Abstract
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.
Highlights
In this paper, we prove the existence of relaxation periodic solutions and make conclusion about their stability
Найдутся такие γ10 > 0 и λ0 > 0, что при γ1 ≥ γ10 и λ ≥ λ0 однородное периодическое решение системы (4) орбитально устойчиво
Summary
В связи с этим здесь предполагается, что функция F (u) является финитной, то есть для некоторого p > 0 имеет место равенство В настоящей работе исследуется поведение решений системы из двух связанных уравнений вида (1) при условиях (2) и (3): u1 + u1 = λF (u1(t − T )) + γ(u2 − u1), u2 + u2 = λF (u2(t − T )) + γ(u1 − u2). Что через некоторый отрезок времени каждое из рассматриваемых решений попадает в множество S(x), где для xполучено асимптотическое при λ → ∞ представление вида x = φ(x) + o(1).
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