Stability predictions through a succession of folds

Abstract

For a discrete, or discretized, conservative gradient system, such as is envisaged in catastrophe theory and arises throughout the physical sciences, it is often necessary to assess the stable regions of an equilibrium path that exhibits a succession of folds. At each fold the degree of instability changes by one, so that as the system evolves from a region of known stability the first fold must represent a loss of stability. At a second fold, however, it is not clear whether the system is suffering a further loss, as we shall see in some examples, or is regaining its original stability as is more common in elasticity. A new theorem involving a conjugate parameter allows all such stability changes to be readily assessed on the basis of the form of the equilibrium paths themselves. The application of the general theory to the external and internal stabilities of an elastic structure under dead and rigid loading is demonstrated. Under the former, the load is the control parameter and the corresponding deflection plays the role of the conjugate parameter, while in a direct analysis of rigid loading these roles are reversed. A supplementary study of rigid loading which uses Lagrange multipliers supplies further theorems relating the dual concepts of external and internal stability. The use of the theorems is demonstrated in the buckling of elastic arches and shallow domes, and in the incipient gravitational collapse of a massive cold star. The possible stabilization of bifurcations by rigid loading is examined, and shows how the results can also be of value in bifurcational instabilities.