A structural system with an unstable post-buckling response that subsequently restabilizes has the potential to exhibit hnmoclinic connections from the fundamenta]! equilibrium state to itself over a range of loads, and heteroclinic connections between fundamental and periodic equilibrium states over a different (smaller) range of loads. It is argued that such equilibrium configurations are impurtant in the interpretation of observed behaviour, and govern the minimum possible post-buckling loads.
To illustrate this, the classical problem of a long thin axially-compressed cylindrical shell is revisited from three different perspectives: asymptotic conjecture, analogy with nonlinear dynamics, and numerical contihuation analysis of a partial spectral decomposition of the underlying equilibrium equations. The nonlinear dynamics analogy demonstrates tha! the structure of the heteroclinic connections is more complicated than that indicated by the asymptutics: this is confirmed by the numerics. However, when the asymptotic portrayal is cornpared to the numerics, it turns out 1o be surprisingly accurate in its Maxwell-load prediction of the practically-significant first minimum to appear in the post-buckling regime.