Delay-differential equations with discrete feedback: Explicit formulae for infinitely many coexisting periodic solutions

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Volumn 87, Issue 7, 2007, Pages 471-479

  Bayer, Wolf ^a   An Der Heiden, Uwe ^b  

^a Hans und Hilde Coppi Gymnasium   (Germany)

^b UNIVERSITY OF WITTEN HERDECKE   (Germany)

Author keywords

Delay; Delay differential equations; Difference differential equations; Mackey Glass equation; Multiple oscillations; Nonlinear feedback

Indexed keywords

EID: 34547636326     PISSN: 00442267     EISSN: 15214001     Source Type: Journal    
DOI: 10.1002/zamm.200210329     Document Type: Article

References (16)

1. W. Bayer and U. an der Heiden, Oscillation types and bifurcations of a nonlinear second order differential-difference equation. J. Dyn. Differ. Equ. (USA) 10, 303-326 (1998).
2. L. M. Fridman, E. M. Fridman, and E. I. Shustin, Steady modes in an autonomous system with break and delay. Differ. Equ. 29, 1161-1166 (1993).
3. K.P. Hadeler and J. Tomiuk, Periodic solutions of difference-differential equations. Arch. Rat. Mech. Anal. 65, 87-95 (1977).
4. J. K. Hale and A. F. Ivanov, On higher order differential delay equations. Math. Anal. Appl. 173, 505-514 (1993).
5. U. an der Heiden and M.C. Mackey, The dynamics of production and destruction: Analytic insight into complex behaviour. J. Math. Biol. 16, 75-101 (1982).
6. U. an der Heiden and H.-O. Walther, Existence of Chaos in Control Systems with Delayed Feedback. J. Diff. Equ. 47, 273-295 (1983).
7. A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in: Dynamic Reported - Expositions in Dynamical Systems, Vol. 1, edited by C. K. R. T. Jones, U. Kirchgraber, and H. O. Walther (Springer Verlag, Berlin, Heidelberg, New York, 1992).
8. A. Longtin and J. G. Milton, Complex oscillations in the human pupil light reflex with 'mixed' and delayed feedback. Math. Biosci. 90, 183-199 (1988).
9. M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems. Science 197, 87-89 (1977).
10. M.C. Mackey and U. an der Heiden, Dynamical diseases and bifurcations: understanding functional disorder in physiological systems. Funkt. Biol. Med. 1, 156-164 (1982).
11. M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors. J. Econ. Theory 48, 497 (1990).
12. J. Mallet-Paret, and R. D. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation. Ann. Mat. Pura Appl. 145, 33-128 (1986).
13. J. Milton and A. Longtin, Clamping the pupil light reflex with external feedback: Evaluation of constriction and dilation from cycling measurements. Vision Res. 30, 515-525 (1990).
14. A. N. Sharkovsky, Yu.L. Maistrenko, and Yu. Romanenko, Difference Equations and their Applications (Kluwer Acad. Publ., Dordrecht, 1993) (1986 in Russian).
15. H.-O. Walther, An invariant manifold of slowly oscillating solutions for x′(t) = -μx(t) + f(x(t - 1)). J. Reine Angew. Math. 414, 67-112 (1991).
16. H.-O. Walther, The 2-Dimensional Attractor of x′(t) = -μ x(t) + f(x(t - 1)), Memoirs of the American Mathematical Society, Vol. 113, Nr. 544 (1995).