Solutions with dense short segments from regular delays

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Walther, Hans-Otto¹ 

[1] Univ Giessen, Math Inst, Arndtstr 2, D-35392 Giessen, Germany

关键词: Delay differential equation;    State-dependent delay;    Complicated motion;   

DOI  :  10.1016/j.jde.2019.11.079

来源: Elsevier

【 摘 要 】

We construct a delay functional d(U) with values in (0, r) and find a positive number h < r such that the negative feedback equation x'(t) = -x(t-d(U)(x(t,r))), with the segment x(t,r) : [-r, 0] -> R given by x(t,r)(s) = x(t+s), has a solution whose short segments x(t,h) = x(t,r)vertical bar[-h,0] are dense in an open subset of the space C-1([-h, 0], R). The domain U of d(U) is open in C-1([-r, 0], R), and the delay differential equation defines a continuous semiflow of continuously differentiable solution operators on the solution manifold {phi is an element of U :phi'(0) = -phi(-d(U)(phi))}. The result implies a kind of chaotic solution behaviour which is not confined to a thin Cantor dust. (C) 2019 Elsevier Inc. All rights reserved.

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