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Dynamics of coupled Van der Pol oscillators. (English. Russian original) Zbl 1504.34090

J. Math. Sci., New York 262, No. 6, 817-824 (2022); translation from Itogi Nauki Tekh., Ser. Sovrem. Mat. Prilozh., Temat. Obz. 168, 53-60 (2019).
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.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34D06 Synchronization of solutions to ordinary differential equations
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