Multidimensional continuation of families of periodic orbits

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

Wang, Yuqing ; Dankowicz ; Harry

关键词: Numerical continuation, ordinary differential equations, periodic orbits, multidimensional manifolds.;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/102392/WANG-THESIS-2018.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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.