Consider the following time series model
\begin{align*} X_t&=f_{1}(X_{t-1}) + f_{2}(Y_{t-1}) + \varepsilon_t^X\\ Y_t&=g_{1}(Y_{t-1}) + g_{2}(X_{t-1}) + \varepsilon_t^Y, \end{align*}
where epsilons are all iid. What should functions $f_1, f_2, g_1, g_2$ satisfy for time series $(X,Y)$ to be stationary? Are there some known results for such a time series models?
Maybe a little easier question is that what should $f$ satisfy for stationarity of $X_t=f(X_{t-1}) +\varepsilon_t$ ? For example, we can assume that $f$ is continous and $liminf_{x\to\pm\infty}|\frac{x}{f(x)}|>1$. Is it enough?