Let $f:[0,1]^2\to\mathbb{R}$ be continuous and let $\delta>0$. Fix initial condition $\vec{x}_0=(x_{10},x_{20})\in [0,1]^2$; time $t$ takes values in $[0,\infty)$. Consider the following system of ODE: \begin{equation} \dot{x}_{\color{red}{1}t} = \begin{cases} \hfill -\delta \boldsymbol{1}_{(0,1]}(x_{\color{red}{1}t}),& \text{if } f(x_{\color{red}{1}t},x_{\color{blue}{2}t})\leq 0 \\ \hfill 0,& \text{if } x_{\color{red}{1}t}=1 \text{ and } f(1,x_{\color{blue}{2}t})\geq \delta \\ \hfill f(x_{\color{red}{1}t},x_{\color{blue}{2}t})-\delta ,& \text{otherwise}\\ \end{cases} \end{equation}
\begin{equation} \dot{x}_{\color{blue}{2}t}=\begin{cases} \hfill -\delta \boldsymbol{1}_{(0,1]}(x_{\color{blue}{2}t}),& \text{if } f(x_{\color{blue}{2}t},x_{\color{red}{1}t})\leq 0 \\ \hfill 0,& \text{if } x_{\color{blue}{2}t}=1 \text{ and } f(1,x_{\color{red}{1}t})\geq \delta \\ \hfill f(x_{\color{blue}{2}t},x_{\color{red}{1}t})-\delta ,& \text{otherwise}\\ \end{cases} \end{equation} Where $\boldsymbol{1}_{\cdot}(\cdot)$ denotes the indicator function. Can this be analytically solved?
The main source of weirdness for me is the fact that the above are piecewise functions. I understand that additional restrictions on $f$ might be needed; if so, please let me know, and I'll add the other assumptions I have in mind. (I chose to omit them for the moment in the interest of not obfuscating my question.)