Basins of attraction in a harmonically excited spherical bubble model

Abstract

Basins of the periodic attractors of a harmonically excited single spherical gas/vapour bubble were examined numerically. As cavitation occurs in the low pressure level regions in engineering applications, the ambient pressure was set slightly below the vapour pressure. In this case the system is not strictly dissipative and the bubble can grow infinitely for sufficiently high pressure amplitudes and/or starting from large initial bubble radii, consequently, the stable bubble motion is not guaranteed. For moderate excitation pressure amplitudes the exact basins of attraction were determined via the computation of the invariant manifolds of the unstable solutions. At sufficiently large amplitudes transversal intersection of the manifolds can take place, indicating the presence of a Smale horseshoe map and the chaotic behaviour of system. The incidence of this kind of chaotic motion was predicted by the small parameter perturbation method of Melnikov.