Non-linear Systems

The bifurcation diagrams are computed probabilistically in a sense, as the last set of points in the iteration process are considered as the critical points for each value of r. There is no direct computation of critical points, so the error could begin to increase as more points appear in the system.

The Lyapunov exponent is calculated using the derivative of the function given as an input, using the equation described in this article . The points above the red line are the ones we associate with chaotic behavior, as the Lyapunov exponent can be thought of as a measure of predictability.