1. Some figures adapted from a 2004 Lecture by Larry Liebovitch, Ph.D. Chaos BIOL/CMSC 361: Emergence 1/29/08

2. Emergence Non-linear Non-linear Coherence Coherence Dynamic Dynamic Self-Organization Self-Organization Complexity Complexity Macro-level Property (Structured) Macro-level Property (Structured) Unexpected Unexpected Unpredictable Unpredictable

3. Properties of Chaos Deterministic Deterministic Small Number of Variables  Complex Output Small Number of Variables  Complex Output –Bifurcations –Attractors Sensitive to Initial Conditions Sensitive to Initial Conditions –Forecasting Uncertainty grows Exponentially Phase Space is Low Dimensional Phase Space is Low Dimensional

4. Deterministic predict that value these values A future state fully determined by previous states  Chaos: future states fully determined by initial state

5. Random A process or system whose behavior is A process or system whose behavior is –Stochastic; –without definite aim, reason or pattern; –Whose outcome is described by a probability distribution. Probability: relative possibility that an event will occur Probability: relative possibility that an event will occur Stochastic: a non-deterministic process Stochastic: a non-deterministic process Deterministic: a prior state fully determines the future state of the process or system Deterministic: a prior state fully determines the future state of the process or system

6. Mean Mean Variance Variance Power Spectrum Power Spectrum R a n d o m C h a o t i c

7. Random Data 1 x(n) = rand()

8. Chaos Data 2

9. Properties of Chaos Deterministic Deterministic Small Number of Variables  Complex Output Small Number of Variables  Complex Output –Bifurcations –Attractors Sensitive to Initial Conditions Sensitive to Initial Conditions –Forecasting Uncertainty grows Exponentially Phase Space is Low Dimensional Phase Space is Low Dimensional

10. Population Growth

12. Bifurcation Diagram Choose a constant starting value for x (x 0 =0.1) Choose a constant starting value for x (x 0 =0.1) Choose a starting value for the rate (r = 0) Choose a starting value for the rate (r = 0) Use the equation to compute successive values of x(n) from prior values up to x(1000) Use the equation to compute successive values of x(n) from prior values up to x(1000) Ignore the first 900 values of x; these are transient values (before system stablizes) Ignore the first 900 values of x; these are transient values (before system stablizes) Plot x(901) to x(1000) on the Y-axis versus the current value for r Plot x(901) to x(1000) on the Y-axis versus the current value for r Change the value of r and repeat Change the value of r and repeat ZeroSteady Chaos

13. Bifurcations

14. Attractors

15. Another Example

16. Lorenz: Convection 1963. J. Atmos. Sci. 20:13-141 1963. J. Atmos. Sci. 20:13-141 COLD Model HOT

17. Lorenz Equations X(t) Z(t) Y(t)

18. Lorenz Equations X = speed, direction of the convection circulation X = speed, direction of the convection circulation –X > 0  clockwise –X < 0  counterclockwise Y = temperature difference between rising and falling fluid Y = temperature difference between rising and falling fluid Z = rate of temperature change through fluid column Z = rate of temperature change through fluid column

19. Lorenz Equations Phase Space Z X Y

20. Sensitivity to Initial Conditions X(t) X= 1.00001 Initial Condition: different same X(t) X= 1. 0 0

21. Lorenz Attractors X < 0 X > 0 cylinder of air rotating counter- clockwise cylinder of air rotating clockwise

22. Why an “Attractor”? Trajectories from outside: pulled TOWARDS it why its called an attractor starting away:

23. Why “Strange”? strange not strange

24. Lorenz Strange Attractor Trajectories on the attractor: pushed APART from each other sensitivity to initial conditions starting on:

25. Properties of Chaos Deterministic Deterministic Small Number of Variables  Complex Output Small Number of Variables  Complex Output –Bifurcations –Attractors Sensitive to Initial Conditions Sensitive to Initial Conditions –Forecasting Uncertainty grows Exponentially Phase Space is Low Dimensional Phase Space is Low Dimensional

26. Sensitivity nearly identical initial values very different final values or...very different behaviors...

27. Sensitivity small change in a parameter one pattern another pattern

28. Non-Chaotic System system outputcontrol parameter

29. Chaotic System system outputcontrol parameter

30. Clockwork Universe Initial Conditions X(t 0 ), Y(t 0 ), Z(t 0 )... Can compute all future X(t), Y(t), Z(t)... Equations

31. Chaotic Universe Initial Conditions X(t 0 ), Y(t 0 ), Z(t 0 )... sensitivity to initial conditions Can not compute all future X(t), Y(t), Z(t)... Equations

32. Shadowing Theorem: Non-Chaotic If the errors at each integration step are small, there is an EXACT trajectory which lies within a small distance of the errorfull trajectory that we calculated. Calculated True

33. Shadowing Theorem: Chaotic There is an INFINITE number of trajectories. We’re on an exact trajectory, just not on the one we thought we were on. Calculated Expected

34. Deterministic: Chaotic X(n+1) = f {X(n)} Accuracy of values computed for X(n): 3.455 3.45? 3.4?? 3.??? ? ? ?

35. Why? Sensitivity to initial conditions means that the conditions of an experiment can be quite similar, but that the results can be quite different  a little initial error has a large impact!

36. 4. We are on a “real” trajectory. 3. Pulled back towards the attractor. 2. Error pushes us off the attractor. 1. We start here. Trajectory that we actually compute. Trajectory that we are trying to compute.

37. Properties of Chaos Deterministic Deterministic Small Number of Variables  Complex Output Small Number of Variables  Complex Output –Bifurcations –Attractors Sensitive to Initial Conditions Sensitive to Initial Conditions –Forecasting Uncertainty grows Exponentially Phase Space is Low Dimensional Phase Space is Low Dimensional

38. Mechanism? Chance Determinism Data x(t) t ?

39. Mechanism? Chance d(phase space set) Determinism d(phase space set) = low Data x(t) t ?

40. Procedure Determine topological properties of the object Determine topological properties of the object –Fractal Dimension High Fractal Dimension  Random High Fractal Dimension  Random Low Fractal Dimension  Chaotic Low Fractal Dimension  Chaotic Fractal Dimension does not equal Fractal Dimension

41. Procedure 3 Determine the Topological Properties of this Object Especially, the fractal dimension. 4 High Fractal Dimension = Random = chance = Random = chance Low Fractal Dimension Low Fractal Dimension = Chaos = deterministic = Chaos = deterministic

42. Fractal Dimension A measure of self similarity A measure of self similarity X time d

44. Fractal Dimension: The Dimension of the Attractor in Phase Space is related to the Number of Independent Variables. X time d x(t) x(t+ t) x(t+2 t)

45. Mechanism that generated the experimental data. DeterministicRandom d = low d The fractal dimension of the phase space set tells us if the data was generated by a random or a deterministic mechanism.