I’m not a mathematician, but I don’t know that there’s a line in the sand where a complex system becomes “chaotic” in a technical sense. I was just responding to the characterization of chaos as randomness rather than a high form of complexity that is still deterministic.
As for this pendulum, it would not be difficult at all to simulate. We use magnetic repulsion sets in AI work a lot, and I could simulate that system in a handful of lines of code. Definitely not very difficult. It’s more a question of how much fidelity you want in the simulation. The longer it runs, the more it depends on the initial conditions- the exact position of his hand, the flex in the wood and cardboard, etc. Simulating a system like this is easy. Reproducing exactly what happened in that video could be challenging without just fudging the variables iteratively until it happens to approximate what happened in that video.
A system doesn’t have to be very complex to behave chaotically. The double pendulum is a good example:
What defines a chaotic system is the sensitivity to the initial condition. That a small perturbation will grow exponentially so that long time prediction is impossible.
Classically chaotic systems are systems in which the evolution is highly dependent on comparatively small changes in initial conditions. It’s a range rather than a bright line. The system is still deterministic, but the evolution quickly becomes too complex to compute. This is why predictions of such systems, weather forecasting for example, grow less reliable the further ahead they’re computed using linear approximations.
Interesting (thanks). I wonder if a system like this which has a highly variable initial input eventually comes to rest in a fixed and determinable state so that well it’s initial behavior is chaotic over time it becomes predictable inside a fixed set.
That’s not quite the correct way to think about it. Gravitational potential energy converts into kinetic energy but also magnetic potential energy, and there is a chaotic interplay between the three, with energy also continually being dissipated by air and apparatus friction (and some small eddy currents, I imagine). Eventually it will find its way to a zero kinetic energy state, at which point nothing more happens because there is no motion to change anything. If the magnets were only steering it (acceleration always perpendicular to direction of travel), kinetic energy + gravitational potential energy would be roughly constant (ignoring the friction), but they aren’t. Instead, gravitational pE+ kinetic + magnetic pE = [constant]. (Edit. I should add, the only thing I am quibbling about with your description is the “steering” bit)
I believe analyzing this ROMP would be considered an n-body problem, but not the n-body problem. I’m not aware of a name that applies specifically to this system, other than ROMP (randomly oscillating magnetic pendulum). You could turn it into a Magic 8 ball- like machine if you placed an answer at each of the attractors.
There are very simple systems that exhibit chaotic behavior, and there are very complex systems that nevertheless exhibit stable behavior.
The mathworld entry, while not providing an exhaustive definition of chaotic systems, does point out
An example of a simple physical system which displays chaotic behavior is the motion of a magnetic pendulum over a plane containing two or more attractive magnets. The magnet over which the pendulum ultimately comes to rest (due to frictional damping) is highly dependent on the starting position and velocity of the pendulum (Dickau). Another such system is a double pendulum (a pendulum with another pendulum attached to its end).
