Homoclinic and heteroclinic bifurcations in the non-linear dynamics of a beam resting on an elastic substrate

The non-linear dynamics of a slender "elastica", fixed at its base and free at the top, resting on an elastic substrate, axially loaded and subjected to periodic excitation, has been analyzed. Taking into account the non-line ar inertial terms, the single-mode dynamics of the systems is governed by a Duffing equation with fifth-order non-linearities, In the considered range of parameters, two qualitatively different phase portraits exist. When the axial load p is less than the Eulerian critical value, there are three cen ters and two saddles (with the related stable and unstable manifolds). Afte r the pitchfork bifurcation, the two saddles and the middle center coalesce in an unique new saddle which has a pair of symmetric homoclinic solutions . Melnikov criteria on the chaotic dynamics of the system are derived on th e basis of analytical expressions for the homoclinic and the heteroclinic o rbits. They involve transverse intersections of the stable and unstable man ifolds that represent the starting point for a subsequent route to a chaoti c dynamics. Numerical simulations which aim to show some effects of the glo bal bifurcations on the actual dynamics are presented. (C) 1999 Elsevier Sc ience Ltd. All rights reserved.