TY - JOUR

T1 - The Basin of Attraction in the Generalized Kapitsa Problem

AU - Kulizhnikov, D. B.

AU - Tovstik, P. E.

AU - Tovstik, T. P.

N1 - Kulizhnikov, D.B., Tovstik, P.E. & Tovstik, T.P. The Basin of Attraction in the Generalized Kapitsa Problem. Vestnik St.Petersb. Univ.Math. 52, 309–316 (2019) doi:10.1134/S1063454119030129

PY - 2019/7/1

Y1 - 2019/7/1

N2 - 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.

AB - 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.

KW - generalized Kapitsa problem

KW - two-scale expansions

KW - vertical basin of attraction

KW - VERTICAL ROD

KW - STABILITY

UR - http://www.scopus.com/inward/record.url?scp=85071953154&partnerID=8YFLogxK

UR - https://link.springer.com/article/10.1134/S1063454119030129

U2 - 10.1134/S1063454119030129

DO - 10.1134/S1063454119030129

M3 - Article

AN - SCOPUS:85071953154

VL - 52

SP - 309

EP - 316

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -