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.[1][2] 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,[3][4][5] complex Ginzburg–Landau equation[6]). 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,[7][8] 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 (even in the case of multistability and coexistence of local attractors with different Lyapunov dimensions).[9][10] 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
  1. A. Eden (1989). An abstract theory of L-exponents with applications to dimension analysis. PhD thesis. Indiana University.
  2. Eden, A. (1989). "Local Lyapunov exponents and a local estimate of Hausdorff dimension". Modélisation Mathématique et Analyse Numérique. 23 (3): 405–413. doi:10.1051/m2an/1989230304051.
  3. Leonov, G.; Lyashko, S. (1993). "Eden's hypothesis for a Lorenz system". Vestn. St. Petersbg. Univ., Math. 26 (3): 15–18.
  4. Leonov, G.A.; Kuznetsov, N.V.; Korzhemanova, N.A.; Kusakin, D.V. (2016). "Lyapunov dimension formula for the global attractor of the Lorenz system". Communications in Nonlinear Science and Numerical Simulation. 41: 84–103. arXiv:1508.07498. Bibcode:2016CNSNS..41...84L. doi:10.1016/j.cnsns.2016.04.032. S2CID 119614076.
  5. Kuznetsov, N.V.; Mokaev, T.N.; Kuznetsova, O.A.; Kudryashova, E.V. (2020). "The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension". Nonlinear Dynamics. 102 (2): 713–732. doi:10.1007/s11071-020-05856-4.
  6. Doering, C.R.; Gibbon, J.D.; Holm, D.D.; Nicolaenko, B. (1987). "Exact Lyapunov dimension of the universal attractor for the complex Ginzburg–Landau equation". Physical Review Letters. 59 (26): 2911–2914. Bibcode:1987PhRvL..59.2911D. doi:10.1103/physrevlett.59.2911. PMID 10035685.
  7. Kuznetsov, N.V. (2016). "The Lyapunov dimension and its estimation via the Leonov method". Physics Letters A. 380 (25–26): 2142–2149. arXiv:1602.05410. Bibcode:2016PhLA..380.2142K. doi:10.1016/j.physleta.2016.04.036. S2CID 118467839.
  8. Kuznetsov, N.V.; Leonov, G.A.; Mokaev, T.N.; Prasad, A.; Shrimali, M.D. (2018). "Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system". Nonlinear Dynamics. 92 (2): 267–285. arXiv:1504.04723. doi:10.1007/s11071-018-4054-z. S2CID 54706479.
  9. Kuznetsov, N.V.; Leonov, G.A.; Mokaev, T.N. (2017). "Finite-time and exact Lyapunov dimension of the Henon map". arXiv:1712.01270 [nlin.CD].
  10. Kuznetsov, Nikolay; Reitmann, Volker (2021). Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Cham: Springer.


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