  We address the issue why the phase diagrams for quasi-one-dimensional systems
are rather simple, while the renormalization group equations behind the scene
are non-linear and messy looking. The puzzle is answered in two steps -- we
first demonstrate that the complicated coupled flow equations are simply
described by a potential $V(h_i)$, in an appropriate basis for the interaction
couplings $h_i$. The renormalization-group potential is explicitly constructed
by introducing the Majorana fermion representation. The existence of the
potential prevents chaotic behaviors and other exotic possibilities such as
limit cycles. Once the potential is obtained, the ultimate fate of the flows
are described by a special set of fixed-ray solutions and the phase diagram is
determined by Abelian bosonization. Extension to strong coupling regime and
comparison with the Zamolodchikov c-theorem are discussed at the end.
