Abstract
In this paper, a first-order equation with state-dependent delay and a nonlinear right-hand side is considered. The conditions of the existence and uniqueness of the solution of the initial value problem are supposed to be executed.The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. The local dynamics depends on real parameters, which are coefficients of the right-hand side decomposition in a Taylor series.The parameter, which is a coefficient at the linear part of this decomposition, has two critical values that determine the stability domain of the zero equilibrium. We introduce a small positive parameter and use the asymptotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov–Hopf bifurcation occurs near the second of these values (provided the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered to be an example. The bifurcation parameter of this equation has the only critical value. A simple sufficient condition of the occurrence of the supercritical Andronov–Hopf bifurcation in the considered equation near a critical value has been obtained as a result of applying the method of normal forms.
Highlights
В работе рассматривается уравнение первого порядка с запаздыванием, зависящим от искомой функции, с нелинейной правой частью
В качестве метода исследования динамики во всех случаях применяется метод нормальных форм
Где τ = εt, а ui(t, τ ) (i = 2, 3) периодические по t функции с периодом 2π/ω0
Summary
Для исследования динамики уравнения (1) построим линеаризованное на нулевом состоянии равновесия уравнение u + u = au(t − T0). Хорошо известно (см., например, [16]), что его характеристический квазиполином λ+1 = ae−λT0 имеет чисто мнимые корни λ = 0 при a = 1 и λ = ±iω0 при a = a0, где ω0 – наименьший положительный корень уравнения ω0 = −tg(ω0T0) и a0 = − 1 + ω02. Уравнение (4) экспоненциально устойчиво при a ∈ (a0, 1) и экспоненциально неустойчиво при a < a0 или a > 1. Нулевое решение уравнения (1) асимптотически устойчиво при a ∈ (a0, 1) и неустойчиво при a < a0 или a > 1. Значения a = a0 и a = 1 являются точками бифуркации нулевого состояния равновесия уравнения (1). Как происходит потеря устойчивости при переходе параметра a через эти значения
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