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tomorrow i am teaching coupled oscillators in my applied dynamical systems class. it all begins with a pair of independent identical pendula, just doing their thing... 1/9
if you couple them -- let each slightly influence the other -- then interesting things happen. the question is to what extent do coupled systems converge to consensus? 2/9
model an oscillator as a simple "spinner" -- an angle, increasing at a fixed rate and "firing" like a neuron every period. if you add a small coupling term to an identical pair of such spinners, then their phase difference tends to zero: they synchronize. 3/9
this raises so many questions! what if you have more than two oscillators? does it matter how you connect them? does the underlying network topology govern behavior? what if the frequencies vary? does it always converge? all great questions... 4/9
let's simulate nine spinners with two different network topologies: (1) a ring, with nearest-neighbor coupling; and (2) an 8-simplex, with all-to-all coupling. with the same initial conditions, these systems sync at different rates. you're seeing different eigenvalues... 5/9
this gets really interesting with a large inhomogeneous network, where you can see the convergence rates vary with connectivity density. 6/9
and if the network is evolving over time? ah, well, that's another story that does not quite fit into this thread. 7/9
in the end, there's a lot of topology that governs what is going on. for example, in the ring network, it does not always synchronize to zero phase. instead, you can get convergent patterns with nonzero cohomology: a wave. 8/9
there is, of course, a lot more to say here. if you're interested in a popular overview, the book "SYNC" by @stevenstrogatz has a wealth of info & applications in nature. 9/9
@stevenstrogatz if you're looking for this thread stitched together, you can find a version of it on my blog with higher quality YT vids right here: profghristmath.com/2019/09/03/cou…
