Bicubic B-Spline Surface Approximation of Invariant Tori

Published

October 20, 2010

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

NIST Pub Series

NIST Interagency/Internal Report (NISTIR)

Pub Type

NIST Pubs

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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 October 8, 2025)