I am looking at the following autonomous dynamical system $\dot{\mathbf x}(t)=\mathbf{f}(\mathbf{x}(t))$ on $\mathbb R^2$, with $\mathbf x=(x,y)$
$\dot x=-3x\sin(y)^2,$
$\dot y=-\frac{1}{x}-\frac{3}{2}\sin(2y),$
where the dot denotes the derivative with respect to $t$. For now I am only interested in the future asymptotics ($t\to\infty$) of solutions with initial data satisfying $x(0)>0$, and as a first simplifying case I could also assume $x(0)$ to be small.
Plotting the solutions numerically, it seems like $\lim_{t\to\infty}x(t)=0$. However the decay of $x$ on a timescale $T$, is superposed by an oscillation caused by the $\sin(y)^2$ and $\sin(2\theta)$ terms, on smaller and smaller timescales. The latter is, since $y$ decays very fast, and seems to go to $-\infty$ – I am not sure if in finite or infinite time.
The point is though, that these oscillations also decay in amplitude, and seem to go to zero asymptotically. So yes, the oscillations get faster and faster, but also their amplitude gets smaller and smaller.
So this is what numerics suggests. Now I am trying to get some analytical grip on that problem. In particular, I would like to show that the oscillations die off, and more importantly, that $x$ ultimately goes to zero.
An interesting experiment is also to naively set up a system for the time averaged solution, by replacing $\sin(y)^2\mapsto\frac{1}{2}$ and $\sin(2y)\mapsto0$. The resulting solution for this averaged $x(t)$ then indeed approximates the average of the original $x(t)$ quite well. But I am not sure how this helps me, if what I want is a rigorous proof of the asymptotics.