Attractors of nonlinear dynamical systems with a weakly monotone measure

This paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure. It is assumed that the measure of a set of points in the phase space is increasing along the flow. We prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions. For a system of autonomous ordinary differential equations, we present attractivity conditions in terms of the divergence with a density function. Unlike previous results in the literature, our approach allows the use of a wider class of density functions if the divergence vanishes on a set of positive measure. As an example, the attitude stabilization problem of a rigid body is solved by using an affine feedback control for the kinematic equations in terms of quaternions.