Dynamics of coupled Van der Pol oscillators (Q2673121)

The problem on the synchronization of Van der Pol oscillators is considered, where the oscillators are identical and constraints between them are weak, for systems \[ \begin{aligned} \ddot{x}_1 -2\varepsilon\dot{x}_1 + x_1 + ax^2_1\dot{x}_1 &= \varepsilon\gamma(\dot{x}_2-\dot{x}_1),\\ \ddot{x}_2 -2\varepsilon\dot{x}_2 + x_2 + ax^2_2\dot{x}_2 &= \varepsilon\gamma(\dot{x}_1-\dot{x}_2), \end{aligned} \] and \[ \begin{aligned} \ddot{x}_1 -2\varepsilon\dot{x}_1 + x_1 + ax^2_1\dot{x}_1 &= \varepsilon\gamma(\dot{x}_3-\dot{x}_1),\\ \ddot{x}_2 -2\varepsilon\dot{x}_2 + x_2 + ax^2_2\dot{x}_2 &= \varepsilon\gamma(\dot{x}_1-\dot{x}_2),\\ \ddot{x}_3 -2\varepsilon\dot{x}_3 + x_3 + ax^2_3\dot{x}_3 &= \varepsilon\gamma(\dot{x}_2-\dot{x}_3), \end{aligned} \] where \(\varepsilon>0\), \(a>0\) and \(\gamma\not= 0\). It is proved that for some \(\varepsilon_0>0\) and all \(\varepsilon \in (0, \varepsilon_0)\), the system of two equations has a synchronous periodic solution \((z_1=z_2)\) and this cycle is orbitally asymptotically stable if \(\gamma>0\); an antiphase cycle (\(z_2=-z_1)\) exists if \(\gamma<1\) and it is stable for \(\gamma<0\) and unstable for \(\gamma>0\). Asymptotic formulas are obtained for these periodic solutions. Similar results are obtained for the system of three coupled oscillators.