Characterization of limiting homoclinic behaviour in a one-dimensional elastic buckling model

Localization of buckle patterns in a long one-dimensional structure is studied when the nonlinearity present ensures subcritical behaviour but subsequently restabilizes the response. Previous works have shown that with a weak restabilizing component localized soliton-like solutions are possible but as the degree of restabilization increases towards a critical threshold value so the localized forms give way to a thoroughly periodic response. Multiple-scale asymptotic analysis is used to describe the evolution of the solution structure through this stage, which is shown to be governed by a pair of second-order amplitude equations containing quintic nonlinearity. These equations are investigated both numerically and asymptotically and their solutions are compared with direct computations using the full governing forms. The agreement between the approaches is found to be very encouraging and suggests that further insight into the processes at work may be obtained by application of more refined analytical techniques.