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.