In the present paper, a theory is developed qualitatively and quantitatively describing the paradoxical behavior of general non-conservative systems under the action of small dissipative and gyroscopic forces. The problem is investigated by the approach based on the sensitivity analysis of multiple eigenvalues. The movement of eigenvalues of the system in the complex plane is analytically described and interpreted. Approximations of the asymptotic stability domain in the space of the system parameters are obtained. An explicit asymptotic expression for the critical load as a function of dissipation and gyroscopic parameters allowing to calculate a jump in the critical load is derived. The classical Ziegler–Herrmann–Jong pendulum considered as a mechanical application demonstrates the efficiency of the theory.