The linear and nonlinear stiffness coupling forces in dynamical oscillators are usually dominated by positive stiffness components. Therefore, plotting the resultant force in y-axis with respect to the change in displacement in x-axis results in an odd symmetry in the first and third quadrants of the xy-plane. However, the appearance of negative stiffness content in coupling elements between dynamical oscillators generates a force that can be dominated by an odd-symmetry in the second and fourth quadrants. The underlying nonlinear dynamical behavior of systems employing this kind of force has not been well-studied in the literature. Accordingly, the considered system here is composed of two linear oscillators (LOs) that are nonlinearly coupled by a force of which the negative stiffness content is dominant. Therefore, the underlying dynamical behavior of the considered system in physical and dimensionless forms is studied on the frequency-energy-plots (FEPs) where many backbone curves of periodic solution have been obtained. It is found that within a wide range of nonlinear frequency levels, the nonlinear coupling force is dominated by a strong negative stiffness content at the obtained FEPs backbones.