Visualization of phase trajectories of the Rikitake dynamic system

Abstract

ГОУ ВПО Норильский индустриальный институт 663310, Россия, г. Норильск, ул. 50 лет Октября, д. 7PotapovVI@norvuz.ru Получено 10 января 2010 г.В статье проведено компьютерное исследование четырехмерной динамической системы Рикитаки с тремя параметрами, адекватно описывающей работу модели спаренных динамо с учетом вязкого терния. Показано, что в этой системе имеется пять состояний равновесия: четыре устойчивых фокуса-узла и одно седло (3, 1). Установлены бифуркации рождений пространственных перекрученных циклов, соответствующие удвоению периода колебаний динамической переменной и приводящие к хаотическим колебаниям при увеличении отно-шения коэффициентов трения.Ключевые слова: система Рикитаки, состояния равновесия, предельные циклы, хаос V. I. Potapov Visualization of phase trajectories of the Rikitake dynamic systemIn paper the computer research of four-dimensional dynamic system with three parameters adequately describing behavior of model coupled Dynamo in view of viscous friction is carried out. Is shown, that in this system there are five equilibrium states: four stable are focusesnode and one is saddle (3, 1). Are established the bifurcations of the spatial overwound cycles appropriate to doubling of the period of oscillations dynamic variable and resulting to chaotic oscillations at increase of the relation of factors friction.

Weakly nonlinear magnetic convection in a nonuniformly rotating electrically conductive medium under the action of modulation of external fields

“…The amplitudes of the perturbed quantities depend only on the slow time τ = ǫ 2 t. For simplicity let us take into account the nonlinear terms in (29) only in the heat balance equation. As it is shown in [23], this approximation is equivalent to applying the Galerkin approximation of the minimum order to the equations ( 29).…”

“…In recent works [28]- [29] was investigated a modified system of Rikitaki equations taking into account friction and not reducing it to a three-dimensional form as for example in [24]. This made it possible to more clearly show that at first the oscillations of the current (or magnetic) variable near a certain stationary state with an increase in amplitude go into oscillations around an another stationary state, which simulated inversions [29]. In [28] it is established that after chaotic behavior the system goes into stable mode.…”

“…According to the authors of [28], such a regime can describe superchrons in the inversion of the geomagnetic field. In contrast to the works [23]- [29], in [11]- [12], [14] it is proposed to model the magnetic field inversion by a dynamic system of equations of Lorentz type, respectively, for (6D) and (8D)-dimensional phase space.…”