1 Correction to: Journal of Dynamics and Differential Equations https://doi.org/10.1007/s10884-021-10000-w

The original version of this article unfortunately contained few typos in equations presented under proof of Proposition 3.5 section.

Initially, it was published incorrectly as

It follows from (3.11) that period L and parameter \({\mathcal {I}}\) are expressed uniquely by

$$\begin{aligned} L = 4 \alpha ^{-1} K(k), \quad {\mathcal {I}} = \alpha ^4 k^2 (1-k^2), \end{aligned}$$
(3.12)

where \(\alpha \) depends on k if \(c \ne 0\). Computing derivatives in k yields

$$\begin{aligned} \frac{d {\mathcal {I}}}{d k} = -\frac{2kc}{(1-2k^2)^3}, \quad \frac{d L}{dk} = \frac{4}{\sqrt{c(2k^2-1)}} \left[ (1-2k^2) \frac{d}{dk} K(k) - 2k K(k) \right] . \end{aligned}$$

If \(c > 0\) and \(k \in (\frac{1}{\sqrt{2}},1)\), then \(\frac{d {\mathcal {I}}}{dk} > 0\) and \(\frac{d L}{dk} < 0\) so that the mapping \({\mathcal {I}} \mapsto L\) is monotonically decreasing. As \(k \rightarrow \frac{1}{\sqrt{2}}\), \(\alpha \rightarrow \infty \) and \(L \rightarrow 0\). As \(k \rightarrow 1\), \(K(k) \rightarrow \infty \) and \(L \rightarrow \infty \).

If \(c < 0\) and \(k \in (0,\frac{1}{\sqrt{2}})\), then \(\frac{d \mathcal {I}}{dk} > 0\) and \(\frac{d L}{dk} < 0\) due to (3.7) and

$$\begin{aligned} \frac{d L}{dk} = \frac{4}{\sqrt{c(2k^2-1)} k (1-k^2)} \left[ (1-k^2) (E(k) - K(k)) - k^2 K(k) \right] < 0, \end{aligned}$$

Now, the corrected equations are presented here:

It follows from (3.11) that period L and parameter \({\mathcal {I}}\) are expressed uniquely by

$$\begin{aligned} L = 4 \alpha ^{-1} K(k), \quad {\mathcal {I}} = \alpha ^4 k^2 (1-k^2), \end{aligned}$$
(3.12)

where \(\alpha \) depends on k if \(c \ne 0\). Computing derivatives in k yields

$$\begin{aligned} \frac{d {\mathcal {I}}}{d k} = \frac{kc^{2}}{(1-2k^2)^3}, \quad \frac{d L}{dk} =- \frac{4}{\sqrt{c(2k^2-1)}} \left[ (1-2k^2) \frac{d}{dk} K(k) - 2k K(k) \right] . \end{aligned}$$

If \(c > 0\) and \(k \in (\frac{1}{\sqrt{2}},1)\), then \(\frac{d {\mathcal {I}}}{dk} < 0\) and \(\frac{d L}{dk} > 0\) so that the mapping \({\mathcal {I}} \mapsto L\) is monotonically decreasing. As \(k \rightarrow \frac{1}{\sqrt{2}}\), \(\alpha \rightarrow \infty \) and \(L \rightarrow 0\). As \(k \rightarrow 1\), \(K(k) \rightarrow \infty \) and \(L \rightarrow \infty \).

If \(c < 0\) and \(k \in (0,\frac{1}{\sqrt{2}})\), then \(\frac{d \mathcal {I}}{dk} > 0\) and \(\frac{d L}{dk} < 0\) due to (3.7) and

$$\begin{aligned} \frac{d L}{dk} = \frac{4}{\sqrt{c(2k^2-1)} k (1-k^2)} \left[ (1-k^2) (E(k) - K(k)) - k^2 E(k) \right] < 0, \end{aligned}$$

The original article has been corrected.