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Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve $\mu^2=\nu^n-1, n\in Z$: ergodicity, isochrony, periodicity and fractalsстатья
Статья опубликована в высокорейтинговом журнале
Информация о цитировании статьи получена из
Web of Science,
Scopus
Статья опубликована в журнале из списка Web of Science и/или Scopus
Дата последнего поиска статьи во внешних источниках: 22 декабря 2014 г.
- Авторы: Grinevich P.G., Santini P.M.
- Журнал: Physica D: Nonlinear Phenomena
- Том: 232
- Номер: 1
- Год издания: 2007
- Издательство: Elsevier BV
- Местоположение издательства: Netherlands
- Первая страница: 22
- Последняя страница: 32
- DOI: 10.1016/j.physd.2007.05.002
- Аннотация: We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different classes of real and autonomous Newtonian dynamics in the plane. The rectilinear motion has been studied in a number of papers, while the cyclic motion is much less understood. For small data, the cyclic time trajectories lead to isochronous dynamics. For bigger data the situation is quite complicated; computer experiments show that, for sufficiently small degree of the monomial, the motion is generically isochronous with integer period, which depends in a quite sensitive way on the initial data. If the degree of the monomial is sufficiently high, computer experiments show essentially chaotic behavior. We suggest a possible theoretical explanation of these different behaviors. We also introduce a two-parameter family of two-dimensional mappings, describing the motion of the center of the circle, as a convenient representation of the cyclic dynamics; we call such a mapping the center map. Computer experiments for the center map show a typical multifractal behavior with periodicity islands. Therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity.
- Добавил в систему: Гриневич Петр Георгиевич