  An optoelectronic oscillator exhibiting a large delay in its feedback loop is
studied both experimentally and theoretically. We show that multiple
square-wave oscillations may coexist for the same values of the parameters
(multirhythmicity). Depending on the sign of the phase shift, these regimes
admit either periods close to an integer fraction of the delay or periods close
to an odd integer fraction of twice the delay. These periodic solutions emerge
from successive Hopf bifurcation points and stabilize at a finite amplitude
following a scenario similar to Eckhaus instability in spatially extended
systems. We find quantitative agreements between experiments and numerical
simulations. The linear stability of the square-waves is substantiated
analytically by determining stable fixed points of a map.
