1. Chaos in Logistic Models
Transition to chaos in a family of logistic growth sequences.
Presented in Elementary Math Models class, April 21, 2015

2. Fish Pop Model Let an = population in 100,000’s
Diff eqn an+1 = m(1 – an) an
Look at various combinations of m and a0
Note that L = 1 so mL = m

3. m = 3.2 a0 = .7 and .4

4. m = 3.4 a0 = .7 and .4

5. m = 3.5 a0 = .7 and .4

7. m = a0 = and

9. New Kind of Graph Transition to Chaos
Compare long term behavior of models differing only in the value of m
Horizontal axis shows values of m
Compress sequence graph to a single vertical line, located above the value of m
Discard first 100 terms, plot next 500, so that only long term behavior is shown
First example: m = 3.2

14. Bifurcation Diagram Detailed survey of results for 0 ≤ m ≤ 4
Shows long term behavior for ALL logistic growth models in one graph
Shows transition to chaos
Shows surprising visual patterns in the chaotic behaviors
A new kind of order – not in the evolution of any one model, but in relations among many different versions of the model

16. Closeup on Chaos
Expanded view for for 3.5 ≤ m ≤ 4

18. High Resolution Image
Expanded view for for 2.8 ≤ m ≤ 4 by Robert Devaney, Boston University