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The Two-Dimensional Attractor of a Differential Equation with State-Dependent Delay

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Abstract

The delay differential equation

$$\dot x(t) = - \mu x(t) + f(x(t - r)), r = r(x(t))$$

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 \(A.A\)\{0} coincides with the set of segments of bounded globally defined slowly oscillating solutions. If \(A\)≠{0}, then \(A\) is homeomorphic to the closed unit disk, and the unit circle corresponds to a periodic orbit.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720, Szeged, Hungary

    Tibor Krisztin

  2. Department of Mathematics, University of Pau, 64000, Pau, France

    Ovide Arino

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  1. Tibor Krisztin
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  2. Ovide Arino
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Krisztin, T., Arino, O. The Two-Dimensional Attractor of a Differential Equation with State-Dependent Delay. Journal of Dynamics and Differential Equations 13, 453–522 (2001). https://doi.org/10.1023/A:1016635223074

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