How do small velocity-dependent forces (de)stabilize a non-conservative system?

The influence of small velocity-dependent forces on the stability of a linear autonomous non-conservative system of general type is studied. The problem is investigated by an approach based on the analysis of multiple roots of the characteristic polynomial whose coefficients are expressed through the invariants of the matrices of a non-conservative system. For systems with two degrees of freedom approximations of the domain asymptotic stability are constructed and the structure of the matrix of velocity-dependent forces stabilizing a circulatory system is found. As mechanical examples the Bolotin problem and the Herrman-Jong pendulum are considered in detail.