We have approached the formation of patterns in nonequilibrium systems through the notion of states that develop via a supercritical linear instability and so saturate at small amplitudes near the threshold of the instability. The resulting patterns retain to some degree the features of the linearly growing mode and this allows many aspects of the pattern formation to be analyzed in a tractable way via the amplitude equation formalism. In nature, however, most nonequilibrium systems are not close to any threshold, and the amplitudes of their corresponding states cannot be considered small. Even near the threshold of a linear instability, structure can emerge via a subcritical bifurcation such that the exponential growth saturates with a large amplitude. What can be said about these strongly nonlinear patterns that are far from the linearly growing mode?

Experiments and simulations indicate that patterns far from threshold can be divided into two classes. One class qualitatively resembles patterns that, at least locally, take the form of stripes or lattices. The other class of patterns far from threshold involves novel states that do not correspond locally to lattice structures.

Far from threshold less can be said about stripe and lattice states with any generality. One question that can be addressed generally is the slow variation of the properties of the stripes or lattices over large distances in a sufficiently big domain. As we explain in Section 9.1.1 these slow dynamics are connected with symmetries of the system.