Given an interconnection of dynamic systems $1,2,\cdots, n$ with $x_1(t), x_2(t), \cdots, x_n(t)$ the corresponding states such that
$\dot{x}_{i+1}(t) = -x_{i+1}(t) + f(x_i(t))$, where $f(x_i(t))$ is monotone increasing and continuous in $x_i(t)$, and $f(x_i) = x_{i+1}$ with $x_i$ the stationary value of the state of the $i$-th system.
Now suppose $x_1(t) = x_1 + e^{-t}$, i.e., $\underset{t \rightarrow \infty}{lim}~ x_1(t) = x_1$. Can we prove that $\underset{t \rightarrow \infty}{lim}~ x_i(t) = x_i$ for all $i \in \{1,2,\cdots, n\}$?