Suppose we have two ODEs:
- $\dot{x}(t) = f(x(t),t)$
- $\dot{y}(t) = f(y(t),t) + g(y(t),t)$
If we have identical starting conditions $x(0) = y(0)$, we see $$y(t) - x(t) = \int_0^t \left[ f(y(t),t)-f(x(t),t) \right] dt + \int_0^t g(y(t),t)dt $$
Is there analysis giving conditions on $f$ and $g$ that permits us to prove that $\lim_t x(t) - y(t) = 0$ or maybe their limits exist and $x(\infty), y(\infty)$ land very close together?
Intuitively for small $g$ (uniformly bounded near 0) and some condition on the derivative of $f$ (like $f'$ has eigenvalues with real part bounded below a negative number and $\nabla f$'s first component is uniformly bounded) we can ensure these end up close together.