Superstability and rigorous asymptotics in singularly perturbed state-dependent delay-differential equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:250

Article

Mallet-Paret, John¹  Nussbaum, Roger D.² 

[1] Brown Univ, Div Appl Math, Providence, RI 02912 USA

[2] Rutgers State Univ, Dept Math, Piscataway, NJ 08854 USA

关键词: Delay-differential equations;    State-dependent delay;    Singular perturbation;    Periodic solution;    Stability;   

DOI  :  10.1016/j.jde.2010.10.024

来源: Elsevier

【 摘 要 】

We study the singularly perturbed state-dependent delay-differential equation epsilon(x) over dot(t) = -x(t) - kx(t - r), r = r(x(t)) = 1 + x(t), (*) which is a special case of the equation epsilon(x) over dot(t) = g(x(t), x(t - r)), r = r(x(t)). One knows that for every sufficiently small epsilon > 0, Eq. (*) possesses at least one so-called slowly oscillating periodic solution, and moreover, the graph of every such solution approaches a specific sawtooth-like shape as epsilon -> 0. In this paper we obtain higher-order asymptotics of the sawtooth, including the location of the minimum and maximum of the solution with the form of the solution near these turning points, and as well an asymptotic formula for the period. Using these and other asymptotic formulas, we further show that the solution enjoys the property of superstability, namely, the nontrivial characteristic multipliers are of size O(epsilon) for small epsilon. This stability property implies that this solution is unique among all slowly oscillating periodic solutions, again for small epsilon. (C) 2010 Published by Elsevier Inc.

【 授权许可】

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