Dust acoustic solitary waves, blow-up solitary waves and periodic waves have been investigated in unmagnetized dusty plasmas with Maxwell-distributed electrons and ions, considering dust charge fluctuations using the bifurcation theory of planar dynamical systems. The basic equations are transformed to an ordinary differential equation involving the electrostatic potential. Applying the bifurcation theory of planar dynamical systems, we have established the existence of solitary, blow-up solitary and periodic waves. Four exact solutions of the solitary, blow-up solitary and periodic waves are derived depending on the physical parameters. Regarding the solitary, blow-up solitary and periodic waves, we have presented the combined effects of the density ratio of electrons and ions (${\it\alpha}$), the temperature ratio of electrons and ions $({\it\beta})$ and the speed of the travelling wave ($v$) on the characteristics of dust acoustic solitary, blow-up solitary and periodic waves.
