I know that there’s something called the Lyapunov exponent. Could we “diminish the chaos” if we use logarithms, like with the Richter scale for earthquakes?
This is a neat question. I think the answer is no, and here’s my attempt to describe why.
The Lyapunov exponent measures the difference between the trajectories over time. If your system is the double pendulum, you need to be able to take two random states of the double pendulum and say how different they are. So it’s not like you’re measuring the speed, or the length, or something like that. And if you have this distance metric on the whole space of double-pendulum states, then you can’t “take the log” of all the distances at the same time (I think because that would break the triangle inequality).
Hopefully, not talking out of my hat, but the difference between the final states of a double pendulum can be typed:
Somewhere in the middle of the pendulum’s journey through space and time. I’ve seen this visually and true there’s divergence. This divergence is based on measurement of the pendulum’s position in space at a given time. So with initial state A, the pendulum at time Tn was at position P1 while beginning with initial stateB(|A−B|≈0), the pendulum at time Tn was at position P2. The alleged divergence is the difference |P1−P2|, oui? Take in absolute terms, |P1−P2|=106, but logarithmically, log|P1−P2|=only 6.
At the very end when the pendulum comes to rest. There’s no divergence there, oui?
This is a neat question. I think the answer is no, and here’s my attempt to describe why.
The Lyapunov exponent measures the difference between the trajectories over time. If your system is the double pendulum, you need to be able to take two random states of the double pendulum and say how different they are. So it’s not like you’re measuring the speed, or the length, or something like that. And if you have this distance metric on the whole space of double-pendulum states, then you can’t “take the log” of all the distances at the same time (I think because that would break the triangle inequality).
Hopefully, not talking out of my hat, but the difference between the final states of a double pendulum can be typed:
Somewhere in the middle of the pendulum’s journey through space and time. I’ve seen this visually and true there’s divergence. This divergence is based on measurement of the pendulum’s position in space at a given time. So with initial state A, the pendulum at time Tn was at position P1 while beginning with initial stateB(|A−B|≈0), the pendulum at time Tn was at position P2. The alleged divergence is the difference |P1−P2|, oui? Take in absolute terms, |P1−P2|=106, but logarithmically, log|P1−P2|=only 6.
At the very end when the pendulum comes to rest. There’s no divergence there, oui?