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The $$2$$-Dimensional Attractor of $$x'(t)=-{\mu } x(t) + f(x(t-1))$$
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Memoirs of the American Mathematical Society
1995; 76 pp; softcover
Volume: 113
ISBN-10: 0-8218-2602-6
ISBN-13: 978-0-8218-2602-7
List Price: US$39 Individual Members: US$23.40
Institutional Members: US\$31.20
Order Code: MEMO/113/544

The equation $$x'(t) = - \mu x(t) + f(x(t-1))$$, with $$\mu \geq 0$$ and $$xf(x) \le 0$$ for $$0\neq x\in {\mathbb R}$$, is a prototype for delayed negative feedback combined with friction. Its semiflow on $$C=C([-1,0],{\mathbb R})$$ leaves a set $$S$$ invariant, which also plays a major role for the dynamics on the full space $$C$$. The main result determines the attractor of the semiflow restricted to the closure of $$S$$ for monotone, bounded, smooth $$f$$. In the course of the proof, Walther derives Poincaré-Bendixson theorems for differential-delay equations. The method used here is unique in its use of winding numbers and homotopies in nonconvex sets.

Researchers and graduate students studying dynamical systems and differential delay equations.

• Introduction
• Notation, preliminaries
• Basic properties of solutions
• Attractors
• Phase space decomposition
• A-priori estimates, phase curves with trivial $$\alpha$$-limit set, and invariant manifolds
• Graph representation
• Transversals
• Angles along projected phase curves
• The Poincaré-Bendixson theorem
• Proof of Theorem 7.1(ii)
• References