# If we perturb an ODE, with the same starting conditions can we show that they converge together over time or at least do not diverge?

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.

• What conditions must $f$ and $g$ satisfy if they are linear? This may give some intuition for the general case. Mar 1 at 17:44
• So if we assume $f(x,t) = Ax(t)$, and $g(x,t) = Bx(t)$, then so long as the eigenvalues($A$) and eigenvalues($A+B$) are in the left half plane, we see both systems converge asymptotically. Further, solving gives $x = e^{At}x(0), y = e^{(A+B)t}y(0)$, so both will converge to $0$ as $t\to\infty$. Which gives the result. I suspect that gives one hypothesis: If $d(f(x(t),t))/dt$ and $d((f+g)(x(t),t))/dt$ have spectra in the LHP for all $t>0$, we likely get the results. Mar 1 at 17:53
• You will get local results about the origin. Showing global stability for nonlinear systems is much more involved. Mar 1 at 17:55
• Is there a good reference or some search terms for this? Mar 1 at 18:01
• Nonlinear stability would be a starting place. There no very general theorems for global stability of nonlinear systems to my knowledge Mar 1 at 20:08