A　transducer　is　a　system　that　couples　two　loads.　For　example,　an　electromechanical　transducer　couples　a　mechanical　force　and　an　electrical　voltage.　A　two-load,　nonlinear　system　can　exhibit　rich　behavior　of　bifurcation,　which　can　be　displayed　in　a　three-dimensional　space,　with　the　horizontal　plane　representing　the　two　loads,　and　the　vertical　axis　representing　the　state　of　the　system.　In　this　three-dimensional　space,　a　state　of　equilibrium　at　fixed　loads　corresponds　to　a　point　on　a　surface.　The　surface　is　smooth,　but　its　projection　to　the　load　plane　results　in　singularities　of　two　types:　fold　and　cusp.　Here　we　identify　the　fold　and　cusp　for　a　dielectric　elastomer　transducer　by　a　combination　of　experiment　and　calculation.　We　conduct　two　kinds　of　experiment:　electrical　actuation　under　a　constant　force　and　mechanical　pulling　under　a　constant　voltage.　The　theory　and　the　experiment　agree　well.　The　fold　and　cusp　are　essential　in　the　design　of　loading　paths　to　avoid　or　harness　the　bifurcation.