Not sure if title phrases my question correctly. Suppose we are given necessary asymptotic stability conditions of a 2D system at the origin \begin{equation} \dot{x}(t) = f(x(t), u(t))\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)\end{equation} where $x(t):=(x_1(t),x_2(t)) \in U\underset{open}\subset\mathbb{R}^2$ is the state, and $u(t) \in \mathbb{R}$ is an external input. The planar system is actually in the form of $$ \begin{aligned}x_1(t) &= g(x_1(t),x_2(t),u(t)) \\ x_2(t) &= h(x_1(t),u(t))\end{aligned}.$$
Now, let system dynamics be augmented by an auxiliary strictly-increasing time-varying input $\phi : [0,\infty)\to\mathbb{R}^+$ as $$ \begin{aligned}x_1(t) &= g(x_1(t),x_2(t),u(t)) \\ x_2(t) &= H(\phi(t),x_1(t),u(t))\end{aligned}.\;\;\;\;\;\;\;\;\;(2)$$ such that $\phi(t)$ is not a necessary signal for stability, that is, there is no function that renders origin of (2) stable when $H$ does not depend on either $x_1(t)$ or $u(t)$. I was wondering if set of necessary conditions for stability of (1) can be translated to necessary conditions for (2). Should I treat (2) as a time varying perturbation to nominal system (1) or would this be too conservative to compromise the translation of necessary conditions?
Special cases of interest are
$\phi(t):=\phi(0)+\int_0^t u(\tau)d\tau$
$H$ is periodic in $\phi$ (may apply some kind of averaging to study this case?)
As an example to motivate the question, consider an example stability condition for system (1) as $h(x_1(t),u(t)) <= \alpha(x_1(t)u(t))$ $\forall x_1(t)$ in the open neighborhood of origin, $\forall u(t)$, and some class K function $\alpha$. The augmented system does not have to satisfy these conditions $\forall \phi(t)$ but my guts feelings still tell me that they must be satisfied for most of $\phi(t)$.