1 Erratum to: Nonlinear Dyn 2016 83:2435–2452 DOI 10.1007/s11071-015-2492-4

The statement given in Lemma 4, part 2 of the original article, namely “For \(\beta > 0\),

$$\begin{aligned} E_{{p,\beta }}(t^{p} J_j)= & {} t^{1-\beta } B_j(t) + C_j (t), \text { hence }~\\&E_{{p,\beta }}(t^{p} A)_{n \times n} = t^{1-\beta } B(t) + C(t),\hbox {''} \end{aligned}$$

is incorrect. The corrected version of the above expression is presented below. Note that Lemma 4 part 1 remains as it is.

For the corrected expression, we introduce the following notations. Notation: Let \(I_{\gamma }\) denote \(n_j \times n_j\) matrix with entries \([I_{\gamma }]_{i,k} = \delta _{i+\gamma ,k},~(\gamma =0,\ldots ,n_j -1,~1 \le i,k\le n_j)\).

Lemma 4

(Part 2 (Corrected)) For \(0< p < 1\), \(j=1,2 \ldots , l\) and \(q \in \mathbb {N}\backslash \{1\},\)

$$\begin{aligned} E_{p,p}(t^{p} J_j) = t^{-p} \widetilde{B_j(t)} + \widetilde{C_j(t)}, \end{aligned}$$
(1)

where \(\widetilde{B_j(t)}\) and \(\widetilde{C_j(t)}\) are \(n_j \times n_j\) matrices defined as

$$\begin{aligned} \widetilde{B_j(t)}&:= 0 \quad j=1,2,\ldots ,s, \nonumber \\ \widetilde{B_j(t)}&:= \widetilde{\Psi }_0(t,\lambda _j) I_0 + \widetilde{\Psi }_1(t,\lambda _j) I_1 + \cdots \nonumber \\&\qquad + \widetilde{\Psi }_{n_j-1}(t,\lambda _j) I_{n_j -1} \quad \nonumber \\&\qquad j=s+1,s+2,\ldots ,l, \end{aligned}$$
(2)
$$\begin{aligned} \widetilde{C_j(t)}&:= \widetilde{\Delta }_0(t,\lambda _j) I_0 +\widetilde{\Delta }_1(t,\lambda _j) I_1 + \cdots + \widetilde{\Delta }_{n_j-1}\nonumber \\&\qquad (t,\lambda _j) I_{n_j-1} \quad j=1,2,\ldots ,l, \end{aligned}$$
(3)

and

$$\begin{aligned} \widetilde{\psi _m} (t, \lambda _j)= & {} \frac{1}{m!} \left[ \dfrac{\partial ^{m+1}}{\partial \lambda ^{m+1}} \exp \left( t \lambda _j^{\frac{1}{p}}\right) \right] ,~~\nonumber \\&m=0,1,2,\ldots ,n_j-1. \end{aligned}$$
(4)
$$\begin{aligned} \widetilde{\Delta _m}(t,\lambda _j):= & {} \dfrac{1}{m!}\nonumber \\&\bigg ( \sum _{k=2}^{q} \dfrac{(-1)^{m+2}(k+m)!}{(k-1)!} \dfrac{ \lambda _j ^{-k-m-1} t^{-pk}}{\Gamma (1-pk)} \nonumber \\&\quad +\, O( |\lambda _{j}|^{-q-m} t^{-p-pq}) \bigg ). \end{aligned}$$
(5)

Let \(\widetilde{B(t)}\) and \(\widetilde{C(t)}\) denote the block diagonal matrices consisting of \(\widetilde{B_j(t)}\) and \(\widetilde{C_j(t)}\) on the diagonal, respectively. Hence \(E_{p,p}(t^{p} A) = t^{-p} \widetilde{B(t)} + \widetilde{C(t)}\).

As a consequence of the above correction, appropriate changes should be made in expressions of Lemma 5, 6, 8 and the step II of the proof of the theorem. Thus in view of these corrections the proof of the local stable manifold theorem given by us in the original article continues to hold true. For details we refer the readers to our article [1].