| Contents |
| We first consider the ODE
\begin{align}\label{PODE}
x'(t) = -f(x(t)) + g(t),
\end{align}
which is a perturbed version of the autonomous ODE $y'(t)=-f(y(t))$. We assume the unperturbed equation has a unique globally stable equilibrium at zero and under the assumption that $f \in RV_0(1)$ we construct, essentially, necessary and sufficient conditions on $g$ under which the rate of convergence to zero is preserved. Analogously, we define the perturbed SDE
\begin{align}\label{PSDE}
dX(t) = -f(X(t))dt + \sigma(t)dW(t),
\end{align}
where $W$ is a one dimensional Brownian motion. Once more, under the assumption that $f$ is regularly varying with unit index, our goal is to characterise conditions under which we preserve almost sure rates of decay to zero. The constructions of the above results are greatly abetted by the study of the auxiliary ODE
\begin{align}\label{IPODE}
z'(t) = -f(z(t)+\gamma(t)),
\end{align}
for a suitable forcing function $\gamma$. We refer to this as the internally perturbed equation and will discuss the utility and novelty of its study in advancing the solutions of our main problems.
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This talk will be based on research that was funded by the Irish Research Council under the project GOIPG/2013/402. |
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