Generalized solutions to degenerate dynamical systems

Abstract

The solutions to degenerate dynamical systems of the form A(x)ẋ=f(x) are studied by considering the equation as a differential inclusion. The set Z={det(A(x))=0}, called the singular set, is assumed to have an empty interior. The reasons leading us to the definition of the sets used for differential inclusion are exposed in detail. This definition is then applied on the one hand to generic cases and on the other hand to the particular cases resulting from physics, which can be found in Saavedra, Troncoso, and Zanelli [J. Math. Phys. 42, 4383 (2001)]. It is shown that generalized solutions may enter, leave, or remain in the singular locus.