Resonance and fractal geometry

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Summaryi. resonance in parametrized systemsii. two oscillators: torus- en circle dynamicsiii. driven oscillators, examples:- Hopf-Neĭmark-Sacker bifurcation- parametric resonance- Hopf saddle-node bifurcation for mapsiv. universal properties for parameter space- resonance ∼ open & dense or residual- quasi-periodicity ∼ nowhere dense or meagrea fractal set of positive measure- and chaos emerging
What is resonance?Resonance: interaction of oscillating subsystems,whererational ratio of frequencies and withcompatible motionvoorbeelden:1 : 1 resonance: Huygens’s clocks,Moon and Earth, Charon and Pluto1 : 2 resonance: Botafumeiro(Santiago de Compostela)2 : 3 resonance: Mercury
Page 1: Resonance and fractal geometryHenk
Page 6 and 7: Tidal resonanceMoon ‘caught’ by
Page 8 and 9: Mathematical programmemodelling in
Page 10 and 11: Example: Arnol ′ d familyϕ ↦
Page 12 and 13: Devil’s staircase0.60.4̺0.20-0.2
Page 15: Hopf-Neĭmark-Sacker Imore general:
Page 18 and 19: Conclusions ‘Arnol ′ d’ tongu
Page 20: Resonance tongues swing665544332211
Page 23 and 24: Quasi-periodic Schrödingertongues
Page 25 and 26: Hopf saddle-node Imap P : R 3 → R
Page 27 and 28: Hopf saddle-node III1n=0.46C1C100-1
Page 29: Conclusions IIfractal geometry with

Resonance and fractal geometry

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