Invariant sets are sets which map themselves into itself. They are studied for at least half century due to interesting conclusions they bring to attention. Most interesting ones are butterfly effect (unpredictable weather) and chaos theory (bifurcations). There is many ways how to find these sets (sometimes known also as attractors):
- subdivision algorithm, which divides state space into smaller and smaller regions (Dellnitz & Hohmann)
- Junge-Kevrekidis approach, which starts with randomly chosen points in state space and minimizes "energy" of the predefined optimization function
- very simple forward method of staying within bounds (for Henon map and chaotic saddle)
Recently, I had time to implement subdivision algorithm (once again) and Junge-Kevrekidis approach. Code is available on GitHub, written in Python (3.6.1).