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The \(2\)-Dimensional Attractor of \(x'(t)=-{\mu } x(t) + f(x(t-1))\)
Hans-Otto Walther
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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
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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.

Readership

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

Table of Contents

  • 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
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