**Abstract**: In this talk, I will look at an old problem of very simple flow on a torus,

x' = wx + g(x,y)

y' = wy

In particular, I will be interested in the phase-locked states for this system. A common class of phase models has the form that

g(x,y) = a(y) sin(x) + b(y) cos(x)

I show that this system is equivalent to a simple map:

r' = m(r)

where m is a fractional linear map (moebius transformation) and thus there can only be robust n:1 locking (n cycles of x to 1 cycle of y) and in particular, no fractional rotation numbers.

I will also revisit systems of the form g(x,y) = gamma P(x)Q(y) and explore n:m locking in these and what is important in the form of P(x) in order to get richer dynamics.

This work is a collaboration with Leon Glass

427 Thackeray Hall