  About twenty years ago, Rabinowitz showed firstly that there exist
heteroclinic orbits of autonomous Hamiltonian system joining two equilibria. A
special case of autonomous Hamiltonian system is the classical pendulum
equation. The phase plane analysis of pendulum equation shows the existence of
heteroclinic orbits joining two equilibria, which coincide with the result of
Rabinowitz. However, the phase plane of discrete pendulum equation is similar
to that of the classical pendulum equation, which suggests the existence of
heteroclinic orbits for discrete pendulum equation also. By using variational
method and delicate analysis technique, we show that there indeed exist
heteroclinic orbits of discrete pendulum equation joining every two adjacent
points of $\{2k\pi+\pi: k\in{\mathbb Z}\}$.
