Dynamics and Chaos. Copyright by Melanie Mitchell

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1 Dynamics and Chaos Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

2 Dynamics: The general study of how systems change over time Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

3 Planetary dynamics P

4 Fluid Dynamics

Dynamics of Turbulence http://www.noaanews.

5 Dynamics of Turbulence

6 Electrical Dynamics Chua Circuit

7 Climate dynamics

8 Crowd dynamics

9 Population dynamics

10 Dynamics of stock prices

11 Group dynamics

Dynamics of conflicts http://www.blogs.va.

12 Dynamics of conflicts uploads/2011/10/afghantenblog.jpg Dynamics of cooperation EFD7-4BB5-845B-9E5722C1CA03_mw1024_n_s.jpg

13 Dynamical Systems Theory: The branch of mathematics of how systems change over time Calculus Differential equations Iterated maps Algebraic topology etc. The dynamics of a system: the manner in which the system changes Dynamical systems theory gives us a vocabulary and set of tools for describing dynamics Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

14 A brief history of the science of dynamics Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

15 Aristotle, BC

16 Nicolaus Copernicus,

17 Galileo Galilei,

18 Isaac Newton,

19 Pierre- Simon Laplace,

20 We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes. Pierre Simon Laplace, A Philosophical Essay on Probabilities

21 Henri Poincaré,

22 If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible...

F = m a F = G m 1 m 2 / d 2 Conference on Complex Systems, September, 2015 Atom s velocity: 3.

23 F = m a F = G m 1 m 2 / d 2 Conference on Complex Systems, September, 2015 Atom s velocity: a Azcolvin429, Wikimedia Commons

24 Images/mission-blue-butterfly_header.jpg hurricane_depth.jpg

25 Dr. Ian Malcolm You've never heard of Chaos theory? Non-linear equations? Strange attractors?

26 Chaos in Nature Dripping faucets Heart activity (EKG) Electrical circuits Computer networks Solar system orbits Weather and climate (the butterfly effect ) Population growth and dynamics Financial data Brain activity (EEG) Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

27 What is the difference between chaos and randomness? Notion of deterministic chaos Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

28 A simple example of deterministic chaos: Exponential versus logistic models for population growth n t +1 = 2n t Exponential model: Each year each pair of parents mates, creates four offspring, and then parents die. Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

29 n Exponential behavior: Population size vs. year n t = 2 t n Year Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

30 Linear behavior: population at current year versus population at next year n t +1 = 2n t Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

31 Linear Behavior: The whole is the sum of the parts Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

32 Linear Behavior: The whole is the sum of the parts Linear: No interaction among the offspring, except pair-wise mating. More realistic: Introduce limits to population growth.

population that the habitat will support, due to limited

33 Logistic model Pierre Verhulst (1838) Notions of: birth rate death rate maximum carrying capacity k (upper limit of the population that the habitat will support, due to limited resources) Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

34 Notions of: Logistic model birth rate and death rate n t +1 = birthrate n t deathrate n t = (b d)n t maximum carrying capacity k (upper limit of the population that the habitat will support due to limited resources) # n t +1 = (b d)n t % $ k n t k # = (b d) kn 2 t n & % t ( $ k ' & ( ' interac(ons between offspring make this model nonlinear

35 Nonlinear Behavior n t +1 = (birthrate deathrate)[kn t n t 2 ]/k

36 Nonlinear behavior of logistic model Nonlinear: The whole is different than the sum of the parts birth rate 2, death rate 0.4, k=32 (keep the same on the two islands)

37 Logistic map x t +1 = Raaa x t (1 x t ) Lord Robert May b n t +1 = (birthrate deathrate)[kn t n t 2 ]/k Mitchell Feigenbaum b Let x t = n t /k Let R = birthrate deathrate Then x t +1 = Rx t (1 x t )

38 LogisticMap.nlogo 1. R = 2 2. R = R = R = 3.1 Notion of period doubling Notion of attractors 5. R = R = R = 4, look at sensitive dependence on initial conditions Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

39 Bifurcation Diagram Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

40 The fact that the simple and deterministic equation [i.e., the Logistic Map] can possess dynamical trajectories which look like some sort of random noise has disturbing practical implications. It means, for example, that apparently erratic fluctuations in the census data for an animal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors; they may simply derive from a rigidly deterministic population growth relationship...alternatively, it may be observed that in the chaotic regime, arbitrarily close initial conditions can lead to trajectories which, after a sufficiently long time, diverge widely. This means that, even if we have a simple model in which all the parameters are determined exactly, long-term prediction is nevertheless impossible Robert May, 1976

41 Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R period 4 R period 8 R period 16 R period 32 A similar period doubling route to chaos is seen in any one-humped (unimodal) map. R period (chaos) Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

42 Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R period 4 R period 8 R period 16 R period 32 Rate at which distance between bifurcations is shrinking: R period (chaos) Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

43 Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R period 4 R period 8 R period 16 R period 32 Rate at which distance between bifurcations is shrinking: R 2 R 1 R 3 R 2 = R 3 R 2 R 4 R 3 = = = R period (chaos) Copyright by Melanie Mitchell R 4 R 3 R 5 R 4 = = % R lim n +1 R n ( ' * n & R n +2 R n +1 ) Conference on Complex Systems, September, 2015

44 Period Doubling and Universals in Chaos (Mitchell Feigenbaum) In other words, each Rate new at which bifurcation distance appears between about R 1 3.0: period 2 times faster bifurcations than the is shrinking: previous one. R period 4 R period 8 R period 16 R 2 R 1 R 3 R 2 = = R This same period rate 32 of R 3 R 2 = occurs in any unimodal map. R 4 R = R period (chaos) Copyright by Melanie Mitchell R 4 R 3 R 5 R 4 = = % R lim n +1 R ( n ' * n & R n +2 R n +1 ) Conference on Complex Systems, September, 2015

45 Amazingly, at almost exactly the same time, the same constant was independently discovered (and mathematically derived by) another research team, the French mathematicians Pierre Collet and Charles Tresser. Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

46 Summary Significance of dynamics and chaos for complex systems Complex, unpredictable behavior from simple, deterministic rules Dynamics gives us a vocabulary for describing complex behavior There are fundamental limits to detailed prediction At the same time there is universality: Order in Chaos Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015

Dynamics: The general study of how systems change over time

Dynamics: The general study of how systems change over time Planetary dynamics P http://www.lpi.usra.edu/ Fluid Dynamics http://pmm.nasa.gov/sites/default/files/imagegallery/hurricane_depth.jpg Dynamics