ANGEO Annales Geophysicae ANGEO Ann. Geophys. 1432-0576 Copernicus Publications Göttingen, Germany 10.5194/angeo-28-993-2010 Modelling solar cycle length based on Poincaré maps for Lorenz-type equations Lundstedt H. 1 Persson T. 2 Swedish Institute of Space Physics, Lund, Sweden Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland 21 04 2010 28 4 993 1002 Copyright: © 2010 H. Lundstedt 2010 This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/ This article is available from https://angeo.copernicus.org/articles/28/993/2010/angeo-28-993-2010.html The full text article is available as a PDF file from https://angeo.copernicus.org/articles/28/993/2010/angeo-28-993-2010.pdf

Two systems of Lorenz-type equations modelling solar magnetic activity are studied: Firstly a low order dynamic system in which the toroidal and poloidal fields are represented by x- and y-coordinates respectively, and the hydrodynamical information is given by the z coordinate. Secondly a complex generalization of the three ordinary differential equations studied by Lorenz. By studying the Poincaré map we give numerical evidence that the flow has an attractor with fractal structure. The period is defined as the time needed for a point on a hyperplane to return to the hyperplane again. The periods are distributed in an interval. For large values of the Dynamo number there is a long tail toward long periods and other interesting comet-like features. These general relations found for periods can further be physically interpreted with improved helioseismic estimates of the parameters used by the dynamical systems. Solar Dynamic Observatory is expected to offer such improved measurements.

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