Complicated dynamics from simple functions
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1 Complicated dynamics from simple functions Math Outside the Box, Oct Randall Pyke This presentation: Presentations Dynamics
2 Discrete dynamical systems: Variables evolve in time in discrete time steps. Some examples: - Temperature on Burnaby Mountain each day at noon - Density of traffic on Highway #1 eastward at Gaglardi overpass each hour - Total amount of sunshine (minutes) each day at Stanley Park - Number of salmon running Wilson creek each day - Closing price of a stock each day - Population of bees each spring (e.g. March 1) in a bee farm in Richmond - Number of people infected with Zika virus each month in Florida
3 An influential paper by biologist Robert May.
4 Deterministic vs random (stochastic) dynamical systems
5 Deterministic vs random (stochastic) dynamical systems
6 Mathematically modelling (deterministic) discrete dynamical systems. Dynamics determined by a function (one dimensional) Initial point (data) Subsequent points; Example: (iteration)
7 Visualizing orbits Time series n Histogram
8 Types of orbits: Periodic orbits, aperiodic orbits Period 1 Period 2 Period 3 Aperiodic
9 Visualizing orbits time series histogram Period 1 Period 2 Period 3
10 Aperiodic orbit ( ergodic orbit) time series histogram
11 The logistic equation:
12 The logistic equation: A summary of the dynamics of the logistic equation:
13 Graphical iteration Attractive (stable) fixed points! (nearby points are attracted to the fixed point)
14 Logistic equation:
15 Graphical iteration: attraction to the period 1 orbit p p
16 Graphical iteration: attraction to a period 2 orbit x1 x2
17 Period 2 orbit of f period 1 orbit of
18 x1 x2 x2 x1
19 A period 4 orbit of x1 x3 x2 x4 Periodic orbits form a right-angle closed curve
20 x1 x3
21 .. is a period 2 orbit of x2 x4 x1 x3
22 and a fixed point of
23 Graphical iteration: aperiodic orbit
24 As varies, the orbital structure of changes..
25 As varies, the orbital structure of changes. We say is a bifurcation point of if the orbital structure of changes at To determine bifurcation points, we can try to find periodic points analytically A bifurcation curve is a plot of the periodic points p as a function of a; p(a). Plotting the bifurcation curves on the a-x plane we obtain a bifurcation diagram.
26 Note that these period 2 points occur only when a>=3
27 x Bifurcation diagram a
28 Graphical analysis of the period doubling bifurcation. Period 2 orbit appears at a = 3 period 2 points period 1 point period 1 point
29 And similarly, the period 2 orbit bifurcates into a period 4 orbit. Period 2 Period 4 a = 3.4 a = 3.5
30 Another way to obtain a kind of bifurcation diagram is to look at the orbits numerically for various values of a and try to identify periodic orbits.. Final State Diagram Here s what we do: Choose an a. Then numerically plot the orbit starting at some point x0. Throw away the first 1000 points in the orbit and then plot the next If there is a (stable) periodic point then the last 1000 points will settle in on it. Here s what the histograms look like; Period 1 Period 2 Period 4
31 Now look down on these, from above. Here s what you see: x 1 point Period 1 x Period 2 2 points Period 4 4 points
32 Line these up, along the a-axis x a
33 Final state diagram Bifurcation diagram
34 Let s do it for more a values;
35 And more;
36 And more;
37 Final state diagram for logisitic equation, 2.8 < a < 4
38 The period doubling bifurcations accumulate to
39 1. Self similarity 2. Shadow lines 3. Ordering of periodic orbits Some features of the final state diagram for the logistic equation:
40 Self-similarity of the final state diagram
41
42 a = 4 a = 3.68
43 Shadow lines Can be computed
44 b c d e a=3.78 Histogram at a=3.78 Shadow lines e d c b What are they? They are peaks in the histogram caused by squeezing of the points in the orbit due to the peak in f(x).
45
46 Periodic orbits as a function of the parameter a
47 From left of have period doublings; 1, 2, 4, 8,. From right of first cusp have all the odd integers; 3, 5, 7,. From right of second cusp, have all 2x(odd) integers; 6, 10, 14,. From right of third cusp, have all 4x(odd) integers; 12, 20, 28,.. If there s an orbit of period k, then there is an orbit of all periods m where m is to the left of k in this ordering. (Once an orbit has appeared, it remains for all larger parameter values.)
48 The Charkovsky ordering of the positive integers; Charkovsky s Theorem: Period 3 implies Chaos
49 first cusp second cusp third cusp
50 The period 3 orbit first appears at a=
51 Final state diagram for
52
53 In fact, this final state diagram is universal for all such ( uni-modal ) functions.
54
55 But there s more; the rate at which the period doubling bifurcations take place is the same!
56 Feigenbaum s constant
57 The Feigenbaum constant specifies the rate at which period doubling bifurcations take place
58 Universal behaviour Feigenbaum s constant is universal
59
60 Qualitative features of the period doubling scenerio for uni-modal maps can be understood by graphical analysis. But the quantitative features, the universality of the rate of bifurcations (Feigenbaum s constant ) needs much more work This was understood by Feigenbaum in 1975 using methods from Renormalization (physics).
61 Qualitative features of the period doubling scenario for uni-modal maps can be understood by graphical analysis. But the quantitative features, the universality of the rate of bifurcations (Feigenbaum s constant ) needs much more work This was understood by Feigenbaum in 1975 using methods from Renormalization (physics). Renormalization in a nutshell.
62 More iteration: Julia sets
63 Complex iteration: More iteration: Julia sets Julia sets are either completely disconnected (`dust ) or are connected (one piece). The values of c for which the Julia set is connected form the Mandelbrot set..
64
65
66 Self-similarity of the Mandelbrot set
67 Relation of the Mandelbrot set with the final state diagram for the logistic function
68 References: Chaos and Fractals, by Peitgen, Jurgens, Saupe Iterated Maps on the Interval as Dynamical Systems, by Pierre Collet and Jean-Pierre Eckmann. (Technical) More resources on my webpage;
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