The two-dimensional attractor of a differential equation with state-dependent delay

被引：31
作者：
Krisztin T. [1 ]
Arino O. [2 ]
机构：
[1] Bolyai Institute, University of Szeged, H-6720 Szeged
[2] Department of Mathematics, University of Pau
来源：
Journal of Dynamics and Differential Equations | 2001年 / 13卷 / 3期
关键词：
Asymptotic expansion; Discrete Lyapunov functional; Global attractor; Negative feedback; Poincaré-Bendixson-type theorem; Slowly oscillating solutions; State-dependent delay;
D O I：
10.1023/A:1016635223074
中图分类号：
学科分类号：
摘要：
The delay differential equation ẋ(t) = -μx(t) + f(x(t-r)), r=r(x(t)) with μ>0 and smooth real functions f, r satisfying f(0)=0, f′ < 0, and r(0) = 1 models a system governed by state-dependent delayed negative feedback and instantaneous damping. For a suitable R ≥ 1 the solutions generate a semiflow F on a compact subset LK of C([-R, 0], ℝ). F leaves invariant the subset S of φ ∈ LK with at most one sign change on all subintervals of [-R, 0] of length one. The induced semiflow on S has a global attractor script A sign.script A sign\{0} coincides with the set of segments of bounded globally defined slowly oscillating solutions. If script A sign ≠ {0}, then script A sign is homeomorphic to the closed unit disk, and the unit circle corresponds to a periodic orbit. © 2001 Plenum Publishing Corporation.
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页码：453 / 522
页数：69
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