Friday 22 April 2022

Attractors, Critical Slowing, and Bifurcations

In this post I want to introduce some ideas of system level analysis connected to attractors and bifurcation.

Consider a solid steel cylindrical piece, placed on its base, sitting on a table. Left to its own devices it will sit there indefinitely, however it can be perturbed, tipped slightly, and it will return to its base. 

For the purposes of analysis we can agree that the body of metal has a centre of gravity, if it is tipped only so much that its centre of gravity remains between the stable centre of the body and the edge in contact with the table when tipped, then it will revert to an upright position. If pushing the body over means that the centre of gravity lies outside of that edge in contact with the table then the block will fall onto its side.

Now, say we apply a perturbation with a finger, that is to say we push the top edge of the body, then we pull our finger away, the block returns to an upright position. But let's consider two scenarios. 1) We push it slightly off centre, when it falls back it barely overshoots and quickly drops back to an upright state. 2) We find out the angle at which it falls onto its side, then we push it to just before this angle and quickly remove the finger, now it falls back but overshoots and rattles back and forth before finally settling. The crucial detail here is that the closer we get the block to tip to the threshold at which it falls onto its side, the longer it takes to recover from such a large perturbation. This is the "Critical slowing of recovery from perturbations", or just plain Critical Slowing.

And that term leads us from the basic mechanistic view of the metal body system on to a broader understanding with more general application.

Now consider the animated gif below. A block of metal is shown, together with a red dot at its centre, the centre of gravity, and a red line being the vector for gravity. To the left is an illustrative graph showing two stable attractor states, the transition between them, and a red dot showing the position of the block in that form of representations for each state of the block.

Attractor 1 is a potential well, the metal block will seek to remain in the lowest local energy state, and this is the bottom of that potential well. Tip it slightly and the red dot will run up the side of the potential well. If taken to just the left side of the hump before the slope marked bifurcation, the system will always return to its lowest energy state. However the further from the bottom of the potential well of Attractor 1 the more wobbling from left to right will happen as the system recovers to that lowest energy state.

Attractor 1 is a mode of stability for the system, there is another stable mode, Attractor 2. Tip the block until its centre of gravity lies outside the edge in contact with the table and it falls on its side. The system has dropped down a large slope of energy potential and landed in the well of Attractor 2. This is an equally 'valid' stable state for the system, albeit one at a lower overall energy state.

Now try to push the block up from being in Attractor 2, on its side, it takes a lot of energy to go back up. To get down from Attractor 1 it only needed the small amount of energy needed to tip it to the start of the bifurcation, then the potential slope is all the way down. To do the reverse you need to give all that energy back into the system, of course in the right way. Warming it up won't put it on its side.

It is worth observing that when you view this system in terms of vertical axis there are attractors and a bifurcation. When you use an equally valid perspective of angle of orientation around the long axis of the block these attractors and bifurcation disappear. Choosing the right representation is vital to seeing what is going on.

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