Stability in large of the systems with mechanisms of negative and quasi-zero stiffness plays an important role for improvement of the infra-low vibration protection. These mechanisms are predisposed to chaotic vibration motion. Analysis of chaotic vibration and comparative selection of the mechanisms are to be reasonable steps before deciding next steps in designing the vibration protection systems. Their dynamic behavior can be diagnosed and predicted by the qualitative and quantitative methods for analysis of chaotic motion. An algorithm has been developed to study chaotic motion of the mechanisms, and the conditions of dynamic stability of the systems with such mechanisms are formulated. The algorithm is based on the Lyapunov largest exponent and Poincare map of phase trajectory methods and includes (a) formulation of chaotic motion models and criterial experiments for the mechanisms and systems, (b) technique of comparative analysis of the models, (c) computation procedure to estimate their dynamic stability in large, (d) formulation of design and functional parameters for providing stable motion of the systems in the infra-frequency range, including near-zero values. Validity of the algorithm is demonstrated through the development of active pneumatic suspensions supplied with passive mechanisms of variable negative stiffness.