In order to judge how close we are from a Steady State $X = \{x_i | i \in [1, n] \}$ we estimate the distance from it by calculating the next Newton step $H = X_{n+1} - X_{n}$ where \begin{equation} X_{n+1} = X_{n} - \left(J(f)(X_{n})\right)^{-1} \; f(X_{n}), \end{equation} $X_n$ is the current solution of the root finding problem and $J(f)(X_{n}))$ is the Jacobian $\delta f_i/\delta x_j |_{X_n}$. Note that Equation (1) involves the inverse of the Jacobian which is not defined for a singular matrix. In order to avoid calculating the matrix inverse we rewrite Equation (1): \begin{equation} J(f)(X_{n}) \; (X_{n+1} - X_{n}) + f(X_{n}) = J(f)(X_{n}) \; H + f(X_{n}) = 0 \end{equation} We do not calculate the exact solution of Equation (2). Instead we are looking for the minimum norm solution: \begin{equation} \mbox{min}_{H} \; \left\lVert J(f)(X_{n}) \; H + f(X_{n}) \right\rVert = 0 \end{equation} We solve this by using the LAPACK routine dgelsy. In case that the Jacobian is rank deficient we compute the $L_2$ norm of the absolute and relative errors. Now that we have computed $H$ minimizing Equation (3) we can compute its relative and absolute $L_2$ norms adding the corresponding errors of the solution in case of rank deficiency. Please note that the resulting $L_2$ norms are in variable space and do not include time which means they are independent from the speed of the process which is not the case for the rate criterion. Finally in order to determine whether the solution $X_n$ is sufficiently close to the Steady State $X$ we compare the maximum of the absolute and relative $L_2$ norms against the desired tolerance.
For the simple non-autonomous ODE $dx/dt = 1/t^2$ the above described criterion will fail since the Jacobian has rank 0. Nevertheless an asymptotic Steady State exists. In general the issue will arise whenever we have a non-autonomous system, i.e., ODEs are explicitly time dependent.
In order to address the above problem we introduce an additional variable $T$ to the ODE $dT/dt = 1$ system. Thus, $T = t + \mbox{const}$. After choosing the constant to be 0 we can replace $t$ in the ODEs with $T$ and have now an autonomous ODE system equivalent to the original. The Jacobian of this system is given by: \begin{equation} \hat{J} = \begin{pmatrix} 0 && 0 \\ \frac{\delta x_i}{\delta T} && \frac{\delta x_i}{\delta x_j} \end{pmatrix} \end{equation} With $\hat{X} = (T, X)$ we solve \begin{equation} \mbox{min}_{\hat{H}} \; \left\lVert \hat{J}(f)(\hat{X}_{n}) \; \hat{H} + f(\hat{X}_{n}) \right\rVert = 0 \end{equation} The resulting $\hat{H}$ is no longer in variable space. In fact the first component explicitly measures time. We can convert time distance by multiplying with the maximum rate $\mbox{max}_i \; f_i(\hat{H})$ to estimate the distance from the Steady State in variable space.
