On the iterates of shifted Euler's function

Let $\phi$ be the Euler’s function and fix an integer $k\ge 0$. We show that, for every initial value $x_1\ge 1$, the sequence of positive integers $(x_n{)}_{n\ge 1}$ defined by $x_{n+1}=\phi (x_n)+k$ for all $n\ge 1$ is eventually periodic. Similarly, for every initial value $x_1,x_2\ge 1$, the sequence of positive integers $(x_n{)}_{n\ge 1}$ defined by $x_{n+2}=\phi (x_{n+1})+\phi (x_n)+k$ for all $n\ge 1$ is eventually periodic, provided that $k$ is even.