The behavior of periodic orbits of the standard mapping near their residues R =1 and R = 0 is studied. There is a sequence of period doubling bifurcations corresponding to the former, the bifurcation ratio δ and scaling factors α and βagree with those obtained from other two-dimensional area-preserving mappings. There are same period bifurcations corresponding to the latter, which is related to the antisymetric nature of the standard mapping. Moreover, by calculating Lyaponov exponents of chaotic orbits, we have found near the accumulation point k∞ of a sequence of period doubling bifurcations a scaling relation λ=λ∞+A(k-k∞)+B(k-k∞)τ with τ≈0.32, it agrees with the result τ=ln(2)/ln(δ)(δ=8.7210972…) conjectured theoretically.