Inverse Jacobi multipliers and first integrals for nonautonomous differential systems

Abstract

In this paper, we consider nonautonomous differential systems of arbitrary dimension and first find expressions for their inverse Jacobi multipliers and first integrals in some nonautonomous invariant set in terms of the solutions of the differential system. Given an inverse Jacobi multiplier V, we find a relation between the Poincaré translation map Π at time T that extends to arbitrary dimensions the fundamental relation for scalar equations, \({V(T, \Pi(x)) = V(0,x)\Pi'(x)}\), found in García et al. (Trans Am Math Soc 362:3591-3612, 2010). The main result guarantees the existence of continua of T-periodic solutions for T-periodic systems in the presence of T-periodic first integrals and inverse Jacobi multipliers.