Controlled Van der Pol system (2)

Using

$z_{i+1,k+1}=y_{k+1}^{(i)}\approx y_{k}^{(i)}+Ty_{k}^{(i+1)}+\frac{T^2}{2}y_{k}^{(i+2)}+\cdots+\frac{T^{r-i}}{(r-i)!}y_{k}^{(r)}+\frac{T^{r-i+1}}{(r-i+1)!}y_{k}^{(r +1)}$

for $i=0, \cdots,r-1$ the controlled Van der Pol system from Controlled Van der Pol system (1) can be rewritten as:

$$\begin{align*} x_{1,k+1} &=x_{1,k}+Tx_{2,k}+\frac{T^2}{2}[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}] \\ &+\frac{T^{2}}{3!}[-c(-cx_{2,k}-d\sin x_{1,k}+u_{1,k})-dx_{2,k}\cos x_{1,k} \\ &\times\{-x_{1,k}+\epsilon(1-x_{1,k}^2)x_{2,k}+u_k] \\ x_{2,k+1} &=x_{2,k}+T[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}] \\ &+\frac{T^2}{2}[-cx_{2,k}-d\sin x_{1,k}+u_{1,k}-dx_{2,k}\cos x_{1,k} \\ &\times\{-x_{1,k}+\epsilon(1-x_{1,k}^2)x_{2,k}+u_k] \\ y_{k} &=x_{1,k} \end{align*}$$