Global dynamics of delay differential equations

A. Aschwanden, A. Schulze-Halberg and D. Stoffer, Stable periodic solutions for delay equations, Disc. Cont. Dynam. Syst., 14 (2006), 721–736.

R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.

Y. Chen, T. Krisztin and J. Wu, Connecting orbits from synchronous periodic solutions to phase-locked periodic solutions in a delay differential system, J. Differential Equations, to appear.

Y. Chen and J. Wu, Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential Integral Equations, 14 (2001), 1181–1236.

O. Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.

B. Fiedler and J. Mallet-Paret, Connections between Morse sets for delay differential equations, J. Reine Angew. Math., 397 (1989), 23–41.

A. Halanay, Differential Equations, Stability, Oscillation, Time Lags, Academic Press, 1966.

J. K. Hale, Theory of Functional Differential Equations, Sringer-Verlag, 1977.

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993.

U. van der Heiden and H.-O. Walther, Existence of chaos in control systems with delayed feedback, J. Differential Equations, 47 (1983), 273–295.

V. Kolmanovski and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer, 1999.

N. Krasovskii, Stability of Motion, Stanford Univ. Press, 1963 (Russian edition 1959).

T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback, Electron. J. Qual. Theory Differ. Equ. (Szeged) (2000), 12 pp.

T. Krisztin, The unstable set of zero and the global attractor for delayed monotone positive feedback, Dynamical systems and differential equations (Kennesaw, GA, 2000). Discrete Contin. Dynam. Systems 2001, Added Volume, 229–240.

T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback, Fields Institute Communications, 29 (2001), 267–296.

T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1–57.

T. Krisztin and J. Wu, The global structure of an attracting set, in preparation.

T. Krisztin, H.-O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, Fields Institute Monograph Series vol. 11, AMS, Providence, 1999.

B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Amer. Math. Soc., 351 (1999), 901–945.

B. Lani-Wayda, Wandering solutions of equations with sine-like feedback, Memoirs of the Amer. Math. Soc., Vol. 151, No. 718, 2001.

B. Lani-Wayda and R. Srzednicki, The Lefschetz fixed point theorem and sybolic dynamics in delay equations, Ergodic Theory Dynam. Systems, 22 (2002), 1215–1232.

B. Lani-Wayda and H.-O. Walther, Chaotic motion generated by delayed negative feedback I. A transversality criterion, Differential and Integral Equations, 8 (1995), 1407–1452.

B. Lani-Wayda and H.-O. Walther, Chaotic motion generated by delayed negative feedback II. Construction of nonlinearities, Mathematische Nachrichten, 180 (1996), 141–211.

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287–289.

J. Mallet-Paret, Morse decompositions for differential delay equations, J. Differential Equations, 72 (1988), 270–315.

J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385–440.

J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441–489.

J. Mallet-Paret and H.-O. Walther, Rapid oscillations are rare in scalar systems governed by monotone negative feedback with a time delay, Preprint, Math. Inst., University of Giessen, 1994.

C. Mccord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, J. Amer. Math. Soc., 9 (1996), 1095–1133.

A. D. Myshkis, Linear Differential Equations with Retarded Argument, Izdat. Nauka, Moscow, 1971 (Russian edition 1949).

M. Polner, Morse decomposition for delay-differential equations with positive feedback, Nonlinear Anal., 48 (2002), 377–397.

G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655–2669.

D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, preprint.

H.-O. Walther, An invariant manifold of slowly oscillating solutions for % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb % qeguuDJXwAKbacfiGaf8hEaGNbaiaarmqr1ngBPrgitLxBI9gBaGGb % aiab+HcaOiab-rha0jab+LcaPiab+1da9iab+jHiTGGbciab9X7aTj % ab-Hha4jab+HcaOiab-rha0jab+LcaPiab+TcaRiab-zgaMjab+Hca % Oiab-Hha4jab+HcaOiab-rha0jab+jHiTiab+fdaXiab+LcaPiab+L % caPaaa!56D2! $$ \dot x(t) = - \mu x(t) + f(x(t - 1)) $$ (t) = −µx(t) + f(x(t − 1)), J. Reine Angew. Math., 414 (1991), 67–112.

H.-O. Walther, A differential delay equation with a planar attractor, Proc. of the Int. Conf. on Differential Equations, Université Cadi Ayyad, Marrakech, 1991.

H.-O. Walther, The 2-dimensional attractor of % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb % qeguuDJXwAKbacfiGaf8hEaGNbaiaarmqr1ngBPrgitLxBI9gBaGGb % aiab+HcaOiab-rha0jab+LcaPiab+1da9iab+jHiTGGbciab9X7aTj % ab-Hha4jab+HcaOiab-rha0jab+LcaPiab+TcaRiab-zgaMjab+Hca % Oiab-Hha4jab+HcaOiab-rha0jab+jHiTiab+fdaXiab+LcaPiab+L % caPaaa!56D2! $$ \dot x(t) = - \mu x(t) + f(x(t - 1)) $$ (t) = −µx(t) + f(x(t − 1)), Memoirs of the Amer. Math. Soc., Vol. 544, Amer. Math. Soc., Providence, RI, 1995.

H.-O. Walther, The singularities of an attractor of a delay differential equation, Funct. Differ. Equ., 5 (1998), 513–548.

H.-O. Walther and M. Yebdri, Smoothness of the attractor of almost all solutions of a delay differential equation, Dissertationes Math., 368 (1997).

E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 194 (1955), 66–87.

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, 1996.