First of all let us try to solve the ODEs. The you can deduce what you wish from it.
$\dot{x}(t)=y(t)$
$\dot{y}(t)= - 4 x(t) + y(t)^2$
The orbits obey the equation
$\frac{d y}{d x}=\frac{y^2-4 x}{y}$
Multiplying by $y$ this can be written as
$\frac{1}{2}\frac{d y^2}{\text{dx}}=y^2-4 x$
and is easily integrated to give
$y^2=c \exp (2 x)+4 x+2$
where $c$ is a constant of integration.
Let the Initial state be $( x(0) = x0$, $y(0) = y0 )$
Then
$c=\left(-4 x(0)+y(0)^2-2\right) \exp (2 (x-\text{x0}))$