• CS423/523 website: https://tinyurl.im/0ZCyG – Also available from cs.unm.edu/~melaniem/Courses.html – Subscribe to [email protected] • SFI summer school applications due 1/29 https://www.santafe.edu/engage/learn/schools/sfi- complex-systems-summer-school • Quick Discussion: Mitchell Chapter 1 • CAS in the news • Logistic Map • Project 1, part 1 • Prep with presenters of the Walker paper k is carrying capacity nt= num rabbits at time t
xt is fraction of carrying capacity at time t
R is intrinsic growth rate: b-d 1 1.0
0.8
0.6 x(t)
0.4
0.2
0 0.0 0 10 20 30 40 50 60 70 80 90 100 t = time 0.0 1.0
Period Doubling Transition to Chaos
X-axis: r Y-axis: period of limit cycle
• As r increases, between ¼ and ¾: – For any given r, system settles into a limit cycle – Successive period doublings (bifurcations) as r increases – The amount that r increases to get to next period doubling gets smaller and smaller for each new bifurcation. (Feigenbaum’s constant) – At the critical value, the dynamical system falls into essentially an infinite-period limit cycle xt,r ∈ [0,1] x = rx (1 x ) t+1 t t 1.0 0.9
0.0 0.8 0.65 1.0 0.85 0.90 Feigenbaum’s constant
1 R1 ~ 3.0 (2 = 2 period attractor) 2 R2 ~ 3.44949 (2 = 4 period attractor) 3 R3 ~ 3.54409 (2 = 8 period attractor) R4 ~ 3.564407 . . . Rinf ~ 3.569946
The rate at which the R values converge is Feigenbaum’s constant ~ 4.6692016
How fast is the next bifurcation relative to the previous bifurcation?
From Flake: dk= ak – ak-1 / (ak+1 – ak) where ak is the value of r at which the logistic map bifurcates into a 2k limit cycle
Feigenbaum’s constant is the same for all unimodal maps (with a parabolic state space as in Mitchell Fig. 2.4 and Flake 10.2b Characteristics of Chaos
• Deterministic • Sensitive • Ergodic • Embedded Characteristics of Chaos
• Deterministic: there is structure in the state space. No randomness. Predictable given perfect information. • Sensitive: dependence on initial conditions, perturbations and numerical precision. Butterfly effect. • Ergodic: the state space trajectory will return to all previous local regions. Life repeats itself. • Embedded: unstable limit cycles are embedded in the chaos. • Additional points from Flake: – Stability of fixed points – Pastry: stretch and fold • Short term prediction is possible, information is lost with each recursive application of the function f(x) • The floating point representation needs m + 1 bits to predict values after m recursive iterations of f. • Table 10.1: you can predict when your predictions will fail Logistic Map Exercise
xt+1 = R * xt * (1 − xt)
X0 = 0.2 or 0.21 R = 2, 3, 3.5 or 3.9
• Do you find: a stable fixed point, periodic cycles or chaos? • Where do you see sensitive dependence on initial conditions? • WIs a population of rabbits growing according to the logistic map a CAS? • Do rabbit populations grow according to the logistic map – What unrealistic simplifications are assumed in the logistic map? – Reductionism is not the only way to simplify Project 1, Part 1 Due Thu Jan 25 on paper in class 1. Plot population vs time for 50 time steps. Demonstrate trajectories with fixed points, periodic cycles and chaotic dynamics. 2. Plot population vs time and demonstrate sensitive dependence on initial conditions. Demonstrate how far into the future you can expect two populations to be correlated for a chaos-generating and a non chaos- generating value of R. 3. Create a bifurcation diagram. You must write code for this yourself (you cannot download it). Label the diagram to show Fiegenbaum’s constant.
4. Create a return map showing xt+1 vs xt for R = 2.75 and for R = 3.85. Create a return map for the population mean of 100 populations with R randomly sampled between 2.5 and 3, and for R between 3.8 and 3.9. 5. Write an introduction, 4 short paragraphs of results and include references following format on the posted template. Provide a caption for each figure. The report is limited to 2 pages. The Project 1 Part 1 report does NOT need to use the template.