‘Chaotic Bouncing of a Droplet on a Soap Film’ Gilet et al.

1. ‘Chaotic Bouncing of a Droplet on a Soap Film’Gilet et al. Maikel Hensen Koen van Gils

2. System overview

3. Force characterization Linear spring:

4. Comparable problem: bouncing ball

5. Bouncing ball Velocity of the (massive) platform Everson gives for the mapping of the bouncing ball: (Reduced acceleration) Trapping Region: Mapping ε=Coefficient of restitution (0 for an perfectly elastic ball)

6. Bouncing ball 1. Simple Orbits 2. Horseshoes 3. Quasi-periodic trajectories

7. Bouncing Ball h t m=2 Periodic orbits (m,n) 1. Ball bounces n times during one period 2. Table has m periods during 1 period of the ball

8. Bouncing ball 1. Simple Orbits (n=1) Simple periodic orbit of order m: Trajectory in which v and phase at n+1 are the same as at n, only m platform oscillations later

9. Bouncing ball m=1 B

10. Bouncing ball 2. Horseshoes mapping (ε +/- 0.4 and B +/- 5.6 If one can prove the excising of a horseshoe, one proves the existence of chaotic trajectories • No stable periodic orbit found Polar coordinates:

11. Bouncing ballHorseshoe mapping Introduction to horseshoe mapping: Most points will follow an orbit that will end up in A.

12. :All x lay in So: =Set of points that are still in A or D after or Ω(Cantor set)

13. Cantor Set

14. f(…BB.DB…)=…BBD.B… Each set Can be labeled with a string of B and D’s: PERIODIC ORBITS! For which: Hyperbolic fixed point Else it is D Homoclinic point 2 points are fixed with f: …BB.BB… and …DD.DD… (Hyperbolic) Points with an infinite string of BBBB at the right and are attracted to the first and Those with an infinite string of DDDD to the second

15. Important properties of Ω: Ω Contains a countable set of periodic orbits. Ω Contains a uncountable set of a-periodic orbits

16. Bouncing ball In the case of our bouncing ball: One can prove there exists a cantor set for: ε =0.4 and B >= 5.6 And there will be chaotic trajectories

17. Bouncing ball 3. Quasi periodic trajectories (Small B) Dominated by fixed point: m=0 Defining: Using power series gives: Accelerates away from fixed point Polar coordinates: Returns to fixed point

18. Bouncing Ball Shows that: If: The ball shoots away from the fixed point, And so the attracting region grows with:

19. Bouncing ball B smaller than 0.25

20. Back to the droplet

21. Bouncing droplet on static soap film

22. Experimental observations Spring force Drag force Driving force Gravity

23. Drag force

24. Simulations in paper

25. MATLAB simulations

26. Effect of step size on accuracy

27. Bifurcation diagram

28. Bouncing mode (2,1)

29. Bouncing mode (3,1)

30. Bouncing mode (4,1)

31. Bouncing mode (9,3)

32. Path following

33. Bouncing mode (4,2)

34. Bouncing mode (4,2)

35. Bouncing mode (4,2)

36. Bouncing mode (8,4)

37. Bouncing mode (16,8)

38. Bouncing mode (32,16)?

39. Grazing bifurcation Occurs if droplet touches the film with impact velocity 0. Grazing has no unique solution, can not be traced back to specific initial conditions Absolute value of derivative around Z=0, greater than 0. No grazing bifurcations seem to occur

40. In practice

41. Period doubling transitions From (1,1) to (2,2)Gamma=1.2

42. Transient bouncing phase – strange attractor If impact velocity and phase differ from attracting modes, the droplet converges to one of the periodic modes Figure:After a transition phase, the droplet enters mode (1,1)

43. Thank you