Nonintegrability of parametrically forced nonlinear oscillators
The authors discuss the nonintegrability of parametrically forced nonlinear oscillators represented by second-order homogeneous differential equations of the form \[ \ddot{x}+f(x,\dot{x},\cos (\omega t),\sin (\omega t))x+g(x,\dot{x},\cos (\omega t),\sin (\omega t))\dot{x}=0, \] where \(\dot{x}=dx/dt\) and \(f\), \(g\) are rational functions satisfying the conditions \(f(0,0,y_1,y_2)=f_0+f_1y_1\), \(g(0,0,y_1,y_2)=g_0+g_1y_1+g_2y_2\). These contain the Duffing and van der Pol oscillators as special cases. Sufficient conditions for their rational nonintegrability are given in the sense of Bogoyavlenskij, using the Kovacic algorithm and an extension of the Morales-Ramis theory. The latter is due to \textit{M. Ayoul} and \textit{N. T. Zung} [C. R., Math., Acad. Sci. Paris 348, No. 23--24, 1323--1326 (2010; Zbl 1210.37076)] and is applied to the associated variational equations. The identity components of the corresponding differential Galois groups are shown to be noncommutative even if the Galois group is triangularizable (i.e., solvable by quadratures).
scientific article; zbMATH DE number 6958430
scientific article