  We analyze the Eckhaus instability of plane waves in the one-dimensional
complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects
arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are
quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability
of plane waves and cease to exist due to saddle-node bifurcations (SN). These
MAWs can be characterized by their average phase gradient $\nu$ and by the
spatial period P of the periodic amplitude modulation. A numerical bifurcation
analysis reveals the existence and stability properties of MAWs with arbitrary
$\nu$ and P. MAWs are found to be stable for large enough $\nu$ and
intermediate values of P. For different parameter values they are unstable to
splitting and attractive interaction between subsequent extrema of the
amplitude. Defects form from perturbed plane waves for parameter values above
the SN of the corresponding MAWs. The break-down of phase chaos with average
phase gradient $\nu$ > 0 (``wound-up phase chaos'') is thus related to these
SNs. A lower bound for the break-down of wound-up phase chaos is given by the
necessary presence of SNs and an upper bound by the absence of the splitting
instability of MAWs.
