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

Result figures

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: