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Quasiperiodicity: Exploring the Approximation of Irrational Numbers

Discover the approximation of irrational numbers by rational fractions, focusing on the Golden Ratio and Fibonacci series. Learn about quasiperiodic patterns in nature like sunflowers and pineapples, and their relation to quasiperiodicity in the human body.

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Quasiperiodicity: Exploring the Approximation of Irrational Numbers

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  1. Periodically kicked rotor:

  2. Winding Number Circle map For the linear map Ω rational → periodic Ω irrational → quasi-periodic

  3. W=5/8 2 7 5 0 4 1 3 6

  4. Continued Fraction nth order approximation: Approximation of irrational number by rational fractions

  5. Golden Ratio → Fibonacci series: 1,2,3.5.8….

  6. flowers

  7. Sunflower

  8. Pineapple

  9. Human body

  10. Quasiperiodicity + ? w2 w1

  11. Quasiperiodicity + Torus

  12. Quasiperiodic

  13. Quasiperiodic

  14. Winding Numbers Frequency-ratio parameter: (Rotation number) number of times the trajectory winds around the small cross-section of the torus after going once around the large circumference.

  15. Hopf Bifurcation

  16. Landau route to turbulence

  17. Ruelle-Taken-Newhouse route to Chaos Ref: Argyris etl

  18. Quasiperiodic route to Chaos Ref: Argyris etl

  19. Ref: Argyris etl

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