TY  - JOUR
ID  - Kennedy2023
LB  - Kennedy2023
AU  - Kennedy, Benjamin B.
TI  - Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback
ST  - Periodic, nonperiodic, and chaotic solutions for a class of difference equations with negative feedback
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2023
VL  - 43
IS  - 4
SP  - 507
EP  - 546
DO  - https://doi.org/10.7494/OpMath.2023.43.4.507
UR  - https://doi.org/10.7494/OpMath.2023.43.4.507
AB  - 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.
KW  - difference equation
KW  - negative feedback
KW  - circle map
SN  - 1232-9274
ER  -