**Memoirs of the American Mathematical Society**

1995;
76 pp;
Softcover

MSC: Primary 34;
Secondary 58

Print ISBN: 978-0-8218-2602-7

Product Code: MEMO/113/544

List Price: $41.00

Individual Member Price: $24.60

**Electronic ISBN: 978-1-4704-0123-8
Product Code: MEMO/113/544.E**

List Price: $41.00

Individual Member Price: $24.60

# The $2$-Dimensional Attractor of $x’(t)=-𝜇 x(t) + f(x(t-1))$

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*Hans-Otto Walther*

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.

#### Table of Contents

# Table of Contents

## The $2$-Dimensional Attractor of $x'(t)=- x(t) + f(x(t-1))$

- Contents v6 free
- Chapter 1. Introduction 18 free
- Chapter 2. Notation, Preliminaries 512 free
- Chapter 3. Basic Properties of Solutions 1118
- Chapter 4. Attractors 1926
- Chapter 5. Phase Space Decomposition 2330
- Chapter 6. A-Priori Estimates, Phase Curves with Trivial α-Limit Set, and Invariant Manifolds 2734
- Chapter 7. Graph Representation 3340
- Chapter 8. Transversals 4148
- Chapter 9. Angles Along Projected Phase Curves 4754
- Chapter 10. The Poincaré-Bendixson Theorem 5764
- Chapter 11. Proof of Theorem 7.1(ii) 6774
- References 7582

#### Readership

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