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Interesting. I looked at bike dynamics a while ago, and my understanding is the non linear equations of motion for a riderless bike are quite well known. Solving the requirrs s bit of computing power, but nothing hard these days.
Basically, because of caster, when you turn the handle bars, the front of the bike drops down, maybe 1cm. This is why the handle bars have a tendency to turn by themselves once any small movement occurs: gravity wants to pull the bike closer to the ground.
When a rider less bike rolling along leans to one side and starts to fall over, the handle bars turn into the fall because of the caster. There’s a forward speed at which the turning handle bars turn fast enough to stop the lean. Because that’s exactly how a bike is balanced, by steering into the fall. If the bike is going faster, the handle bars don’t turn fast enough, and the bike just rolls over. If the bike is going very slowly, the handle bars turn to fast, and over correct. Basically, the bike falls over on the other side to it’s initial lean.
But a bike with a rider is very complex, because the rider can shift the centre of gravity, turn the handle bars, and resist the bikes natural desire for the bars to turn. I don’t think anyone has found it possible to write the equations of motion for a bike with a rider.
There were some great discussions at the last hand made bike show on bike dynamics, and what makes a bike pleasant to ride. Custom bike builder Bastion have basically put the equations of motion for a riderless bike into a spread sheet, and can plot on a graph where any bike falls compared to a desirable square region.