|
ИСТИНА |
Войти в систему Регистрация |
ИСТИНА ЦЭМИ РАН |
||
Application of the classical theory of solitons to study the dynamics in passive-dispersed and active-relaxation media during the formation of twist in a toroidal plasmaтезисы доклада
- Авторы: Belyakin S.T., Степанов Аександр Викторович
- Сборник: V International Conference 'Topological methods in dynamics and related topics'
- Серия: V
- Том: 1
- Тезисы
- Год издания: 2022
- Место издания: Topological methods in dynamics and related topics NNovgorod
- Первая страница: 7
- Последняя страница: 8
- Номер статьи: 3
- Аннотация: This paper presents a dynamic classical soliton model of the conditions for the formation of a swirl in a toroidal vortex of a stellarator. It is represented by the geometric construction of the evolution of the Smale-Williams hyperbolic attractor. Two coupled Van der Pol generators are used as the main model qn+1={2qn}. Based on non-linear dynamics, the Bernoulli map represents the evolution of the torus. The use of the dynamic classical soliton model makes it possible to find the conditions for the formation of a swirl in a toroidal vortex. Mathematical theory of chaos based on a rigorous mathematical approach deals with strange attractors of hyperbolic type. In such attractor all belonging orbits in phase space are of saddle type, and the invariant sets of trajectories approaching the original one in forward or backward time (stable and unstable manifolds) intersect transversally, i.e. without touch. The object forming in a limit of infinite iterations is called the Smale - Williams solenoid. The transversal structure of it is a Cantor-like set. Introducing an angle coordinate Q as shown in the picture, it is easy to see that under iterations of the Smale - Williams procedure it will be governed by a Bernoulli map: Qn+1={2Qn}. Two rest directions correspond to contraction of the phase volume. Hence, an appropriate candidate for realization of the Smale - Williams attractor is a system of non-autonomous oscillators discussed on a page devoted to the Bernoulli map. Let us turn to the following scheme [S.P.Kuznetsov, Y.P.Seleznev, JETP, 2006, No 2].Let us consider a one-dimensional map xn+1={2xn}where the braces designate a fractional part of the number. The plot of the map and the iteration diagram for several steps are shown in the picture. It is convenient to represent the variable x in a binary notation, then the digit 0 at the first position after the dot corresponds to residence of the state of the model in the left part of the unit interval, and 1 to residence in the right part. Such a transformation of the binary sequence is called the Bernoulli shift.It is clear that a small perturbation of initial condition with one step of the iterations grows twice. Hence, the Lyapunov exponent for this mapping equals ln 2=0.693.
- Добавил в систему: Белякин Сергей Тихонович
Прикрепленные файлы
| № | Имя | Описание | Имя файла | Размер | Добавлен |
|---|---|---|---|---|---|
| 1. | Полный текст | V International Conference 'Topological methods in dynamics and related topics' | TMDaRT_-_V_l47Ow6n.pdf | 413,1 КБ | 24 июня 2022 [Sergey111] |
| 2. | Полный текст | Application of the classical theory of solitons to study the dynamics in passive-dispersed and active-relaxation media during the formation of twist in a toroidal plasma | template.pdf | 58,0 КБ | 24 июня 2022 [Sergey111] |