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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 papers, and nowadays various different approaches to the definition of Lyapunov dimension are used. Remark that the attractors with noninteger Hausdorff dimension are called strange attractors. Since the direct numerical computation of the Hausdorff dimension of attractors is often a problem of high numerical complexity, estimations via the Lyapunov dimension became widely spread. The Lyapunov dimension was named after the Russian mathematician Aleksandr Lyapunov because of the close connection with the Lyapunov exponents.
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References
- ↑ Sommerfeld, A. (1902). "Beiträge zum dynamischen Ausbau der Festigkeitslehre". Zeitschrift des Vereines Deutscher Ingenieure (in German). 46: 391–394.
- ↑ Eckert, M. (2013). Arnold Sommerfeld: Science, Life and Turbulent Times 1868–1951. Springer.
- ↑ Blekhman, I. (1953). "Self-synchronization of vibrators for some vibration machines". Ingenernyj Sbornik (in Russian). 16: 49–72.
- ↑ Kononenko, V. (1969). Kolebatelʹnye sistemy s organichennym vozbuzhdeniem[Vibrating systems with a limited power supply] (in Russian). Krasnopolskaya /Shvets.
- 1 2 Kiseleva, M.A.; Kuznetsov, N.V.; Leonov, G.A. (2016). "Hidden attractors in electromechanical systems with and without equilibria" (PDF). IFAC-PapersOnLine. 49 (14): 51–55. doi: 10.1016/j.ifacol.2016.07.975 .
- ↑ Leonov G.A.; Kuznetsov N.V.; Kiseleva M.A.; Solovyeva E.P.; Zaretskiy A.M. (2014). "Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor". Nonlinear Dynamics. 77 (1–2): 277–288. doi:10.1007/s11071-014-1292-6. S2CID 121638758.