Instability of a parameter varying system whose parameters belong to a compact set Suppose, there is a system
$$\dot{x}=f(t, \gamma_p(t), x)$$ with $x\in\mathbb{R}^2$.
For my specific case, parameter vector $\gamma_p$ is a scalar and known monotonic function with a compact image set (for example $\gamma_p(t) = e^{-t}$, thus $\gamma_p \in [1,0)$).
I would like to show instability of this system.
Well known techniques (specifically, averaging) are applicable to show system's instability for all frozen parameter values, that is a simplified system $\tilde{f}$ with
$$\dot{x}=f^\star(t, x) = f(t, \gamma_p(t^\star), x) \quad \forall \; t^\star \in [0, \infty).$$
Can instability of all frozen parameter cases let me conclude anything about the instability of the original system?
Specifics of my problem
$f$ takes the form
$$ f(t,\gamma_p(t),x) = \varepsilon \begin{bmatrix}\tfrac{\alpha(1-\gamma_p(t))}{\gamma_p(t)} \\ 1\end{bmatrix} L\left(t, x_2, (x_1-\alpha x_2)\gamma_p(t)+\alpha x_2\right)$$
where $\varepsilon > 0$ is a small parameter of the system, $\alpha$ is a positive coefficient and $L$ is a $2\pi$ periodic function over only the first argument $t$ (thus not necessarily periodic when $\gamma_p$ is changing) with $$\mathrm{sign}\left[\int_{0}^{2\pi}L(t, x_2, \underbrace{(x_1-\alpha x_2)\gamma_p(t^\star)+\alpha x_2}_{y(t^\star)})\mathrm{d}t\right] = \mathrm{sign}\left[y(t^\star)\right]\quad \forall \; t^\star \in [0,\infty).$$
Substituting $\gamma_p(t^\star)$ gives the frozen parameter system
$$ f^\star(t,x) = \varepsilon \begin{bmatrix}\tfrac{\alpha(1-\gamma_p(t^\star))}{\gamma_p(t^\star)} \\ 1\end{bmatrix} L\left(t, x_2, (x_1-\alpha x_2)\gamma_p(t^\star)+\alpha x_2\right)$$
With $\gamma_p(t^\star)$ now fixed, dynamics become periodic with period $2\pi$ which allows me to apply averaging. Specifically, we can determine the stability/instability through the averaged system
$$ \dot{x}\approx\tilde{f}^\star(x) = \varepsilon \begin{bmatrix}\tfrac{\alpha(1-\gamma_p(t^\star))}{\gamma_p(t^\star)} \\ 1\end{bmatrix}\int_{0}^{2\pi}L(t, x_2, (x_1-\alpha x_2)\gamma_p(t^\star)+\alpha x_2)\mathrm{d}t$$
Using further details of the system, I can show the system grows unboundedly in the direction $[\gamma_p(t^\star), (1-\gamma_p(t^\star)]$ in the space spanned by $(x_1,x_2)$.
 A: I don't think it will be the case in general (I mean without taking advantage of some particular structure of your problem).
Just looking at the question "Can instability of all frozen parameter cases let me conclude anything about the instability of the original system?" the answer is: not always.
For example, consider this system:
$$
\dot{x} = \begin{bmatrix}
0 &1\\ \gamma_1(t) & \gamma_2(t)
\end{bmatrix}x
$$
where $\gamma_1,\gamma_2$ are periodic functions with period 2, and comply
$\gamma_1(t)=-1, \gamma_2(t)=0.01$ for $0\leq t\leq 1$ and $\gamma_1(t)=0.01, \gamma_2(t)=-1$.
Hence, this system switches between
$$
\dot{x} = \begin{bmatrix}
0 &1\\ -1 & 0.01
\end{bmatrix}x, \ t\in[0,1)
$$
and
$$
\dot{x} = \begin{bmatrix}
0 &1\\ 0.01 & -1
\end{bmatrix}x, \ t\in[1,2)
$$
and so on, periodically.
Note that in each case, by analyzing the frozen parameters, the matrices $A_1=\begin{bmatrix}
0 &1\\ -1 & 0.01
\end{bmatrix}$ and $A_2=\begin{bmatrix}
0 &1\\ 0.01 & -1
\end{bmatrix}$ are not Hurwitz, so each system is unstable on its own.
Despite this, the switching system is asymptotically stable. Look at what happens in the switching instants. For example, at $t=1$:
$$
x(1) = \exp(A_1)x(0)
$$
and at $t=2$:
$$
x(2) = \exp(A_2)x(1) = \exp(A_2)\exp(A_1)x(0)
$$
and so on, so that we can look at every two jumps to obtain:
$$
x(k+2)=\exp(A_2)\exp(A_1)x(k) = \tilde{A}x(k)
$$
for $k$ even. Interestingly, the matrix $\tilde{A}=\exp(A_2)\exp(A_1)$ has eigenvalues in side the unit disk, namely $0.1067 + 0.6002i, 0.1067 - 0.6002i$. Hence, the sequence $x(0),x(2),x(4),\dots$ converge to the origin asymptotically. A similar thing happens to $x(1),x(3),x(5),\dots$ and its not hard to see that $x(t)$ in the intervals between converge to the origin as well. The reason is that the unstable behavior of $A_1$ is corrected by $A_2$ and viceversa. Through an average argument you might also see that the average of the two systems is stable.
In summary, in general you won't be able to conclude instability of the parameter-varying system by looking at the frozen parameter systems. However, you might take advantage of some particular structure of your setting, namely that $\gamma(t)$ is scalar or monotone. Hence, without more context/details it is hard to know.
