Constructing Galerkin's approximations of invariant tori using MACSYMA

Abstract

Invariant tori of solutions for nonlinearly coupled oscillators are generalizations of limit cycles in the phase plane. They are surfaces of aperiodic solutions of the coupled oscillators with the property that once a solution is on the surface it remains on the surface. Invariant tori satisfy a defining system of nonlinear partial differential equations. This case study shows that with the help of a symbolic manipulation package, such as MACSYMA, approximations to the invariant tori can be developed by using Galerkin's variational method. The resulting series must be manipulated efficiently, however, by using the Poisson series representation for multiply periodic functions, which makes maximum use of the list processing techniques of MACSYMA. Three cases are studied for the single van der Pol oscillator with forcing parameter e ffi 0.5, 1.0, 1.5, and three cases are studied for a pair of nonlinearly coupled van der Pol oscillators with forcing parameters ~ --0.005, 0.5, 1.0. The approximate tori exhibit good agreement with direct numerical integrations of the systems.

Bicubic b-spline surface approximation of invariant tori

“…In this study we introduce a bicubic B-spline computational algorithm that, at least for a certain class of nonlinear systems, computes invariant tori to a high degree of accuracy for a larger range of parameters. In particular, following the work of Gilsinn [6,7], we shall consider the system of coupled Van der Pol oscillatorsẍ…”