Integrity measures quantifying the erosion of smooth and fractal basins of attraction

Abstract

The phase portraits of a dissipative dynamical system are characterized by the existence of one or more stable attractors, which typically include point equilibria, periodic oscillations (harmonic and subharmonic), quasi-periodic solutions and chaotic attractors. Each attractor is embedded in its own domain or basin of attraction, bounded by a separatrix' associated with an unstable saddle solution. Under the variation of a control parameter, as the attractors move and bifurcate, the basins also undergo corresponding changes and metamorphoses. These changes in size and shape are usually continuous but can be discontinuous as when an attractor vanishes, along with its basin, at a saddle-node bifurcation. Associated with the homoclinic tangling of the invariant manifolds of the saddle solution, basin boundaries can also change in nature from smooth to fractal. In this paper, the escape of a driven oscillator from a cubic potential well, as an archetypal example, is used to explore the engineering significance of the basin erosions that occur under increased forcing. Various measures of engineering integrity of the constrained attractor are introduced: a global measure assesses the overall basin area; a local measure assesses the distance from the attractor to the basin boundary; and a velocity measure is related to the size of impulse that could be sustained without failure. Since engineering systems may be subjected to pulse loads of finite duration, attention is given to both the absolute and transient basin boudaries. The significant erosion of these at homoclinic tangencies is particularly highlighted in the present study, the fractal basins having a severely reduced integrity under all three criteria.