Indeterminate sub-critical bifurcations in parametric resonance

Abstract

In this paper we show that an indeterminate sub-critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems. This occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub-critical bifurcation. Here, in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co-existing attractor the system will settle.