How to Interpret Solutions to Simple Chaotic Systems

Question

Note: I'm a non-physicist so please take any misunderstandings/poor notation on my end with patience.

I remember watching a pop-physics video around 5 years ago which discussed the solution space of a magnetic pendulum. The pendulum was used as a simple example of a chaotic system. In the video, the speaker suggested that if one mapped the function for the final pendulum position at rest $S(X_0,Y_0) = S_f$ where

$$S_f\in(X_1, Y_1)\lor(X_2, Y_2)\lor(X_3, Y_3)$$

For $X_1,Y_1,X_2,Y_2,X_3,Y_3$ where the pendulum's potential energy function $U$ is at a local minimum:

$$\frac{\partial U(X_i,Y_i)}{\partial X} = \frac{\partial U(X_i,Y_i)}{\partial Y} = 0$$

$$\frac{\partial_2 U(X_i,Y_i)}{\partial X_2}\frac{\partial_2 U(X_i,Y_i)}{\partial Y_2} - \frac{\partial_2 U(X_i,Y_i)}{\partial X \partial Y}^2 > 0$$

$$\frac{\partial_2 U(X_i,Y_i)}{\partial X_2} > 0$$

$$\frac{\partial_2 U(X_i,Y_i)}{\partial Y_2} > 0$$

That the system's high reliance upon initial conditions meant that the solution map would be a "fractal". I interpreted this as meaning that while the mapping was continuous (there was always a solution for $S(X,Y)$ for values of $(X,Y) \in X^2+Y^2\le L$) but that $S$ was not differentiable such that $S(X,Y)$ has no explicit correlation to $S(X+\delta X,Y+\delta Y)$.

Is this an appropriate interpretation of how such a mapping would behave?

Given that there are three stable local potential wells (those where the attraction to magnet 1,2,or 3 balances with gravity & tension from pendulum arm) we know there certainly exist regions of our map which will certainly be differentiable, as their solutions are constant and known (the example of starting the pendulum in the potential well of a magnet with no initial velocity).

This makes me question whether there is a critical point of total energy which would transform the system into being chaotic and everything outside of the basic solution state becomes entirely non-differentiable, or whether the mapping actually always maintains pockets of local differentiability, just on such a small scale it appears to be totally chaotic.

It seems apparent in this computational solution that there are large swaths which have constant (therefore differentiable) solutions outside of the base local minimum. I wonder if this is physical or a remnant of the computational method not being truly continuous or if there are consistent solutions we can look to in these systems. Would a true physical system actually show more of a probability density rather than a guaranteed final solution? Is this problem analogous to other similar simple chaotic systems?

Any help in interpreting this problem would be greatly appreciated!

https://www.ioc.ee/~dima/YFX1520/flyer_chaotic_pendulum.pdf

https://beltoforion.de/en/magnetic_pendulum/

Answer 1

The mapping $S_f$ is not continuous because it takes only three possible values, and cannot take values in between these. There are points where $S_f$ is constant in a small enough neighbouring area, and there are other points where $S_f$ takes all three possible values in every neighbouring area, no matter how small. In fractal terminology these two sets of points are known as the Fatou set and the Julia set of the fractal (named after French mathematicians Pierre Fatou and Gaston Julia).

All of this analysis is based on a mathematical idealisation of the system. In an actual physical system, unavoidable sources of error will tend to "smear out" the fractal and blur its fine details.

Answer 2

It probably helps thinking about where exactly the discontinuities on your map lie and what they arise from. If you have a discontinuity on your map it means that the trajectory starting from this initial condition is slowly going perfectly straight between two magnets (say red and yellow) at some point of its future¹. On one side of the discontinuity, you have the case of the pendulum turning towards the yellow magnet (and staying there) and on the other side you have it turning towards the red magnet (and staying there). The trajectory at the discontinuity corresponds to the case where the pendulum comes to rest in an unstable equilibrium between the yellow and red magnet².

Our discontinuity describes a line roughly corresponding to the direction of time evolution (called separatrix): For example, if an initial condition ends in the unstable equilibrium between two magnets² and you go 1 s into the future from that initial condition, you end up at another initial condition that has the same fate³.

Now, if we move orthogonally to that line, we are entering a region where the pendulum will ultimately turn towards the yellow magnet. The next discontinuity in that direction is when the pendulum is missing the yellow magnet on the other side and instead makes another swing before settling on a magnet (involving further decisions of left, right, and overshoot, explaining the fractal nature of the basins). So, when moving between the two discontinuities, we are moving from just reaching the yellow magnet from the right over hitting it directly to just reaching it from the left.

This should make it clear that the two discontinuities must have some distance: There must be a finite region where we hit the magnet.

¹ … or a similar situation ultimately deciding the resting place of the pendulum.
² … or all three magnets. I don’t know from the top of my head how the unstable equilibria are situated and it doesn’t really matter for my answer.
³ This is actually a big simplification since the classical diagram you show only considers initial conditions where the pendulum is at rest while the actual phase space of the pendulum is four-dimensional (including two further dimensions for the speed/momentum of the pendulum) and our line/separatrix becomes a manifold.

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With that, let’s turn to your individual questions:

This makes me question whether there is a critical point of total energy which would transform the system into being chaotic and everything outside of the basic solution state becomes entirely non-differentiable, or whether the mapping actually always maintains pockets of local differentiability, just on such a small scale it appears to be totally chaotic.

As elaborated above, you always have finitely sized regions that map to the same basin; they just become smaller.

It seems apparent in this computational solution that there are large swaths which have constant (therefore differentiable) solutions outside of the base local minimum. I wonder if this is physical or a remnant of the computational method not being truly continuous or if there are consistent solutions we can look to in these systems.

These larger areas correspond to something like the pendulum taking swinging through the middle, taking a left turn, swinging through the middle again, taking the right turn and then settling on the blue magnet. The more of these swings (or similar) you have, the more smaller the area usually becomes.

Would a true physical system actually show more of a probability density rather than a guaranteed final solution?

Sure, at some point you have things like the movement of individual air molecules affecting the behaviour of the pendulum near the separatrix, possibly pushing it to the other side. At this point, you only have probabilities and very quickly the final state is completely random.

This is the classical butterfly effect: At some point tiny changes to the system change the system’s long-term behaviour completely.