A modular network for legged locomotion Golubitsky, Stewart, Buono, Collins 1998

This is an awfully well cited paper. Application of group theory to unravel symmetries and come to relatively strong and concrete conclusions abbou tthe architectures of locomotor controllers. this result is as relevant to the study of actual locomotion as it is to the design of robots. This is about as 'analysis of distributed systems' as you can get. in the end it helped me decie that my interest isn't necesarily this analytic. i think i am leaning in a slightly more concrete direction of Ijspeert, at least for now. that said:

kHere are some symmetries of oscillator networks.

• The oscillators are identical ('a tacit symmetry')
• the coupling patterns have symmetry (however "it is possible, for example, to use vraribable couplings designe dto enforace particular phase relationships" but they don't really approve of that. I don't even know what that means)
• The modules/segments of insects betray a symmetry.

It is ok to approximate an almost symmetric system as symmetric. That is what they say . The symmetry will drive the dynamics and everything else is a minor perturbation. So the symmetry descirbe a limit cycle? I need to get my head around this more.

"Graphically we represent such a network as a set of nodes joined by lines marked with arrows" My notes say "Yay!". three years ago I said my academic goal was to study dots with lines pointing between them. And here i am.

What is a bijective mapping? Understand conjugate solutions. and all the equivariance. and symmetric Hopf bifurcationss.

"From the point of fview of equvariant bifurcation theory, the most important aspect of the network is its symmetry group."

There results end up predicting networks with twice the number of nodes as the nubmer of glegs. they get this as an analytic result and are able to predict all kinds of gaits observed in nature, every thhing from trots, and canters to bucking and pronks. I will have to take a closer look to understand why.