  In this paper, we consider the Cauchy problem for a two-component Novikov
system on the line. By specially constructed initial data $(\rho_0, u_0)$ in
$B_{p, \infty}^{s-1}(\mathbb{R})\times B_{p, \infty}^s(\mathbb{R})$ with
$s>\max\{2+\frac{1}{p}, \frac{5}{2}\}$ and $1\leq p \leq \infty$, we show that
any energy bounded solution starting from $(\rho_0, u_0)$ does not converge
back to $(\rho_0, u_0)$ in the metric of $B_{p, \infty}^{s-1}(\mathbb{R})\times
B_{p, \infty}^s(\mathbb{R})$ as time goes to zero, thus results in
discontinuity of the data-to-solution map and ill-posedness.
