The Basin of Attraction in the Generalized Kapitsa Problem

Abstract

This paper considers the basin of attraction of a stable vertical position of a rod in the Kapitsa problem and its generalizations. A long enough flexible rod with a free upper end and a clumped lower end is shown to lose the vertical position under its own weight. The conditions at which harmonically vertical vibrations favor the vertical position stability of a rod have recently been obtained. The basin of attraction of a vertical position under vibrations is discussed in the case of its instability in lack of vibrations. Firstly, the basin of attraction is found in the context of a classic Kapitsa problem. A rigid rod with an elastically secured lower end is then studied to simulate the problem of flexible rod. The asymptotic method of two-scale expansions is also used. It has been established that the transition into a vertical position depends on the initial phase of perturbation. The basin of attraction is found to consist of two parts. In one of them, the transition into a vertical position remains indifferent to the initial phase, whereas in another one, some domains exhibit a dependence on the initial phase.