## recursion relation for system of ODE with a quadratic term

We have 4 smooth functions in t: G, H, g, h; t \geq 0.

These functions are smooth, bounded by 1. Initial conditions: G_0, H_0, g_0, h_0 are given (numbers)

It is also known that $H(t), h(t) \rightarrow 0$ as $t \rightarrow \infty$.

The ODE system I have is: $G' - H' = -G^2 \times h'$, $g' - h' = -g^2 \times H'$.

I'd like to be able to find the limits $G(t), g(t)$ as $t \rightarrow \infty$. Any ideas?

For example, if $G = g$ and $H = h$, then this reduces to $G' = (1-G^2)\times H'$, and this leads to the relation $-H(t) + arctanh(G(t)) = -H(0) + arctanh(G(0))$, which then allows me to get the limit $\lim_{t \rightarrow \infty}G(t)$.

(Motivations for this come from a certain physical system, but I don't think it's relevant here).

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