Eden's conjecture


In the mathematics of dynamical systems, Eden's conjecture states that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.
The validity of the conjecture was proved for a number of well-known systems having global attractor. It is named after Alp Eden, who proposed it in 1987.

Kuznetsov–Eden's conjecture

For local attractors, a conjecture on the Lyapunov dimension of self-excited attractor, refined by N. Kuznetsov, is stated that for a typical system, the Lyapunov dimension of a self-excited attractor does not exceed the Lyapunov dimension of one of the unstable equilibria, the unstable manifold of which intersects with the basin of attraction and visualizes the attractor. The conjecture is valid, e.g., for the classical self-excited Lorenz attractor; for the self-excited attractors in the Henon map. For a hidden attractor the conjecture is that the maximum of the local Lyapunov dimensions is achieved on an unstable periodic orbit embedded into the attractor.