1. دانشگاه صنعتي اميركبير دانشكده مهندسي پزشكي آشوب و بررسی آن در سیستم های بیولوژیکی ارائه راحله داودی استاد دكتر فرزاد توحيدخواه دی 1388

2. What is talked in this seminar: • Introduction to chaos • Chaos properties • History • Fractals • Chaos and stochastic process • Logistic Map

3. What is talked in this seminar: (continue) • Biological models producing chaos • Chaos in heart sign of healthy or disease? • Application: • Model of heart rate • Applying chaos theory to a cardiac arrhythmia

4. What chaos is: • One behavior of nonlinear dynamic systems • Unpredictable for long time but limited to a specific area (attractor) • Seems to be random while it happens in deterministic systems • Highly sensitive to initial condition

5. Chaos Properties: • Fractal (Self Similarity) • Liapunove Exponent (Divergence) • Universality

6. History • Henri Poincaré - 1890 • while studying the three-body problem, he found that there can be orbits which are non-periodic, and yet not forever increasing nor approaching a fixed point.

7. Poincare &Three body problem The problem is to determine the possible motions of three point masses m1,m2,and m3, which attract each other according to Newton's law of inverse squares.

8. History … In 1977, Mitchell Feigenbaum published the noted article “ Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic maps. Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different phenomena.

9. History … Edward Lorenzwhose interest in chaos came about accidentally through his work on weather prediction in 1961. • small changes in initial conditions produced large changes in the long-term outcome. Predictability:Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?

10. Butterfly Effect The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

11. History of Fractals The father of fractals: Gaston Julia. 1900 There were some other works out there, such as Sierpinski’s triangle and Koch’s curve. Mandelbrot 1970 :Mandelbrot Set.

12. Fractals …

13. Fractals … Koch’s curve

14. Fractals …

15. Fractals …

16. Types of Attractors:

17. Types of Attractors:

20. Self Similarity in Chaos

21. Chaos and stochastic process • mean • variance • power spectrum

22. Similar time series

23. RANDOM random x(n) = RND

24. CHAOS Deterministic x(n+1) = 3.95 x(n) [1-x(n)]

28. How to recognize chaos from random • Power spectra • Structure in state space • Dimension of dynamics • Sensitivity to initial condition • Lyapunov Exponents • Predictive Ability • Controllability of Chaos

29. Structure in state space Poincare Section

31. divergence

32. Divergence

33. Divergence

34. Divergence

35. Predictive Ability

36. Logistic Map

40. Feigenbaum Number

41. Biological models producing chaos • Nonlinearity • Time delay • Compartment Cascades • Forcing Functions

42. Chaos in Biology

43. why chaos is so important in Biology? • Chaotic systems can be used to show : • rhythms of heartbeats • walking strides

44. why chaos is so important in Biology? • Fractals can be used to model: • Structures of nerve networks • circulatory systems • lungs • DNA

47. Evidence for chaotic healthy hearts

50. Applying chaos theory to a cardiac arrhythmia