Dissipative systems: Relative roughness, nonroughness of various degrees, and integrability

This paper is devoted to the study of the relative structural stability (the relative roughness) of dynamical systems considered not on the whole space of dynamical systems, but only on a certain subspace of it. Moreover, the space of deformations of dynamical systems also does not coincide with the whole space of admissible deformations. In particular, we consider dissipative systems of differential equations that arise in the rigid-body dynamics and the theory of oscillations; dissipation in such systems may by positive or negative. We examine the relative roughness of such systems and, under certain conditions, their relative nonroughness of various degrees. We also discuss problems of integrability of these systems in finite combinations of elementray functions.