Instability, asymptotic trajectories and dimension of the phase space

Suppose the origin $x=0$ is a Lyapunov unstable equilibrium position for a flow in $\mathbb{R}^n$. Is it true that there always exists a solution $t\mapsto x(t)$, $x(t)\ne 0$ asymptotic to the equilibrium: $x(t)\to 0$ as $t\to -\infty$? The answer to this and similar questions depends on some details including the parity of $n$ and the class of smoothness of the system. We give partial answers to such questions and present some conjectures.