An unstable periodic orbit of period p is identified empirically by finding a point and its pth iterate that are closer than a tolerance eps1, with the additional criterion that ith iterates for i<p do not come within eps2 of the reference point (which would indicate a period i orbit). The values eps1,eps2 must be juggled so that the desired periodic orbit is found, and not confused with lower period orbits (e.g. a period 2 orbit iterated twice appears as a period 4 orbit).
After the periodic orbits are found, the eigenvectors and eigenvalues for the linearization of the map near the periodic points are calculated (see instructions and text).
This is demonstrated here for period 1,2 and 4 orbits for the Henon map.
Period 1 | |
Period 2 | |
Period 4 |
Try varying eps1, eps2 to see how these values affect the success of the algorithm. Also see how consistent the results for the eigenvalues and eigenvectors are for repetitions. Change the number of points points used to construct the linear fit. Is there an optimum value?