Self-excited nonlinear oscillator

Self-excited nonlinear oscillator:

$$\begin{align*} \dot{x}_1 &=x_2,\\ \dot{x}_2 &=-\omega_1^2\sin{y_1}-\varrho x_2 + k_p \arctan(k_c y_2),\\ \dot{x}_3 &=\omega_2 \cdot (x_1 - x_3), \\ y_1 &=x_1, \\ y_2 &=x_1-x_3, \end{align*}$$

where $y(t)\in \mathbb{R}^2$ is the sensor output vector (to be transmitted over the communication channel), $\omega_1, \omega_2, \varrho, k_p, k_c$ are system parameters, $x=[x_1,x_2,x_3]^{\mathrm T}\in\mathbb{R}^3$ of the system state vector $x(t)$ based on the signals $y_1(t),y_2(t)$, transmitted over the communication channel.

The system has the form

$\dot{x}(t)=Ax(t)+\varphi(y(t)),$ $y(t)=Cx(t),$

where

$A=\begin{bmatrix}0 & 1 & 0 \\ 0 & -\varrho & 0 \\ \omega_2 & 0 & -\omega_2\end{bmatrix},$

$C = \begin{bmatrix}1, & 0, & 0\\ 1, & 0, & -1\end{bmatrix},$

$ \varphi(y) = \begin{bmatrix} 0\\ \omega_1^2 \sin{y_1} + k_p \arctan{(k_c y_2)} \\ 0 \end{bmatrix}.$