Mechanical metamaterials, designed to possess unique properties not achieved in conventional materials, offer broad possibilities for applications, ranging from vibration isolation to seismic wave manipulation. Among these, negative stiffness metamaterials emerge as a promising research area due to their unique deformation capabilities, allowing for greater deformation compared to traditional materials. However, lack of comprehensive mathematical modeling and experimental testing still hamper a full comprehension of their mechanical behavior and a mindful design. Specifically, we present a preliminary analysis of the frequency response curves for a negative stiffness metamaterial whose hysteretic behavior is simulated by using the Vaiana-Rosati hysteretic model. Through simulations, we examine its stability and bifurcation characteristics.