Homoclinic orbits in the near-integrable double discrete sine-Gordon equation

We establish the existence of homoclinic orbits for the near{integrable double discrete sine-Gordon (dDSG) equation under periodic boundary conditions. The hyperbolic structure and homoclinic or- bits are constructed through the Backlund transformation and Lax pair. A geometric perturbation method based on Mel'nikov analysis is used to establish necessary criteria for the persistent of tem- porally homoclinic orbits for the class of dDSGequations with dissipative perturbations.