1 Correction to: Nonlinear Dyn https://doi.org/10.1007/s11071-022-07370-1

The article was published with errors in equations (35), (36) and (37). To derive the standard slow–fast normal form near the folded singularity Q, we use the transformation \(X=x-x_3\), \(Y=y-y_3\), \(\mu =b-b^*\) and the linear scaling \(X'=-\frac{1}{k\sqrt{a}}X\), \(Y'=-\frac{1}{ka}Y\), \(t'=\sqrt{a}t\). The equations (35a)–(35b) then should appear as

$$\begin{aligned} \frac{dX'}{dt'}= & {} -Y'h_1(X',Y',\mu ')+X'^2h_2(X',Y',\mu ')\nonumber \\&+ \epsilon h_3(X',Y',\mu '), \end{aligned}$$
(35a)
$$\begin{aligned} \frac{dY'}{dt'}= & {} \epsilon \left( X'h_4(X',Y',\mu ')-\mu ' h_5(X',Y',\mu ')\right. \nonumber \\&\left. + Y' h_6(X',Y',\mu ')\right) , \end{aligned}$$
(35b)

where \(h_1=1, h_2=1, h_3=0\), \(h_4=1+4a^{\frac{3}{2}}X'+\mathcal {O}(|X', Y', \mu '|^2)\), \(h_5=1+ \mathcal {O}(X', Y', \mu ')\), \(h_6=-\frac{1}{\sqrt{a}}+ \mathcal {O}(X', Y', \mu ')\), \(\mu '=\frac{\mu }{ka^{\frac{3}{2}}}\). Correspondingly, the corrected equations (36a)–(36f) will be as follows

$$\begin{aligned} a_1= & {} \frac{\partial h_3}{\partial X'}(0,0,0)=0, \end{aligned}$$
(36a)
$$\begin{aligned} a_2= & {} \frac{\partial h_1}{\partial X'}(0,0,0)=0, \end{aligned}$$
(36b)
$$\begin{aligned} a_3= & {} \frac{\partial h_2}{\partial X'}(0,0,0)=0, \end{aligned}$$
(36c)
$$\begin{aligned} a_4= & {} \frac{\partial h_4}{\partial X'}(0,0,0)=4a^{\frac{3}{2}}, \end{aligned}$$
(36d)
$$\begin{aligned} a_5= & {} h_6(0,0,0)=-\frac{1}{\sqrt{a}}, \end{aligned}$$
(36e)
$$\begin{aligned} A= & {} -a_2+3a_3-2a_4-2a_5=\frac{8b^*}{k\sqrt{a}}(1+2a)>0.\nonumber \\ \end{aligned}$$
(36f)

The singular Hopf bifurcation and maximal canard curves are then given by \(\mu =\mu _H\left( \sqrt{\epsilon }\right) =\frac{ka\epsilon }{2}+\mathcal {O}(\epsilon ^{3/2})\), \(\mu =\mu _c\left( \sqrt{\epsilon }\right) =\frac{ka}{4}\left( 1+4a^2\right) \epsilon + \mathcal {O}(\epsilon ^{3/2})\), and the equations (37a)–(37b) should be read as

$$\begin{aligned} b_H\left( \sqrt{\epsilon }\right)= & {} b^*+\frac{ka\epsilon }{2}+\mathcal {O}(\epsilon ^{3/2}), \end{aligned}$$
(37a)
$$\begin{aligned} b_c\left( \sqrt{\epsilon }\right)= & {} b^*+\frac{ka}{4}\left( 1+4a^2\right) \epsilon + \mathcal {O}(\epsilon ^{3/2}). \end{aligned}$$
(37b)