  In this work we study the local coupled Kuramoto model with periodic boundary
conditions. Our main objective is to show how analytical solutions may be
obtained from symmetry assumptions, and while we proceed on our endeavor we
show apart from the existence of local attractors, some unexpected features
resulting from the symmetry properties, such as intermittent and chaotic period
phase slips, degeneracy of stable solutions and double bifurcation composition.
As a result of our analysis, we show that stable fixed points in the
synchronized region may be obtained with just a small amount of the existent
solutions, and for a class of natural frequencies configuration we show
analytical expressions for the critical synchronization coupling as a function
of the number of oscillators, both exact and asymptotic.
