# A Newton-Raphson method for numerically constructing invariant curves

Marty, Wolfgang. A Newton-Raphson method for numerically constructing invariant curves. 2009, University of Zurich, Faculty of Science.

## Abstract

This thesis is concerned with the numerical construction of simply closed invariant curves of maps defined on the plane. We develop and discuss a Newton-Raphson method that is based on solving a linear functional equation. By using formal power series analytic solutions are derived and conditions for the existence of a unique 2π-periodic continuous solution are established. In order to approximate this particular solution a basis of functions is introduced and an infinite system of linear equations for the coefficients of the basis is considered. We solve a sequence of finite subsystems with increasing dimension. By using B-splines and Fourier series an algorithm for approximating the invariant curve is derived. The algorithm is tested with explicitly given maps, followed by the application to the Van-der-Pol equation and the logistic map. The implementation is checked extensively and the efficiency of the method is illustrated.

This thesis is concerned with the numerical construction of simply closed invariant curves of maps defined on the plane. We develop and discuss a Newton-Raphson method that is based on solving a linear functional equation. By using formal power series analytic solutions are derived and conditions for the existence of a unique 2π-periodic continuous solution are established. In order to approximate this particular solution a basis of functions is introduced and an infinite system of linear equations for the coefficients of the basis is considered. We solve a sequence of finite subsystems with increasing dimension. By using B-splines and Fourier series an algorithm for approximating the invariant curve is derived. The algorithm is tested with explicitly given maps, followed by the application to the Van-der-Pol equation and the logistic map. The implementation is checked extensively and the efficiency of the method is illustrated.

Detailed statistics

Item Type: Dissertation Sauter Stefan, Werner Bodo 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2009 14 Aug 2012 08:19 05 Apr 2016 15:54 213 http://opac.nebis.ch/F/?local_base=EBI01&con_lng=GER&func=find-b&find_code=090&request=001841407
Permanent URL: https://doi.org/10.5167/uzh-63930