In　this　paper,　we　derive　the　global　bifurcation　diagrams　of　a　SD　oscillator　which　exhibits　both　smooth　and　discontinuous　dynamics　depending　on　the　value　of　a　parameter　a.　We　research　all　possible　bifurcations　of　this　system,　including　Pitchfork　bifurcation,　degenerate　Hopf　bifurcation,　homoclinic　bifurcation,　double　limit　cycle　bifurcation,　Bautin　bifurcation　and　Bogdanov–Takens　bifurcation.　Besides,　we　show　that　the　system　has　five　limit　cycles,　including　four　small　limit　cycles　and　one　large　limit　cycle.　At　last,　we　give　all　numerical　phase　portraits　to　illustrate　our　results.　©　2017,　Springer　Science+Business　Media　B.V.,　part　of　Springer　Nature.