Abstract: Both the symmetric period n-2 motion and asymmetric one of a one-degree-of-freedom impact oscillator were considered.The theory of bifurcations of the fixed point was applied to such model,and it was proved that the symmetric periodic motion only has pitchfork bifurcation by the analysis of the symmetry of the Poincar map.The numerical simulation shows that one symmetric periodic orbit could bifurcate into two antisymmetric ones via pitchfork bifurcation.While the control parameter changes continuously,the two antisymmetric periodic orbits will give birth to two synchronous antisymmetric period-doubling sequences,and bring about two antisymmetric chaotic attractors subsequently.If the symmetric system is transformed into asymmetric one,bifurcations of the asymmetric period n-2 motion can be described by a two-parameter unfolding of cusp,and the pitch-fork changes into one unbifurcated branch and one fold branch.