A 1-D smooth map constructed from DOG wavelet function is discussed in this paper. With analysis on the fixed points and the constructed iterative curves，its dynamical characteristics are thoroughly studied. It is found that the number of the fixed points will increase or decrease depending on the dilation and translation operation of the wavelet and thus the stable or unstable cross points and tangent point or zero points are produced. Numerical calculations are performed to obtain the dynamic behavior，bifurcation diagrams and Lyapunov spectra. Some nonlinear phenomena，such as period-doubling bifurcation，tangent bifurcation，boundary crisis bifurcation，periodic window，and imperfect Feigenbaum-tree，are revealed and investigated.