A classification of the difference linear systems of the second order

A clcissification of the difference linear systems of the second order is given. To a system of each type (here corresponds a certain set in the plane of the parameters $\operatorname{tr}F$, $\det F$ ($F$ is the matrix of the system). A certain difference equation having the linear-fractional first return function is associated with the system. The cycles of the system are described.