Existence Of Quasi-periodic Solutions Of Two Coupled Van Der Pol Equations

In this paper, we consider two coupled Van der Pol equation. There are many results on two coupled Van der Pol equations. In particular, there are great value in terms of mechanical engineering, electronics, bio-sciences and engineering.In this paper, we discuss the existence of quasi-periodic solutions of two coupled Van der Pol equations via KAM theory. We get the average equation of the original system by polar coordinates transformation, then get invariant 2-torus of the average equation. Our main aim is to analyse the existence of invariant 2-torus of the original system.Through a series of reversible transformations, we get a normal form in which the KAM theory can be used, and then the normal form is reduced by the Newton iteration method. In the iterative procedure there will be small divisor problems. Then the parameters must meet the Diophantine condition and we have to estimate the measure. Finally for most of parameters within the considered parameter set, the normal form exists quasi-periodic solutions. Because the coordinate transformations are reversible, the original system exists quasi-periodic solutions for most of the parameters.