In this paper, we consider a generalization of a double discrete sine-Gordon equation. The generalization is done by introducing a number of parameters in the Lax-pair matrices. By restricting to the traveling wave solution, we derive a three-parameter family of discrete integrable dynamical systems using the so-called staircase methods. Special focus is on the cases where the resulting family of dynamical systems is of low dimension, i.e., two-dimensional. In those cases, the dynamics and bifurcation in the system is described by means of analyzing the level sets of the integral functions. Local bifurcation such as period-doubling bifurcation for map has been detected. Apart from that, we have observed nonlocal bifurcations which involve collision between heteroclinic and homoclinic connection between critical points.

DD-sine-Gordon equation, two-dimensional mapping, critical points, integrable systems, bifurcations.