  Stationary periodic patterns are widespread in natural sciences, ranging from
nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical
and biological media and to macro-scale vegetation and cloud patterns. Their
formation is usually due to a primary symmetry breaking of a uniform state to
stripes, often followed by secondary instabilities to form zigzag and
labyrinthine patterns. These secondary instabilities are well studied under
idealized conditions of an infinite domain, however, on finite domains, the
situation is more subtle since the unstable modes depend also on boundary
conditions. Using two prototypical models, the Swift-Hohenberg equation and the
forced complex Ginzburg-Landau equation, we consider bounded domains with no
flux boundary conditions transversal to the stripes, and reveal a distinct
mixed-mode instability that lies in between the classical zigzag and the
Eckhaus lines. This explains the stability of stripes in the mildly zigzag
unstable regime, and, after crossing the mixed-mode line, the evolution of
zigzag stripes in the bulk of the domain and the formation of defects near the
boundaries. The results are of particular importance for problems with large
time scale separation, such as bulk-heterojunction deformations in organic
photovoltaic and vegetation in semi-arid regions, where early temporal
transients may play an important role.
