Chaotic motion generated by delayed negative feedback part I: A transversality criterion

Bernhard Lani-Wayda, Hans Otto Walther, Roger Nussbaum

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20 Scopus citations

Abstract

We study the equation x’(t) = g(x(t-1)) (g) for smooth functions g: ℝ → R satisfying ξg(ξ) < 0 for ξ ≠ 0, and the equation x’(t) = b(t)x(t-1) (b) with a periodic coefficient b: ℝ → (-∞, 0). Equation (b) generalizes variational equations along periodic solutions y of equation (g) in case g’(ξ) < 0 for all ξ ∈ y(ℝ). We investigate the largest Floquet multipliers of equation (b) and derive a characterization of vectors transversal to stable manifolds of Poincaré maps associated with slowly oscillating periodic solutions of equation (g). The criterion is used in Part II of the paper in order to find g and y so that a Poincaré map has a transversal homoclinic trajectory, and a hyperbolic set on which the dynamics are chaotic.

Original languageEnglish (US)
Pages (from-to)1407-1452
Number of pages46
JournalDifferential and Integral Equations
Volume8
Issue number6
StatePublished - Jul 1995

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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