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

• Autorzy: Kennedy Benjamin B.

Identyfikatory

DOI

10.7494/OpMath.2023.43.4.507

• Języki publikacji: EN

Abstrakty

EN

We study the scalar difference equation [formula], 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 N 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.

Słowa kluczowe

EN

difference equation; negative feedback; circle map

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2023
• Tom: Vol. 43, no. 4
• Strony: 507--546
• Opis fizyczny: Bibliogr. 21 poz., rys., wykr.

Twórcy

autor

Kennedy Benjamin B.

• Department of Mathematics, Gettysburg College, 300 N. Washington St., Gettysburg, PA 17325, U.S.A.

Bibliografia

• [1] D. Burago, Y. Burago, S. Ivanov, A Course in Metric Geometry, Graduate Studies in Mathematics 33, American Mathematical Society, 2001.
• [2] K.L. Cooke, A.F. Ivanov, On the discretization of a delay differential equation, J. Differ. Equations Appl. 6 (2000), no. 1, 105–119.
• [3] O. Diekmann, S.A. Van Gils, S.M. Verduyn Lunel, H.-O. Walther, Delay Equations: Funtional-, Complex-, and Nonlinear Analysis, Applied Mathematical Sciences, vol. 110, Springer-Verlag, 1995.
• [4] X. Ding, W. Li, Stability and bifurcation of numerical discretization Nicholson blowflies equation with delay, Discrete Dyn. Nat. Soc. 2006, Art. ID 19413.
• [5] Á. Garab, C. Pötzsche, Morse decomposition for delay difference equations, J. Dynam. Differential Equations 31 (2019), no. 2, 903–932.
• [6] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993.
• [7] K. in ’t Hout, C. Lubich, Periodic orbits of delay differential equations under discretization, BIT 38 (1998), no. 1, 72–91.
• [8] A. Ivanov, On the comparative dynamics of a differential delay equation and its discretization, Proceedings of the 6th International Conference on Differential Equations and Dynamical Systems (2009), 169–173, Watam Press 2009.
• [9] J.L. Kaplan, J.A. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal. 6 (1975), 268–282.
• [10] A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995.
• [11] B.B. Kennedy, Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics, AMS/MAA Textbooks, vol. 70, MAA Press, Providence, RI, 2022.
• [12] T. Koto, Naimark-Sacker bifurcations in the Euler method for a delay differential equation, BIT 39 (1998), 110–115.
• [13] T. Koto, Periodic orbits in the Euler method for a class of delay differential equations, Comput. Math. Appl. 42 (2001), no. 12, 1597–1608.
• [14] Y.A. Kuznetsov, Elements of Applied Bifurcation Theory. Second Edition, Applied Mathematical Sciences, vol. 112, Springer-Verlag, New York, 1998.
• [15] Z. Li, Q. Zhao, D. Liang, Chaotic behavior in a class of delay difference equations, Adv. Difference Equ. 2013 (2013), Article no. 99.
• [16] J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Differential Equations 72 (1988), no. 2, 270–315.
• [17] J. Mallet-Paret, G.R. Sell, The Poincaré–Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations 125 (1996), 441–489.
• [18] J. Mallet-Paret, G.R. Sell, Differential systems with feedback: Time discretizations and Lyapunov functions, J. Dynam. Differential Equations 15 (2003), no. 2–3, 659–698.
• [19] R.D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. 101 (1974), 263–306.
• [20] H. Peters, Chaotic behavior of nonlinear differential-delay equations, Nonlinear Anal. 7 (1983), no. 12, 1315–1334.
• [21] H.-W. Siegberg, Chaotic behavior of a class of differential-delay equations, Ann. Mat. Pura Appl. 138 (1984), 15–33.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)