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PreprintCC BY 4.0HAL

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

  • Matthew J. Colbrook
Open at sourcePDFCite

Abstract

We disprove the unrestricted form of Eden's conjecture, which asserts that the maximal local Lyapunov dimension on a compact global attractor is attained at an equilibrium or a periodic orbit. For each c>4, we construct a polynomial differential equation on ℝ5 with uniformly negative divergence. Its compact global attractor has Lyapunov dimension 4+4/c, attained precisely on an invariant two-torus carrying an irrational linear flow. The maximum over all equilibria and periodic orbits is $3$. These conclusions hold both for the local dimension defined by singular values at every positive time and for the asymptotic Kaplan--Yorke dimension. The proof uses an exact singular-value identity on the whole attractor. The results of this paper are also formally verified in Lean.

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Cite

BibTeXRIS
BibTeX
@misc{colbrook2026aperiodic,
  title = {{Aperiodic maximisers of Lyapunov dimension: A counterexample to the unrestricted form of Eden's conjecture}},
  author = {Matthew J. Colbrook},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://hal.science/hal-05757958v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Aperiodic maximisers of Lyapunov dimension: A counterexample to the unrestricted form of Eden's conjecture
AU  - Matthew J. Colbrook
PY  - 2026
DA  - 2026/09/21
PB  - HAL
UR  - https://hal.science/hal-05757958v1
AB  - We disprove the unrestricted form of Eden's conjecture, which asserts that the maximal local Lyapunov dimension on a compact global attractor is attained at an equilibrium or a periodic orbit. For each c>4, we construct a polynomial differential equation on ℝ^5 with uniformly negative divergence. Its compact global attractor has Lyapunov dimension 4+4/c, attained precisely on an invariant two-torus carrying an irrational linear flow. The maximum over all equilibria and periodic orbits is $3$. These conclusions hold both for the local dimension defined by singular values at every positive time and for the asymptotic Kaplan--Yorke dimension. The proof uses an exact singular-value identity on the whole attractor. The results of this paper are also formally verified in Lean.
ER  -