Deterministic Chaos
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math

An example of an obviously chaotic system.

I have written before about chaos but was thinking of it again today and came up with a more obviously chaotic system than the bouncing balls on an oscillating plate mentioned last time. It’s a system you can simulate yourself.

Pick a real number, any real number you want. We’ll call that number the “‍state‍” of the system. Then repeatedly do the following:

1. Throw away the integer part of the state (e.g., 3.1419… becomes 0.1419…).

2. Multiply the state by ten (e.g., 0.1419… becomes 1.419…).

That’s it. Deterministic chaos can be very simple.

Let’s work an example. Suppose you start with state 2.45618435047…. Knowing this, I know your next state will be 4.5618435047…, then 5.618435047…, then 6.18435047…, 1.8435047…, 8.435047…, 4.35047…, 3.5047…, 5.047…, 0.47…, 4.7…, etc. Still, no matter how accurately I measure your initial state it doesn’t take very many steps before I have no idea what your state will be.

Now suppose we add a third step,