Suppose there is a continuous dynamic system of order $n+1$ given by $$\begin{align} \dot{x}_1 &= Ax_1 + F(t)x_1 + G_1(t)x_2 \\ \dot{x}_2 &= kx_2 + G_2(t)x_1 \end{align}$$ where $x_1\in\mathbb{R}^n$ is a vector and $x_2\in\mathbb{R}$ is a scalar. Moreover, we know that matrices $A, F(t), G_1(t), G_2(t)$ are bounded and such that for $k < 0$ the system is asymptotically stable what can be shown by the Lyapunov analysis. Specifically, $A\in\mathbb{R}^{n\times n}$ is Hurwitz.
Question: Can we say anything specific about the behavior of this system for $k>0$? Specifically, I would like to show that the system is unstable and moreover its evolution is such that $x_2$ diverges and (at least after some initial transient state) does not oscillate (i.e. does not change its sign).
Side question: If nothing can be said in the general case, what tools could be useful in analysis of the specific system (assuming I know $A, F(t), G_1(t), G_2(t)$)? I am familiar with multiple approaches to show stability of the system, but hardly any to analyze the evolution of unstable one.
Without further assumptions, one cannot say anything.
Consider the case $G_2 = 0$. In that case the system clearly becomes unstable if $k > 0$.
On the other hand if $n = 1, A = \alpha < 0, F = 0, G_1 = -G_2 = \gamma > 0$, then the system is stable if $k + \alpha < 0, \, \gamma^2 + k\alpha > 0$. These conditions may be satisfied for arbitrary large positive $k$, if also $\gamma$ and $|\alpha|$ are sufficiently large.