On the non-integrability of a family of Duffing-van der Pol oscillators

Abstract

We investigate the non-integrability of a family of Duffing-van der Pol oscillators x+ alpha x(x²-1)+x+ beta x³= gamma cos omega t by studying the analytic properties of the dynamics in complex time. We find that the solutions of (∗) have no worse than algebraic singularities at t, with only (t-t∗)½ terms present in their series expansions, unlike, for example, the alpha =0 Duffing case, where, typically, log(t-t∗) terms arise. Still, when integrating (∗) around long enough contours, a remarkably intricate pattern of square root singularities emerges, on different sheets, which appears to prevent solutions from ever returning to the original sheet. Such evidence of infinitely-sheeted solutions, termed the ISS property, has also been observed in a number of Hamiltonian systems and is illustrated here on a simple example of a single, first-order differential equation. We suggest that the ISS property is a necessary condition for non-integrability, i.e. non-existence of a complete set of analytic, single-valued constants of the motion, which would permit the complete integration of a dynamical system in terms of quadratures.