The paper describes asymptotic properties of a strongly nonlinear system $\dot{x}=f(t,x)$, $(t,x)\in ℝ\times {ℝ}^n$. The existence of an $\lfloor n/2\rfloor$ parametric family of solutions tending to zero is proved. Conditions posed on the system try to be independent of its linear approximation.

In the paper it is shown that each solution $u(r,\alpha )$ ot the initial value problem (2), (3) has a finite limit for $r\to \infty$, and an asymptotic formula for the nontrivial solution $u(r,\alpha )$ tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions $u(r,\overline{\alpha })$, $u(r,\widehat{\alpha })$.