<p>This study demonstrates that a modified Kelvin–Voigt oscillator incorporating a nonlinear spring element governed by the double-well Duffing hyperelastic potential—representative of elastomeric seismic bearings composed of hard rubber—can exhibit chaotic vibration behavior under certain excitation conditions. This result reveals that the chaotic responses contain pronounced subharmonic and broadband continuous spectral components, which may amplify structural motion and compromise the protective effectiveness of seismic isolation systems. To investigate this phenomenon, the system dynamics are formulated as an autonomous set of three coupled first-order differential equations and solved numerically using the Adams–Bashforth–Moulton predictor–corrector scheme, well suited for stiff nonlinear systems. The analysis provides new insight into the complex vibrational behavior of rubber-based seismic and vibration isolation devices modeled by Duffing-type hyperelastic potentials, highlighting the importance of nonlinear material effects in the dynamic stability of protected structures.</p>