In this paper,we investigate the multi-buckling and multi-parameter codimension phenomena for a geometrical nonlinear oscillator comprising a lumped mass and a pair of springs,which is a rig-coupled SD oscillator [1]-[5].Even the springs are linear,the system is strong nonlinear with irrationalities due to the geometrical con_gurations.The equilibrium stabilities and the bifurcations of the unperturbed system are provided to demonstrate the complex equilibrium bifurcations and the transitions which result in the multiple buckling phenomena.This study also presents the high co-dimension bifurcation phenomena with multiple parameters dominating the behaviour of the system.