Two van der Pol oscillators coupled via a bath (2)

Consider the model described in Two van der Pol oscillators coupled via a bath (1).

The current model is using a slightly different notation:

$$\begin{align*} \dot{\xi}_{1}^{1} &= \dot{x}_1 \\ \dot{\xi}_{2}^{1} &= \dot{x}_2 \\ \dot{\xi}_{1}^{2} &= \dot{x}_3 \\ \dot{\xi}_{2}^{2} &= \dot{x}_4 \\ \dot{\eta}_{1} &= \dot{x}_5 \end{align*}$$

Note that this system is decouplable by static state feedback because the decoupling matrix of this system is

$D_{1}(\xi,\eta)=\left[\matrix{1 &0 \cr 0 &1}\right]$

The authors have proposed the following Yuz and Goodwin type approximate model which is more accurate than the Euler model.

$$\begin{align*} x_{1,k+1}&=x_{1,k}+T_{x_{2,k}}+\frac{T^2}{2}\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{2,k+1}&=x_{2,k}+T\{u_{1,k}-x_{1,k}+\epsilon\{1-x_{1,k}^{2}\}x_{2,k}+k(x_{5,k}-x_{1,k})\} \\ x_{3,k+1}&=x_{3,k}+T_{x_{4,k}}+\frac{T^2}{2}\{u_{2,k}-x_{1,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{1,k})\} \\ x_{4,k+1}&=x_{4,k}+T_{x_{4,k}}+T\{u_{2,k}-x_{2,k}+\epsilon\{1-x_{3,k}^{2}\}x_{4,k}+k(x_{5,k}-x_{3,k})\} \\ x_{5,k+1}&=x_{5,k}+T\{k(x_{1,k}-x_{5,k})+k(x_{3,k}-x_{5,k})\} \\ y_{1,k}&=x_{1,k} \\ y_{2,k}&=x_{3,k} \end{align*}$$