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.

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.

## Downloads

## Additional indexing

Item Type: | Dissertation |
---|---|

Referees: | Sauter Stefan, Werner Bodo |

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2009 |

Deposited On: | 14 Aug 2012 08:19 |

Last Modified: | 05 Apr 2016 15:54 |

Number of Pages: | 213 |

Related URLs: | http://opac.nebis.ch/F/?local_base=EBI01&con_lng=GER&func=find-b&find_code=090&request=001841407 |

## Download

TrendTerms displays relevant terms of the abstract of this publication and related documents on a map. The terms and their relations were extracted from ZORA using word statistics. Their timelines are taken from ZORA as well. The bubble size of a term is proportional to the number of documents where the term occurs. Red, orange, yellow and green colors are used for terms that occur in the current document; red indicates high interlinkedness of a term with other terms, orange, yellow and green decreasing interlinkedness. Blue is used for terms that have a relation with the terms in this document, but occur in other documents.

You can navigate and zoom the map. Mouse-hovering a term displays its timeline, clicking it yields the associated documents.