Basin boundary metamorphoses in the canonical escape equation

A study is made of the escape boundary, in the space of the starting values of displacement and velocity, for a sinusoidally driven nonlinear oscillator. The oscillator has constant inertia and linear viscous damping, with a restoring force that corresponds to a cubic potential well. This cubic form of the total potential energy has the universal canonical form encountered by all metastable mechanical systems as they approach a generic fold catastrophe under a single control parameter. The basin boundary metamorphoses are observed, and an engineering integrity diagram is proposed, quantifying the rapid erosion of area that is triggered under increasing excitation by a homoclinic tangency. The pictures show clearly the fractal basin associated with the developed homoclinic tangle. The boundary crisis, at which the final chaotic attractor loses its stability, is examined in terms of an accessible saddle point, which is here a directly unstable subharmonic of order six.