Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Bicubic B-Spline Surface Approximation of Invariant Tori

Published

Author(s)

Sita Ramamurti, David E. Gilsinn

Abstract

The invariant torus of a coupled system of Van der Pol oscillators is approximated using bicubic B-splines. The paper considers the case of strong nonlinear coupling. In particular, the shapes of invariant torii for the Van der Pol coupling parameter $\lambda$ are computed in the range [0.1, 2.0]. Comparisons are given with results obtained by the MATLAB differential equation solver ode45. Very good normed residual errors of the determining equations for the approximate tori for the cases $\lambda = 0.1,\ 0.6$ are shown. At the upper limit of $\lambda = 2.0$ memory errors occured during the optimization phase for solving the determining equations so that full optimization for some knot sets was not achieved, but a visual comparison of the resulting invariant torus figure showed a close similarity to the solution using ode45.
Citation
NIST Interagency/Internal Report (NISTIR) - 7731
Report Number
7731

Keywords

bicubic B-splines, determining equations, invariant torus, large parameter case, optimization, Van der Pol oscillators

Citation

Ramamurti, S. and Gilsinn, D. (2010), Bicubic B-Spline Surface Approximation of Invariant Tori, NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=906906 (Accessed May 27, 2024)

Issues

If you have any questions about this publication or are having problems accessing it, please contact reflib@nist.gov.

Created October 20, 2010, Updated June 2, 2021