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Bicubic B-Spline Surface Approximation of Invariant Tori



Sita Ramamurti, David E. Gilsinn


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.
NIST Interagency/Internal Report (NISTIR) - 7731
Report Number


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


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], (Accessed May 27, 2024)


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Created October 20, 2010, Updated June 2, 2021