<p>Nonlinear oscillators with multiple potential wells are widely used in practical structures such as vibration absorbers and energy harvesters due to their lower energy thresholds. The study of the chaotic characteristics of various homoclinic oscillators is of great significance for further understanding multi-stable systems, since homoclinic orbits characterize the boundary between different types of motion. The proper design of triple smooth-discontinuous (SD) oscillators can achieve four-well characteristics, and the elastic restoring force of the system can be approximated as a piecewise linear function. Unlike bistable and tri-stable oscillators, this oscillator exhibits multiple types of homoclinic orbits, and analytical expressions for these types of orbits are derived by analyzing the geometric structure of phase space. By utilizing the energy relationships of the Hamiltonian system and extending the Melnikov method, the threshold curves of these three types of homoclinic orbits are derived. Finally, the relationship between these threshold curves and the chaotic characteristics of the system is investigated through numerical simulations.</p>