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For the real-line case, it is shown that both the Camassa--Holm \cite{Guo}and Novikov equations \cite{Li-arx} are ill-posed in $B_{\infty,1}^{1}$. Inthis paper, by presenting a new construction of initial data which leads to thenorm inflation phenomena, we prove that both the periodic Camassa--Holm andNovikov equations are also ill-posed in $B_{\infty,1}^{1}$.

Ill-posedness for the periodic Camassa--Holm type equations in the end-point critical Besov space $B^{1}_{\infty,1}$