Opuscula Math. 43, no. 4 (2023), 507-546

Opuscula Mathematica

Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback

Benjamin B. Kennedy

Abstract. We study the scalar difference equation \[x(k+1) = x(k) + \frac{f(x(k-N))}{N},\] where \(f\) is nonincreasing with negative feedback. This equation is a discretization of the well-studied differential delay equation \[x'(t) = f(x(t-1)).\] We examine explicit families of such equations for which we can find, for infinitely many values of $ and appropriate parameter values, various dynamical behaviors including periodic solutions with large numbers of sign changes per minimal period, solutions that do not converge to periodic solutions, and chaos. We contrast these behaviors with the dynamics of the limiting differential equation. Our primary tool is the analysis of return maps for the difference equations that are conjugate to continuous self-maps of the circle.

Keywords: difference equation, negative feedback, circle map.

Mathematics Subject Classification: 39A12, 39A23, 39A33.

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  • Benjamin B. Kennedy
  • Department of Mathematics, Gettysburg College, 300 N. Washington St., Gettysburg, PA 17325, U.S.A.
  • Communicated by Petr Stehlík.
  • Received: 2023-01-13.
  • Revised: 2023-04-05.
  • Accepted: 2023-04-14.
  • Published online: 2023-06-13.
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Cite this article as:
Benjamin B. Kennedy, Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback, Opuscula Math. 43, no. 4 (2023), 507-546, https://doi.org/10.7494/OpMath.2023.43.4.507

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