Nonintegrability of Nonlinear Oscillators with Parametric Forcing

We discuss nonintegrability of parametrically forced nonlinear
oscillators which are represented by second-order homogeneous
differential equations with trigonometric coefficients and contain the
Duffing and van der Pol oscillators as special cases. Specifically, we
give sufficient conditions for their rational nonintegrability in the
meaning of Bogoyavlenskij, using Kovacic's algorithm as well as an
extension of the Morales-Ramis theory due to Ayoul and Zung. In
application of the extended Morales-Ramis theory, for the associated
variational equations, the identity components of their differential
Galois groups are shown to be not commutative even if the differential
Galois groups are triangularizable, i.e., they can be solved by
quadratures. The obtained results are very general and reveal their
rational nonintegarbility for the wide class of parametrically forced
nonlinear oscillators. We also give two examples for the van der Pol
and Duffing oscillators to demonstrate our results.