{"title":"The Spherical Kapitza – Whitney Pendulum","authors":"Ivan Yu. Polekhin","doi":"10.1134/S1560354722010075","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we study the global dynamics of the inverted spherical pendulum with a vertically rapidly vibrating suspension point in the presence of an external horizontal periodic force field. We do not assume that this force field is weak or rapidly oscillating. Provided that the period of the vertical motion and the period of the horizontal force are commensurate, we prove that there always exists a nonfalling periodic\nsolution, i. e., there exists an initial condition such that, along the corresponding solution, the rod of the pendulum always remains above the horizontal plane passing through the pivot point. We also show numerically that there exists an asymptotically stable nonfalling solution for a wide range of parameters of the system.</p></div>","PeriodicalId":752,"journal":{"name":"Regular and Chaotic Dynamics","volume":"27 1","pages":"65 - 76"},"PeriodicalIF":0.8000,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Regular and Chaotic Dynamics","FirstCategoryId":"4","ListUrlMain":"https://link.springer.com/article/10.1134/S1560354722010075","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 1
Abstract
In this paper we study the global dynamics of the inverted spherical pendulum with a vertically rapidly vibrating suspension point in the presence of an external horizontal periodic force field. We do not assume that this force field is weak or rapidly oscillating. Provided that the period of the vertical motion and the period of the horizontal force are commensurate, we prove that there always exists a nonfalling periodic
solution, i. e., there exists an initial condition such that, along the corresponding solution, the rod of the pendulum always remains above the horizontal plane passing through the pivot point. We also show numerically that there exists an asymptotically stable nonfalling solution for a wide range of parameters of the system.
期刊介绍：
Regular and Chaotic Dynamics (RCD) is an international journal publishing original research papers in dynamical systems theory and its applications. Rooted in the Moscow school of mathematics and mechanics, the journal successfully combines classical problems, modern mathematical techniques and breakthroughs in the field. Regular and Chaotic Dynamics welcomes papers that establish original results, characterized by rigorous mathematical settings and proofs, and that also address practical problems. In addition to research papers, the journal publishes review articles, historical and polemical essays, and translations of works by influential scientists of past centuries, previously unavailable in English. Along with regular issues, RCD also publishes special issues devoted to particular topics and events in the world of dynamical systems.
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