| Contents |
| Consider the nonlinear differential system
\[
x_i'= \delta a_i(t) F_i(x_{i+1}), \qquad i=1, \ldots, n, \ t \in [a, \infty)
\]
where $\delta\in \{-1, 1\}$, $a_i: [a, \infty) \to [0, \infty)$ and $F_i: \mathbb R \to \mathbb R$, are continuous functions, with $uF_i(u)>0$ for every $u \neq 0$ and $i=1, \ldots, n$. Here $x_{n+1}=x_1$. The existence and the asymptotic expansion of strongly increasing and strongly decreasing solutions will be analyzed under the assumptions that the coefficients $a_i$ are regularly varying at infinity with index $\sigma_i \in \mathbb R$, and the nonlinear terms $F_i$ are regularly varying (at zero or at infinity) with index $\alpha_i>0$; a subhomogeneity condition is also assumed. In particular, if $F_i(u)=|u|^{\alpha_i} \mathrm{sgn }(u)$, we find conditions under which all extreme solutions are regularly varying. |
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