1 Erratum to: Nonlinear Dyn DOI 10.1007/s11071-016-3096-3

The authors regret there is a negligence in Sects. 2.1 and 3 in the original publication.

The solution of Eq. (2) in Sect. 2.1 can be expressed in the integral form when \(\mathbf{A}(t)\) is a constant matrix

$$\begin{aligned} \mathbf{x}(t)= & {} \hbox {e}^{\mathbf{A}(t)\cdot (t-t_1 )}{} \mathbf{x}(t_1 )\nonumber \\&+\sum _{j=1}^s {\int _{t_1 }^t {\hbox {e}^{\mathbf{A}(t)\cdot (t-\xi )}{} \mathbf{B}_j (\xi )\cdot \mathbf{x}(\xi -\tau _j )} \hbox {d}\xi } , \end{aligned}$$
(3)

where \(\mathbf{x}(t_1 )\) denotes the state value at \(t=t_{1}\).

At each small subinterval \([t_{i-1} ,t_i ]\), Eq. (3) is represented as

$$\begin{aligned} \mathbf{x}(t)= & {} \hbox {e}^{\mathbf{A}(t)\cdot (t-t_{i-1} )}\mathbf{x}(t_{i-1})\nonumber \\&+\sum _{j=1}^s \int _{t_{i-1} }^t \hbox {e}^\mathbf{A}(t)\cdot (t-\xi )\mathbf{B}_j (\xi )\nonumber \\&\cdot \mathbf{x}(\xi -\tau _j ) \hbox {d}\xi , \quad t\in [t_{i-1} ,t_i ]. \end{aligned}$$
(4)

It is assumed that \(\mathbf{A}(t)\) is a constant matrix on the small interval \(t\in [t_{i-1} ,t_i ]\). For convenience, \(\mathbf{A}(t)\) is replaced with \(\mathbf{A}(t_{i})\) as an approximation. For the corresponding discretized time points \(t_i =t_1 +( {i-1} )h\) (\(i=1,\ldots ,n+1)\), \(\mathbf{x}(t_i )\) is approximated by Newton–Cotes formulas according to Ding et al. [2].

(5)

Since the Newton–Cotes formula has the local truncation error \(\mathcal{O}(h^{3})\), the second term of right-hand side of Eq. (5) has the local truncation error \(\mathcal{O}(h^{3})\). Therefore, the discretization error of the proposed method is \(\mathcal{O}(h^{3})\), which can also be verified via the convergence of critical eigenvalues.

Equation (39) in Sect. 3 can be expressed in the integral form when \(\mathbf{A}(t)\) is a constant matrix

$$\begin{aligned} \mathbf{x}(t)= & {} \hbox {e}^{\mathbf{A}(t)\cdot (t-t_1 )}{} \mathbf{x}(t_1 )\nonumber \\&+\int _{t_1 }^t {\hbox {e}^{\mathbf{A}(t)\cdot (t-\xi )}\left[ {\int _{-\sigma }^0 {\mathbf{B}(\theta )\mathbf{x}(\xi +\theta )\hbox {d}\theta } } \right] } \hbox {d}\xi .\nonumber \\ \end{aligned}$$
(40)

The period interval T is also equally discretized with a time step h. At each small subinterval \([t_{i-1} ,t_i ]\), Eq. (40) is represented as

$$\begin{aligned} \mathbf{x}(t)= & {} \hbox {e}^{\mathbf{A}(t)\cdot (t-t_{i-1} )}{} \mathbf{x}(t_{i-1} )\nonumber \\&+\int _{t_{i-1} }^t {\hbox {e}^{\mathbf{A}(t)\cdot (t-\xi )}\left[ {\int _{-\sigma }^0 {\mathbf{B}(\theta )\mathbf{x}(\xi +\theta )\hbox {d}\theta } } \right] }\hbox {d}\xi , \nonumber \\&\quad t\in [t_{i-1} ,t_i ]. \end{aligned}$$
(41)

It is assumed that \(\mathbf{A}(t)\) is a constant matrix on the small interval \(t\in [t_{i-1} ,t_i ]\). For simplicity, \(\mathbf{A}(t)\) can be replaced with \(\mathbf{A}(t_{i})\) as an approximation. At the corresponding discretized time points \(t_i =t_1 +( {i-1} )h\) (\(i=1,\ldots ,n+1)\), we approximate \(\mathbf{x}(t_i )\) by using Newton–Cotes formulas, leading to

$$\begin{aligned} \mathbf{x}(t_i )= & {} \hbox {e}^{\mathbf{A}(t_i )\cdot (t_i -t_{i-1} )}{} \mathbf{x}(t_{i-1} )+\int _{t_{i-1} }^{t_i } \hbox {e}^{\mathbf{A}(t_i )\cdot (t_i -\xi )}\nonumber \\&\left[ {\int _{-\sigma }^0 {\mathbf{B}(\theta )\mathbf{x}(\xi +\theta )\hbox {d}\theta } } \right] \hbox {d}\xi , \quad t\in [t_{i-1} ,t_i ] \nonumber \\\approx & {} \hbox {e}^{\mathbf{A}(t_i )\cdot h}{} \mathbf{x}(t_{i-1} )\nonumber \\&+\,\frac{h}{2}\left\{ \left. {\left[ {\int _{-\sigma }^0 {\mathbf{B}(\theta )\mathbf{x}(t+\theta )\hbox {d}\theta } } \right] } \right| _{t=t_i }\nonumber \right. \\&\left. +\,\hbox {e}^{\mathbf{A}(t_i )\cdot h}\cdot \left. {\left[ {\int _{-\sigma }^0 {\mathbf{B}(\theta )\mathbf{x}(t+\theta )\hbox {d}\theta } } \right] } \right| _{t=t_{i-1} } \right\} .\nonumber \\ \end{aligned}$$
(42)

However, it does not affect the results and conclusions of the paper. The authors would like to apologize for any inconvenience caused to the readers.