Understanding Chaos Theory: Unpredictability in Deterministic Systems

[Luke77]

What is chaos theory? I know this is a broad subject, so feel free to direct me to books or links.

 

[Bhumble]

All I remember about chaos theory is finding unstable equilibrium points for potential and free energies that could result in moving towards a lower energy in multiple ways. Then the more this is compounded eventually a situation of non-repetition is reached so that the system cannot be described as periodic.
That's pretty much all I remember about it and problems with pendulums with the exact amount of energy to flip upright (does it fall to the left or the right?)...

 

[Pythagorean]

First, let's talk about deterministic systems. In a deterministic system, you have a set of equations that describes how something changes in time. In order to see how the system evolves, you first have to give it initial conditions. Given a set of initial conditions, the system will always evolve the same way in time.

Chaos theory was discovered in the last century as a way that deterministic systems can be unpredictable. In a chaotic system, a very tiny change in the initial conditions can lead to completely different behavior. This makes the systems fundamentally unpredictable because initial conditions can be irrational (in which case you can never express the initial conditions on a computer and you can never enter the conditions into a calculator to compute the final result). That means you have to use a number close by that actually has an end to it... thus, you have changed the initial conditions slightly. As time goes on, the solution of the system will diverge from where it would have gone if you used the irrational number.

So basically, chaos is described as "a sensitivity to a change in initial conditions" or... "a sensitivity to perturbation". It results in irregular spatial and/or temporal patterns (chaotic systems might never repeat patterns, especially systems exhibiting a flavor of chaos called "transient chaos").