An alternative Lyapunov-like instability proof

Question

Let $\gamma$ and $\omega$ be two positive real numbers. Consider the following 1-dimensional linear time-varying system $$ \dot{x}(t)=\underbrace{\left(-\frac{1}{2}(1+\gamma)-\cos(\omega t) + \frac{1}{2}\sqrt{4\cos^2(\omega t) +(1-\gamma)^2}\right)}_{=:a(t)}x(t), \quad x(0)\in\mathbb{R}. $$

By direct computation of the solution of the above differential equation, it is easy to check that, for $\gamma$ sufficiently small, the origin of the above system is an unstable equilibrium point.

However, I'd like to prove instability of the origin via a Lyapunov approach. More precisely, supposing that $\gamma$ is sufficiently small, I'd like to find a function $v(x,t):=q(t)x(t)^2$ such that

1. The norm of $q(t)$ is bounded in $t$, i.e. $\|q(t)\|\le k$, $k>0$, for all $t\ge 0$;
2. The derivative of $v(x,t)$ along the trajectories of the system is strictly negative for all $t$, that is $2a(t)q(t)+\dot{q}(t)<0$ for all $t$;
3. There exists (at least) a $\bar{t}>0$ such that $q(\bar{t})<0$.

So my question is:

Is it actually possible to find such a Lyapunov function $v(x,t)$?

I made several attempts but no one worked. So every comment/suggestion would be greatly appreciated. Thanks!