Let (varphi^t_t≥ 0,U) be a dynamical system on an open set U of a finite‑dimensional Euclidean space (or, more generally, of a Hilbert space), where each map varphi^t:U→ U is differentiable, and suppose the system possesses a global attractor A⊂ U, i.e. a compact invariant set attracting all bounded sets of U. For u∈A and t>0 let alpha_1(t,u) ≥ alpha_2(t,u) ≥ … be the singular values of the differential D varphi^t(u) arranged in non‑increasing order, and define the local Lyapunov exponents at u by mu_i(u) = limsup_t→+∞frac1t,ln alpha_i(t,u),qquad i=1,2,… Let j(u) = max Big,j : mu_1(u)+…+mu_j(u) ≥ 0,Big, and define the local Lyapunov dimension (Kaplan–Yorke formula) at u by dim_L(u) = j(u) + fracmu_1(u)+…+mu_j(u)(u)big|mu_j(u)+1(u)big|, with dim_L(u)=0 if no such j exists and dim_L(u)=j(u) if mu_j(u)+1(u)=0 is excluded by the convention mu_1(u)+…+mu_j(u)(u)=0. The Lyapunov dimension of the attractor is dim_LA = sup_u∈Adim_L(u). Then there exists a point u_0∈A with dim_L(u_0) = sup_u∈Adim_L(u) such that u_0 is either a stationary point of the system, i.e. varphi^t(u_0)=u_0 for all t≥ 0, or a point of an unstable (not Lyapunov stable) periodic orbit contained in A, i.e. varphi^T(u_0)=u_0 for some T>0.

Eden's conjecture

About 39 years old · traced to 1987

Let $(\{\varphi^{t}\}_{t\ge 0},U)$ be a dynamical system on an open set $U$ of a finite‑dimensional Euclidean space (or, more generally, of a Hilbert space), where each map $\varphi^{t}:U\to U$ is differentiable, and suppose the system possesses a global attractor $\mathcal{A}\subset U$ , i.e. a compact invariant set attracting all bounded sets of $U$ .

For $u\in\mathcal{A}$ and $t>0$ let

\alpha_{1}(t,u)\ \ge\ \alpha_{2}(t,u)\ \ge\ \cdots

be the singular values of the differential $D\varphi^{t}(u)$ arranged in non‑increasing order, and define the local Lyapunov exponents at $u$ by

\mu_{i}(u)\ =\ \limsup_{t\to+\infty}\frac{1}{t}\,\ln\alpha_{i}(t,u),\qquad i=1,2,\dots

Let

j(u)\ =\ \max\Big\{\,j\ :\ \mu_{1}(u)+\cdots+\mu_{j}(u)\ \ge\ 0\,\Big\},

and define the local Lyapunov dimension (Kaplan–Yorke formula) at $u$ by

\dim_{L}(u)\ =\ j(u)\ +\ \frac{\mu_{1}(u)+\cdots+\mu_{j(u)}(u)}{\big|\mu_{j(u)+1}(u)\big|},

with $\dim_{L}(u)=0$ if no such $j$ exists and $\dim_{L}(u)=j(u)$ if $\mu_{j(u)+1}(u)=0$ is excluded by the convention $\mu_{1}(u)+\cdots+\mu_{j(u)}(u)=0$ . The Lyapunov dimension of the attractor is

\dim_{L}\mathcal{A}\ =\ \sup_{u\in\mathcal{A}}\dim_{L}(u).

Then there exists a point $u_{0}\in\mathcal{A}$ with

\dim_{L}(u_{0})\ =\ \sup_{u\in\mathcal{A}}\dim_{L}(u)

such that $u_{0}$ is either a stationary point of the system, i.e. $\varphi^{t}(u_{0})=u_{0}$ for all $t\ge 0$ , or a point of an unstable (not Lyapunov stable) periodic orbit contained in $\mathcal{A}$ , i.e. $\varphi^{T}(u_{0})=u_{0}$ for some $T>0$ .

References

Primary source

Wikipedia

Additional references

Wikipedia, Eden's conjecture, the article this problem comes from.

dynamical-systems unsolved-problem

Posted 2 months ago · edited 6 hours ago

Progress summary

Refreshed 5 days ago

Claimed progress

A submitted counterexample argues that the conjecture is false, but this claim has not been independently checked.

Eden’s conjecture, proposed by Alp Eden in 1987, asserts that the largest local Lyapunov dimension on a global attractor is attained at an equilibrium or an unstable periodic orbit.

Known results

The conjecture has been verified only for particular systems, including the Lorenz and complex Ginzburg–Landau systems.
For the classical Lorenz parameters, the global attractor’s dimension is attained at the zero equilibrium, with value approximately $2.4013$ (2019).
Numerical studies report system-specific confirmations for selected Rössler systems (2018).

October 8, 2026 community counterexample

A submitted construction claims a polynomial system on $\mathbb{R}^{5}$ with a compact global attractor whose maximum local Lyapunov dimension is $\frac{9}{2}$ on an invariant irrational-flow torus, while every equilibrium or periodic orbit has dimension at most $\frac{7}{2}$ . If correct, this refutes the stated conjecture, but the submission is unverified.

Current status (as of October 2026): The conjecture remains open; a community-submitted counterexample is the only reported general advance and is unverified.

Sources

CounterexampleThis statement is false with a fairly conspicuous counterexample. Evan Li discovered the counterexample, and Lily Zhang (me) helped out with the argument for the proof of the refutation. We used different definitions, but the definitions agree for our counterexample. Our counterexample also excludes the vacuous case in which the supremum is never attained, as well as refuting the weaker version of the conjecture obtained when "unstable" is deleted.See full solutionHide full solution

For the full formalized proof, the lean repository may be found here:
https://github.com/Eval-stack/edens-conjecture
Another disproof of Eden's Conjecture for global attractors was public September 21, 2026. Our public GitHub commit for the global attractor formulation was timestamped September 13, 2026.

We plan to publish the arXiv preprint of the counterexample once we refine the informal proof, and refute 3 other conjecture of Eden (and Kuznetsov) concerning dynamical systems. We may update the GitHub repository once this preprint comes out to be more in line with our paper's arguments.

Here is the solution:

Consider the polynomial system on $\mathbb{R}^5 \cong \mathbb{C}^2 \times \mathbb{R}$ given by

w_1 = (g(|w_1|^2)+i)w_1,\\ \dot w_2 = (g(|w_2|^2)+i\sqrt{2})w_2,\\ \dot z = -24z,

where $g(q)=(q-1)(4-q)$ .

Every forward solution exists for all time, and the set

K=\{(w_1,w_2,0):|w_1|\le2,\ |w_2|\le2\}

is a compact global attractor. The supremum of the local Lyapunov dimensions on $K$ is $\frac92$ , attained precisely on

T=\{(w_1,w_2,0):|w_1|=|w_2|=1\}.

However, the only equilibrium is the origin, whose local Lyapunov dimension is $0$ , and every periodic orbit has local Lyapunov dimension at most $\frac72$ . Thus no equilibrium or periodic orbit attains the supremum.

Writing $w_j=r_je^{i\theta_j}$ , we obtain

\dot r_j=r_j(r_j^2-1)(4-r_j^2),\qquad \dot\theta_1=1,\qquad \dot\theta_2=\sqrt{2}.

The radial equations show that $K$ is invariant and attracts every bounded set. The asymptotic logarithmic growth rates of the singular values for each complex coordinate are $(-4,-4)$ , $(6,0)$ , or $(0,-24)$ , according as its initial radius belongs to $[0,1)$ , equals $1$ , or belongs to $(1,2]$ . The remaining coordinate contributes $-24$ . Computing the local Lyapunov dimensions from these rates gives the maximum $\frac92$ , attained exactly when both radii equal $1$ .

No point of $T$ is periodic, since its angular frequencies have irrational ratio $\sqrt{2}$ . Every other periodic orbit has exactly one nonzero complex coordinate, with radius $1$ or $2$ , giving local Lyapunov dimension $\frac72$ or $1$ , respectively.