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Invariant Sets

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
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).

Few results:
Chaotic saddle
Hénon map
Lorenz attractor
Rössler attractor

Or you can look at videos which came from simple forward method:



And there are many other cool attractors and more:


Some people even study their dimension. Enjoy free code!