Eden’s conjecture on Lyapunov-dimension maximizers

Must the maximum Lyapunov or Kaplan–Yorke dimension of a dissipative system’s attractor occur at an equilibrium or periodic orbit? Eden’s proposal can be tested against an invariant torus carrying an irrational flow.

Why it matters and what counts as progress

Why it matters: A maximum on an irrational-flow torus would limit equilibrium and periodic-orbit methods for computing attractor dimension.

What counts as progress: Construct an attractor whose dimension maximum lies away from every equilibrium and periodic orbit, or establish conditions making the proposal valid.

How the problem developed

1989 · Foundational result [1]
Alp Eden Developed local Lyapunov exponents as a tool for estimating Hausdorff dimension of attractors, in work underlying the later maximizer conjecture.
21 Sept 2026 · Counterexample claim [2]
Matthew Colbrook The preprint reports polynomial ODE examples in R⁵ with uniformly negative divergence whose attractor’s maximum local Lyapunov dimension is attained on an irrational invariant torus rather than at equilibria or periodic orbits.

AI contribution

Aperiodic maximisers of Lyapunov dimension: A counterexample to the unrestricted form of Eden's conjecture

Matthew Colbrook

For each c>4, Colbrook’s polynomial ODE on ℝ⁵ has uniformly negative divergence and a compact global attractor. Its finite-time local Lyapunov and asymptotic Kaplan–Yorke dimension maxima are 4+4/c on an irrational-flow invariant two-torus, while equilibrium and periodic-point maxima are 3.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (Lean 4.33.1 with Mathlib and Lean Comparator + human)

Human understandingFull