The Lorenz system and its generalizations as dynamo models with memory

Abstract

The one of the known applications of the classical Lorenz system is an axisymmetric αω-dynamo with a dynamical quenching of the α-effect by the helicity. In this paper we consider generalizations of the Lorentz system, which are the models of α2 - and α2ω-dynamo. The cases of finite and infinite memory in the quenching functional are considered. The conditions for the existence of stationary dynamo regimes and regimes of regular and chaotic inversions are analytically and numerically studded.

Two-mode dynamo as a oscillator with memory

“…The paper analyzes the previously proposed by the author model of a two-mode dynamo [6] from the point of view of the theory of oscillators with hereditary potential. It is shown that with the help of a qualitative analysis of the oscillator, it is possible to establish what…”

“…The simplest two-mode model of temporal variations of an axisymmetric magnetic field can be written in the following dimensionless form [6]:…”

“…The integro-differential system (1) is a simplest model of the α 2 ω-dynamo. In particular, the system describes: α 2 ω-dynamo, if ξ 0 and s 2 σ > ξD; α 2 -dynamo, if ξ 0 and s 2 σ = ξD; αω-dynamo, if ξ = 0 [6]. In this paper, we restrict ourselves to the αω-dynamo case only, i.e.…”

“…Earlier, we considered some integro-differential generalization of the Lorenz system as a model of a two-mode dynamo with hereditary quenching of α-effect [6]. In this paper, we will study it as an oscillator.…”

Hereditary Oscillator Associated with the Model of a Large-Scale αω-Dynamo

“…A rigorous mathematical derivation of the model is carried out. The general idea of building such a model was previously described in the work of the author [33]. 2.…”