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):
- simple simulations using Runge Kutta for example and observing where the system settles (for continuous cases)
- 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
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).