A direct method for the numerical calculation of quasiperiodic solutions applied to coupled van der Pol oscillators

Abstract

In the following paper we consider nonlinear dynamical systems depending on a parameter λ ∈ ℝ. They are described by autonomous systems of ODEs

$$\frac{{dx}}{{dt}}{\text{ }} = {\text{ }}f(x,\lambda )\,,\,f\,:\,{R^n}x\,R{\text{ }} \to {\text{ }}{R^n}{\text{ }}$$

(1.1)

where f ∈ C ^r, r ≥ 1. The widespread periodically forced nonautonomous sytems

$$\frac{{dx}}{{dt}}{\text{ }} = {\text{ }}f(t,x,\lambda )\,,\,f\,:(t + T,x,\lambda ){\text{ }} = {\text{ }}f(t,x,\lambda ){\text{ }}$$

(1.2)

with known period T can be rewritten as autonomous systems in the phase space S ¹ x ℝⁿ and dealt with like equation (1.1) in principle.