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  • Numerical methods for finding periodic points in discrete maps. - High order islands chains and noble barriers in a toroidal magnetic configuration


We first remind usual physical and mathematical concepts involved in the dynamics of Hamiltonian systems, and namely in chaotic systems described by discrete 2D maps (representing the intersection points of toroidal magnetic lines in a poloidal plane in situations of incomplete magnetic chaos in Tokamaks).

Finding the periodic points characterizing chains of magnetic islands is an essential step not only to determine the skeleton of the phase space picture, but also to determine the flux of magnetic lines across semi-permeable barriers like Cantori.

The report discusses several computational methods used to determine periodic points in N dimensions, which amounts to solve a set of N non-linear coupled equations: Newton method, minimization techniques, Laplace or steepest descend method, conjugated direction method and Fletcher-Reeves method. The last method has been improved in an important way, without modifying its useful double-exponential convergence. This improved method has been tested and applied to finding periodic points of high order m in the 2D Tokamap mapping, for values of m along rational chains of winding number n/m converging towards a noble value where a Cantorus exists. Such precise positions of periodic points have been used in the calculation of the flux across the Cantorus. us.

Additional information

Authors: STEINBRECHER G, University of Craiova, Physics Faculty, Department of Theoretical Physics, Craiova (RO);REUSS J-D, CEA Cadarache, Saint-Paul-lez-Durance (FR);MISGUICH J H, CEA Cadarache, Saint-Paul-lez-Durance (FR)
Bibliographic Reference: Report: EUR-CEA-FC-1719 (2001), pp.24
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