Open Research Index

A counterexample to the unrestricted form of Eden's conjecture

Matthew J. Colbrook

Palomar Registrypalomar/PALOMAR-2026-09-22-000001Sep 22, 2026math.DSscore 0

For every c > 4, an explicit polynomial ODE on Euclidean five-space has a compact global attractor and uniformly negative divergence. Its finite-time local Lyapunov dimension at every positive real time and its asymptotic Kaplan--Yorke dimension have maximum 4 + 4/c, attained precisely on an invariant two-torus with irrational angular flow. The maximum over equilibria and periodic points is 3. All singular-value exponents exist, and local dimensions converge pointwise. This disproves the unrestricted form of Eden's conjecture; the attractor is not assumed transitive.

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