eBook ISBN:  9781470401238 
Product Code:  MEMO/113/544.E 
List Price:  $41.00 
MAA Member Price:  $36.90 
AMS Member Price:  $24.60 
eBook ISBN:  9781470401238 
Product Code:  MEMO/113/544.E 
List Price:  $41.00 
MAA Member Price:  $36.90 
AMS Member Price:  $24.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 113; 1995; 76 ppMSC: Primary 34; Secondary 58
The equation \(x'(t) =  \mu x(t) + f(x(t1))\), 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 PoincaréBendixson theorems for differentialdelay equations. The method used here is unique in its use of winding numbers and homotopies in nonconvex sets.
ReadershipResearchers and graduate students studying dynamical systems and differential delay equations.

Table of Contents

Chapters

1. Introduction

2. Notation, preliminaries

3. Basic properties of solutions

4. Attractors

5. Phase space decomposition

6. Apriori estimates, phase curves with trivial $\alpha $limit set, and invariant manifolds

7. Graph representation

8. Transversals

9. Angles along projected phase curves

10. The PoincaréBendixson Theorem

11. Proof of Theorem 7.1(ii)


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The equation \(x'(t) =  \mu x(t) + f(x(t1))\), 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 PoincaréBendixson theorems for differentialdelay 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.

Chapters

1. Introduction

2. Notation, preliminaries

3. Basic properties of solutions

4. Attractors

5. Phase space decomposition

6. Apriori estimates, phase curves with trivial $\alpha $limit set, and invariant manifolds

7. Graph representation

8. Transversals

9. Angles along projected phase curves

10. The PoincaréBendixson Theorem

11. Proof of Theorem 7.1(ii)