I am looking at a dynamical system of the following form (prime denotes derivation with respect to t):
$H' = -(1+2S^2+A(3\sin(\alpha)^2-1))H$
$S' = -\big(2(1-S^2)-A(3\sin(\alpha)^2-1)\big)S+1-S^2-A$
$A' = 2\big(2S^2-(1-A)(3\sin(\alpha)^2-1)\big)A$
$\alpha' = -\frac{1}{H}-\frac{3}{2}\sin(2\alpha)$
I am interested in solutions with an initial H greater 0. Further, physically relevant solutions are constrained by the equation
$0<A = 1-S^2-M$
where $M>0$ yet another quantity. Hence the maximum range of $S$ is $(-1,1)$ and that of $A$ is $(0,1)$. The above system is decoupled from the evolution equation for $M$, which is why I didn't list the latter here.
The goal is to determine the future asymptotics of the system.
From solving this system numerically, it looks like for large t one gets the following asymptotic values: $H\to0,S\to1/2,A\to0,\alpha\to\pm\infty$, where the sign of $\alpha$ depends on the initial condition. However, because of $\alpha$ blowing up, the numerical results become unstable after some time, and I cannot look at the solution at sufficiently large times to see if this is really the asymptotics or not.
The solution functions, in particular $A(t)$, typically look like some decaying function, which is superposed with an oscillation with ever increasing frequency, but which also has decaying envelope. So I would guess that the oscillation is not significant for determining the overall asymptotics of the system.
This may very well turn out to be wrong, but I never the less thought I would like to set up a system for which the solution would be the average functions, i.e. the functions where the small and fast oscillations are smoothed out, and what is left is only the decay, which happens on larger time scales.
I am however not sure how to construct such a system. Naively replacing $sin(\alpha)^2$ by $1/2$ seems to simple, but in a first experiment it looks like it would actually work!
Is this a legitimate way to do it? If not, how would I do it rigorously?
Cheers for ideas!