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A long structural system with an unstable (subcritical)post-buckling response that subsequently restabilizes typically deformsin a cellular manner, with localized buckles first forming and thenlocking up in sequence. As buckling continues over a growing number ofcells, the response can be described by a set of lengthening homoclinicconnections from the fundamental equilibrium state to itself. In thelimit, this leads to a heteroclinic connection from the fundamentalunbuckled state to a post-buckled state that is periodic. Under suchprogressive displacement the load tends to oscillate between twodistinct values. The paper is both a review and a pointer tofuture research. The response is described via a typical system, asimple but ubiquitous model of a strut on a foundation which includesinitially-destabilizing and finally-restabilizing nonlinear terms. Anumber of different structural forms, including the axially-compressedcylindrical shell, a typical sandwich structure, a model of geologicalfolding and a simple link model are shown to display such behaviour. Amathematical variational argument is outlined for determining the globalminimum postbuckling state under controlled end displacement (rigidloading). Finally, the paper stresses the practical significance of aMaxwell-load instability criterion for such systems. This criterion,defined under dead loading to be where the pre-buckled and post-buckledstate have the same energy, is shown to have significance in the presentsetting under rigid loading also. Specifically, the Maxwell load isargued to be the limit of minimum energy localized solutions asend-shortening tends to infinity.