Geodesic
Dynamics of a system of two simplest oscillators with finite non-linear feedbacks
Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 6, pp. 841-849.

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In this paper, we consider a singularly perturbed system of two differential equations with delay which simulates two coupled oscillators with nonlinear feedback. Feedback function is assumed to be finite, piecewise continuous, and with a constant sign. In this paper, we prove the existence of relaxation periodic solutions and make conclusion about their stability. With the help of the special method of a large parameter we construct asymptotics of the solutions with the initial conditions of a certain class. On this asymptotics we build a special mapping, which in the main describes the dynamics of the original model. It is shown that the dynamics changes significantly with the decreasing of coupling coefficient: we have a stable homogeneous periodic solution if the coupling coefficient is of unity order, and with decreasing the coupling coefficient the dynamics become more complex, and it is described by a special mapping. It was shown that for small values of the coupling under certain values of the parameters several different stable relaxation periodic regimes coexist in the original problem.
Keywords: asymptotics, stability, large parameter, periodic solution.
Mots-clés : relaxation oscillation 
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A. A. Kashchenko. Dynamics of a system of two simplest oscillators with finite non-linear feedbacks. Modelirovanie i analiz informacionnyh sistem, Tome 23 (2016) no. 6, pp. 841-849. http://geodesic.mathdoc.fr/item/MAIS_2016_23_6_a12/

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