1. Periodically kicked rotor:

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

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

4. nth order approximation:
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. Frequency-ratio parameter:
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