Two van der Pol oscillators coupled via a bath (1)

Consider nonlinear van der Pol oscillators coupled via a bath. The normal form of the system is expressed by

$$\begin{align*} \dot{\xi}_{1}^{1}&=\xi_{2}^{1} \\ \dot{\xi}_{2}^{1}&=-\xi_{1}^{1}+\epsilon\{1-(\xi_{1}^{1})^{2}\}\xi_{2}^{1}+k(\eta_{1}-\xi_{1}^{1})+u_{1} \\ \dot{\xi}_{1}^{2}&=\xi_{2}^{2} \\ \dot{\xi}_{2}^{2}&=-\xi_{1}^{2}+\epsilon\{1-(\xi_{1}^{2})^{2}\}\xi_{2}^{2}+k(\eta_{1}-\xi_{1}^{2})+u_{2} \\ \dot{\eta}_{1}&=k(\xi_{1}^{1}-\eta_{1})+k(\xi_{1}^{2}-\eta_{1}) \\ y_{1}&=\xi_{1}^{1} \\ y_{2}&=\xi_{1}^{2}, \end{align*}$$

where parameters $\epsilon$ and $k$ are positive constants. The system has the relative degrees $r_1 = 2, r_2 = 2$ and the following zero dynamics:

$ \dot{\eta}_{1}=-2k\eta_{1}.$