Phase Space and Strange Attractors
Phase diagram for pendulum: https://youtu.be/MjPFHWul2J0?t=29
circles => equilibrium
spirals friction => stable attractor
Fox + rabbit, dynamical systems
Poincaré-Bendixson theorem (equilibrium, limit cycle)
Vector fields, equilibrium 2D => simple, deterministic, 3D => not!
Water mill
Sinaï-Ruelle-Bowen measure / statistics / forecasting
- https://www.youtube.com/watch?v=xQ3mJxr2NAY
- https://bl.ocks.org/gcalmettes/c3363abb74fe219b283782f055b72386
Mathematicians use the concept of a “phase space” to describe the possible behaviours of a system geometrically. Phase space is not (always) like regular space - each location in phase space corresponds to a different configuration of the system. The behaviour of the system can be observed by placing a point at the location representing the starting configuration and watching how that point moves through the phase space.
In phase space, a stable system will move predictably towards a very simple attractor (which will look like a single point in the phase space if the system settles down, or a simple loop if the system cycles between different configurations repeatedly). A chaotic system will also move predictably towards its attractor in phase space - but instead of points or simple loops, we see “strange attractors” appear - complex and beautiful shapes (known as fractals) that twist and turn, intricately detailed at all possible scales.
Another feature of chaotic systems is that they can suddenly flip from one behaviour to another, very different, one. Think about how some kinds of debt can suddenly become a risk rather than an asset to banks. Such a sudden change is known as a phase transition. One way to see a phase transition in action is to play with a tap. Turn it on very slowly and you hear a regular drip, drip, drip. Turn it up slightly and the drips become more frequent and not all the same size. A tiny bit more, and the drips become completely irregular and unpredictable: chaos sets in. Keep going, and the water settles down to a smooth column. However once you increase too much, the water starts twisting, frothing and spiralling: turbulence begins. By steadily increasing the flow we see regularity and chaos, without leaving the world of determinism.