One of the simple models which lead to chaos is the logistic growth process. Using ThetaML, a technology designed for modeling and analyzing stochastic processes, we can implement a very short logistic growth model and get more insight than usual.
Logistic growth model
Suppose, we insert a simple logistic model:
This can be done in ThetaML by:
model logisticGrowth export X R = [2.9:0.1:4] loop r, x : R, X x = rand() end loop 100 Theta 1 loop r, x : R, X x = r*x*(1-x) end end end
Running this model in Theta Suite now produces the following output graphics:
This graphic shows that the values where the logistic process starts oscillating for some value of r>2.9. Another way of looking at it is the bifurcation graph, which we take from wikipedia here: