan–Yorke formula in terms of sums of Lyapunov exponents. Identifying where the supremum in this formula is attained is central to connecting dynamical stability properties with fractal geometry of attractors.

The authors reference the Eden conjecture, which predicts that the maximum in the Kaplan–Yorke expression for global Lyapunov dimension is realized on non-chaotic invariant sets (equilibria or periodic orbits). Confirming or refuting this conjecture remains a fundamental question in the theory of dynamical systems.