Universal perturbations of linear differential equations

Let $X:[0,\infty)\to L(\mathbb{R} ^n)$ be a fundamental solution of $x'=A( t)x$with $X$ and $X^{-1}$ bounded on $[0,\infty)$. We prove that there exist arbitrary small matrix functions $B:[0,\infty)\to L(\mathbb{R} ^n)$ with limit $0$ as $t\to \infty$ such that $y'=(A(t)+B(t))y$ has solutions with $y([0,\infty))$ dense in $\mathbb{R} ^n$.