Let
\(B_\ell (n)\) denote the number of
\(\ell \)-regular bipartitions of
n. In this paper, we prove several infinite families of congruences satisfied by
\(B_\ell (n)\) for
\(\ell \in {\{5,7,13\}}\). For example, we show that for all
\(\alpha >0\) and
\(n\ge 0\),
$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$
and
$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$