Efforts to simulate turbulence in the financial markets include experiments with the logistic x(t) = kx(t - 1)[1 - x(t - 1)], with 0 < x(t) < 1 and 0 £ k < 4. Visual investigation of the logistic equation shows the various stability and instability regimes for the various values of the Feigenbaum number k. Visualizations for t = 20 observations provide clear demonstrations of the stability regimes. The author algebraically analyzes all these regimes in more detail. For 0 < k < 3, the process settles to a unique stable equilibrium. For 3 £ k < 3.6, the process bifurcates, or, as colored visualization shows but not black-and-white, its pitchfork bifurcation branches “bang-bang” switch between two regimes. For 3.6 £ k =< 4, the process becomes chaotic, i.e., deterministically random. In this regime are windows of stability, e.g., at k = 3 + = 3.8284. At k = 4, pure chaos, the process is extremely sensitive to initial values, which is clearly demonstrated visually. The author increases the number of observations to t = 1000, and computes the homogeneous Hurst exponent of the process at k = 4: H = 0.004, indicating that x(t) is blue noise, i.e., extremely antipersistent. A histogram shows a highly platykurtic distribution of x(t), with an imploded “mode”, with extremely fat tails higher than the “mode”, against the reflecting values at x = 0 and x = 1. Several plots of the state directory of the system in the (x(t), x(t - l)) space trace out the parabolic strange attractor. Although the strange attractor is a well-defined parabole, the points on the attractor set are deterministically random and unpredictable.
