In the mathematics of

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...

, Eden's conjecture states that the supremum of the local

Lyapunov dimension In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of paper ...

s 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 (e.g. for the global attractors in the Lorenz system, complex Ginzburg–Landau equation). It is named after

Alp Eden Osman Alp Eden (born 1958) is a Turkish mathematician, scientist and professor of mathematics. He is a retired member of the Boğaziçi University Mathematics Department in İstanbul, Turkey. Education Alp Eden was born in İstanbul in 1958. He fi ...

, 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 ''Phyllostachys nigra'', commonly known as black bamboo or purple bamboo ( zh, 紫竹), is a species of bamboo, native to Hunan Province of China, and is widely cultivated elsewhere. Growing up to tall by broad, it forms clumps of slender arc ...

(even in the case of multistability and coexistence of local attractors with different Lyapunov dimensions). 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.

References

{{DEFAULTSORT:Eden's conjecture Dynamical systems Chaos theory Hidden oscillation