Consider the following general set of 2 ODE's
$$\dot{x}=\Theta(\dot{x} )f_1(x,y)+(1-\Theta(\dot{x}))f_2(x,y)$$ $$\dot{y}=(1-\Theta(\dot{y}))g_1(x,y)+\Theta(\dot{y})g_2(x,y)$$
where $\Theta(x)$ is the unit step function.
what it means is when $\dot{x}>0$ then the dynamics are govern by $f_1$ and when $\dot{x}<0$ then the dynamics are govern by $f_2$. With $\dot{y}$ it is the other way around.
Now let's consider that I can make in $f_1,f_2,g_1,g_2$ the transformations $x-y\rightarrow d$ and $y-y\rightarrow 0$, which leaves me with
$$\dot{x}=\Theta(\dot{x} )f_1(d)+(1-\Theta(\dot{x}))f_2(d)$$ $$\dot{y}=(1-\Theta(\dot{y}))g_1(d)+\Theta(\dot{y})g_2(d)$$
I would like to apply the transformation over the derivatives, such that I get an equation for $\dot{d}$ as well
$$\dot{d}=\dot{x}-\dot{y}=\Theta(\dot{x} )f_1(d)+(1-\Theta(\dot{x}))f_2(d)-(1-\Theta(\dot{y}))g_1(d)-\Theta(\dot{y})g_2(d)$$
If I apply the transformation on the unit step function as well, it seems that I lose information about the system
$$\dot{d}=\Theta(\dot{d} )f_1(d)+(1-\Theta(\dot{d}))f_2(d)-(1-\Theta(0))g_1(d)-\Theta(0)g_2(d)$$
The unit step can be defined when $\Theta(0)=\frac{1}{2}$ or $\Theta(0)=1$. In both cases I lose the information about $\dot{y}$.
Is there any way to make a transformation of such a system to $\dot{d}$?