Controlling chemical chaos

1. Controlling chemical chaos Vilmos Gáspár Institute of Physical Chemistry University of Debrecen Debrecen, Hungary Tutorial lecture at the ESF REACTOR workshop „Nonlinear phenomena in chemistry” Budapest, 24-27 January, 2003

2. This lecture is dedicated to the memory of Professor Endre Kőrös

3. Chaos* “Chaos  A rough, unordered mass of things” Ovid, “Metamorphoses” “What’s in a name?” Shakespeare, “Romeo and Juliet” The answer is nothing and everything. Nothing because “A rose by any other name would smell as sweet.” And yet, without a name Shakespeare would not have been able to write about that rose or distinguish it from other flowers that smell less pleasant. So also with chaos. *Ditto, W.L.; Spano, M. L. Lindner, J. F.: Physica D, 1995, 86, 198.

4. Chaos The dynamical phenomenon we call chaos has always existed, but until its naming we had no way to distinguishing it from other aspects of nature such as randomness, noise and order. chaosMath Stochastic behaviour occurring in a deterministic system. Royal Society,London,1986 From this identification then came the recognition that chaos is pervasive in our word. Orbiting planets, weather patterns, mechanical systems (pendula), electronic circuits, laser emission, chemical reactions, human heart, brain, etc. all have been shown to exhibit chaos. Of these diverse systems, we have learned to control all of those that are on the smaller scale. Systems on a more universal scale (weather and planets) remain beyond our control.

5. A simulation of the Milky Way/Andromeda Collision showing complex (chaotic) motion of heavenly bodies can be seen on the web page of John Dubinski Dept. of Astronomy and AstrophysicsUniversity of Toronto, CANADA http://www.cita.utoronto.ca/~dubinski/movies/mwa2001.mpg

6. Outline • Chaotic dynamics of discrete systems  the Henon map • The idea of controlling chaos • Fundamental equations for chaos control (ABC) • OGY and SPF methods for chaos control • Application of SPF method to chemical systems • Other methods and perspectives - come to my poster

7. Henon map Michele Henon, astronomer, Nice Observatory, France. During the 1960's, he studied the dynamics of stars moving within galaxies. His work was in the spirit of Poincare’sapproach to the classisical three-body problem: What important geometric structures govern their behaviour? The main property of these systems is their unpredictable, chaotic dynamics that are difficult to analyze and visualize. During the 1970's he discovered a very simple iterated mapping that shows a chaotic attractor, now called Henon's attractor, which allowed him to make a direct connection between deterministic chaos and fractals.

8. Henon map Dissipative system- the contraction of volume in the state space • The asymptotic motion will occur on sets that have zero volumes • A set showing stability against small random perturbations: attractor • Chaotic attractor- locally exponential expansion of nearby points on • the attractor

9. CAP1 http://www.robert-doerner.de/en/Henon_system/henon_system.html

10. CAP2

11. CAP3

12. CAP4

13. CAP5

14. CAP6

15. CAP7

16. CAP8

17. CAP9

18. CAP20

19. Henon map Two fundamental characteristics of chaotic systems that makes them unpredictable: Sensitivity dependence on the initial conditions This causes the systems having the same values of control parameters but slightly differing in the initial conditions to diverge exponentially (on the average) during their evolution in time. Ergodicity A large set of identical systems which only differ in their initial conditions will be distributed after a sufficient long time on the attractor exactly the same way as the series of iterations of one single system (for almost every initial condition of this system).

20. CAP1 http://www.robert-doerner.de/en/Henon_system/henon_system.html

21. CAP2

22. CAP3

23. CAP4

24. CAP5

25. CAP6

26. CAP7

27. CAP8

28. CAP9

29. CAP10

30. CAP11

31. CAP12

32. CAP13

33. CAP14

34. CAP15

35. CAP16

36. CAP17

37. CAP18

38. CAP19

39. CAP20

40. The idea of controlling chaos “All stable processes, we shall predict.All unstable processes, we shall control.” John von Neumann, circa 1950. Freeman Dyson: Infinite in All Directions, Chapter „Engineers Dreams”, Harper: N.Y., 1988: “A chaotic motion is generally neither predictable nor controllable. It is unpredictable because a small disturbance will produce exponentially growingperturbation of the motion. It is uncontrollable because small disturbances lead only to other chaotic motions and not to any stable and predictable alternative. Von Neumann’s mistake was to imagine that every unstable motion could be nudged into a stable motion by small pushes and pulls applied at the right places.” So it happened.

41. The idea of controlling chaosHenon map - Bifurcation diagram x http://mathpost.la.asu.edu/~daniel/henon_bifurcation.html

42. The linearized equation of motion of the system around the fixed point zF: ABC of Chaos Control y x For chaos control we apply a small parameter perturbation pn≠ po if and when the system approaches the fixed point.

43. During chaos control – for simplicity – we apply parameter perturbation that is linearly proportional to the system’s distance from the fixed point, where CTis the control vector. From equations (1) and (2) we get the linearized equation of motion around the fixed point when chaos control is attempted: Shinbrot, T.; Grebogi, C.; Ott, E.; Yorke, J. A.: Nature, 1993,363, 411.

44. Chaos control is successful if the new system state zn+1(po+δpn) lies on the stable manifold of the fixed point zF (po) of the unperturbed system.

45. The just described strategy for chaos control implies the followings: • For a successful chaos control, therefore, one has to know: • the dynamics of the system around the fixed point • the system’s distance from the fixed point • the right value of control vector CT • the eigenvalue of the fixed point in the stable direction

46. Numerical exampleHenon map ! The linearized equation of motion around the fixed point zF when pn≠ poparameter perturbation is applied:

47. Let’s find the control vector CTsuch that Numerical exampleHenon map The eigenvalues of the fixed point of the unperturbed system are calculated by solving the following equation: resulting in

48. Suppose: The control vector can be calculated by solving for the new eigenvalues, and by applying the rules of the control strategy. Numerical exampleHenon map

49. which gives Note that C contains parameters characteristics of the system’s dynamics only. Numerical exampleHenon map

50. which gives Numerical exampleHenon map