  Stationary to pulsating soliton bifurcation analysis of the complex
Ginzburg-Landau equation (CGLE) is presented. The analysis is based on a
reduction from an infinite-dimensional dynamical dissipative system to a
finite-dimensional model. Stationary solitons, with constant amplitude and
width, are associated with fixed points in the model. For the first time,
pulsating solitons are shown to be stable limit cycles in the
finite-dimensional dynamical system. The boundaries between the two types of
solutions are obtained approximately from the reduced model. These boundaries
are reasonably close to those predicted by direct numerical simulations of the
CGLE.
