Multidimensional continuation of families of periodic orbits

In this work, we develop an atlas algorithm for continuation of piecewise polynomial discretizations of periodic orbits of ordinary differential equations. Such an algorithm generates a discretized representation of a manifold of such orbits embedded in a larger variable space. Each chart associated with the discretized atlas is defined in terms of a base point on the manifold and a basis for the local tangent space. The goal of any such algorithm is to cover all parts of the manifold without leaving any holes behind, and to do so efficiently without covering areas more than once. The current implementation of atlas algorithms in the continuation package COCO fails in both regards when applied to continuation of solutions to general periodic boundary value problems. This failure arrises due to the fact that COCO treats the variable space in which the manifold is embedded as Euclidean space, e.g., the distance between charts is calculated in terms of the Euclidean norm of the vector between the charts’ base points. For two charts with base points corresponding to the same periodic orbits with two different phases, the result is a non-zero distance even as the intent may be to treat them as the same orbit. Since such distances are used to calculate suitable directions of continuation at each step of the algorithm, an incorrectly computed distance may result in continuation along an inappropriate direction. In this thesis, we overcome this problem by projecting the representation of individual charts to a phase-invariant Fourier representation in which suitable directions of continuation may be identified. We use two examples to illustrate our methodology: continuation along 1- and 2-dimensional manifolds of periodic orbits of two nonlinear dynamical systems. It is observed that the manifolds generated using the proposed algorithm are well-organized and repetitive covering is minimized.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2018

Author dc:creator

• Wang, Yuqing

Contributors dc:contributor

• Dankowicz, Harry

• Numerical continuation, ordinary differential equations, periodic orbits, multidimensional manifolds.

Statement dc:rights

• Copyright 2018 Yuqing Wang

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/102392

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/102392