What makes it "chaotic" isn't just that it'll turn right if you start it leaning a tiny bit right and left if you start it leaning a tiny bit left. What makes it chaotic is that, if you start it in position X, it might make the series of turns RRLRLLLRL... while if you start it in position X+epsilon, it might make the series of turns RRLRLLLLR... or LLRLRRRLR... or some other completely different sequence. It may even make the first hundred turns exactly the same, and then diverge at turn #101.
In other words, any tiny change in the initial conditions, even if it doesn't change the first turn, can change the pattern of turns farther down the line. In chaos, sensitive dependence on initial conditions doesn't just happen at a single point, but throughout the system. It's not just a matter of making or missing the train, but of talking to different people or having slightly different mannerisms because you were 3 minutes and 2.8 seconds early instead of 3 minutes and 2.7 seconds early, and those tiny changes eventually adding up to lead you to Boise instead of Miami.
Nicely put. I often consider this problem when I think counterfactually: "What would have happened if I missed the train? Where would I be right now if I hadn't lost my pencil two days ago?"
I have an instinct to just dismiss this kind of musing, because I think to myself that even the smallest changes propagate in wildly unpredictable ways, with potentially vast consequences -- think "the butterfly effect"-- and so it's pointless to even try to reason counterfactually.
But of course it's not pointless, and our minds do this naturally all the time: holding everything else equal, we're constantly twisting this knob and that one, imagining all kinds of possible worlds, exploring the space of what could have been.
In other words, any tiny change in the initial conditions, even if it doesn't change the first turn, can change the pattern of turns farther down the line. In chaos, sensitive dependence on initial conditions doesn't just happen at a single point, but throughout the system. It's not just a matter of making or missing the train, but of talking to different people or having slightly different mannerisms because you were 3 minutes and 2.8 seconds early instead of 3 minutes and 2.7 seconds early, and those tiny changes eventually adding up to lead you to Boise instead of Miami.