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Periodic solutions for differential equations with state-dependent delay and positive feedback. (English) Zbl 1028.34062
The author explores the differential equation with state-dependent delay \[ \dot{x}(t)=\mu x(t)+f(x(t-r)),\quad r=r(x(t)), \] where \(\mu>0\), \(f\) and \(r\) are smooth real functions with \(r(0)=1\) and \(f'(0)>0\). It is proved that the delay differential equation admits a nontrivial periodic orbit and a homoclinic orbit connecting \(0\) to the periodic orbit. The paper is well written and involves some interesting techniques.

MSC:
34K13 Periodic solutions to functional-differential equations
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[1] Alt, W., Some periodicity criteria for functional differential equations, Manuscripta math., 23, 295-318, (1978) · Zbl 0367.34049
[2] Arino, O.; Hadeler, K.P.; Hbid, M.L., Existence of periodic solutions for delay differential equations with state dependent delay, J. differential equations, 144, 263-301, (1998) · Zbl 0913.34057
[3] Bartha, M., Convergence of solutions for an equation with state-dependent delay, J. math. anal. appl., 254, 410-432, (2001) · Zbl 0980.34074
[4] J. Bélair, Population models with state-dependent delays, Lecture Notes in Pure and Applied Mathematics, Vol. 131, Dekker, New York, 1991, pp. 165-176. · Zbl 0749.92014
[5] Bélair, J.; Mackey, M.C., Consumer memory and price fluctuations on commodity marketsan integrodifferential model, J. dynamics differential equations, 1, 299-325, (1989)
[6] Brokate, M.; Colonius, F., Linearizing equations with state-dependent delays, Appl. math. optim., 21, 45-52, (1990) · Zbl 0694.34050
[7] M. Büger, M.R.W. Martin, The escaping disaster: a problem related to state dependent delay, preprint.
[8] Cooke, K.L.; Huang, W., On the problem of linearization for state-dependent delay differential equations, Proc. amer. math. soc., 124, 1417-1426, (1996) · Zbl 0844.34075
[9] Diekmann, O.; van Gils, S.A.; Verduyn Lunel, S.M.; Walther, H.-O., Delay equations, functional-, complex-, and nonlinear analysis, (1995), Springer New York · Zbl 0826.34002
[10] Driver, R.D., Existence theory for a delay-differential system, Contrib. differential equations, 1, 317-336, (1963)
[11] Driver, R.D., A two-body problem of classical electrodynamicsthe one-dimensional case, Ann. phys., 21, 122-142, (1963) · Zbl 0108.40705
[12] R.D. Driver, A functional differential system of neutral type arising in a two-body problem of classical electrodynamics, International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, Academic Press, New York, 1963, pp. 474-484. · Zbl 0134.22601
[13] R.D. Driver, The “backwards” problem for a delay-differential system arising in classical electrodynamics, Proceedings of the Fifth International Conference on Nonlinear Oscillations, Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev, 1970, pp. 137-143. · Zbl 0261.34043
[14] Driver, R.D.; Norris, M.J., Note on uniqueness for a one-dimensional two-body problem of classical electrodynamics, Ann. phys., 42, 347-351, (1967)
[15] F. Hartung, J. Turi, Stability in a class of functional-differential equations with state-dependent delays, in: Qualitative Problems for Differential Equations and Control Theory, World Scientific Publishing, River Edge, NJ, 1995, pp. 15-31. · Zbl 0840.34083
[16] T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback, in: Fields Institute Communications, Vol. 29, Amer. Math. Soc., Providence, RI, 2001, pp. 267-296. · Zbl 0988.34058
[17] T. Krisztin, The unstable set of zero and the global attractor for delayed monotone positive feedback, (Kennesaw, 2000) 229-240, Proc. 3rd Int. Conf. Dynamical Systems and Differential Equations, J. Du and S. Hu, (Eds.) An added volume to Discrete and Continuous Dynamical Systems, 2001. · Zbl 1301.34091
[18] T. Krisztin, A local unstable manifold for differential equations with state-dependent delay, preprint 2002. · Zbl 1048.34123
[19] Krisztin, T.; Arino, O., The 2-dimensional attractor of a differential equation with state-dependent delay, J. dynamics differential equations, 13, 453-522, (2001) · Zbl 1016.34075
[20] Krisztin, T.; Walther, H.-O., Unique periodic orbits for delayed positive feedback and the global attractor, J. dynamics differential equations, 13, 1-57, (2001) · Zbl 1008.34061
[21] T. Krisztin, H.-O. Walther, J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, Fields Institute Monographs, Vol. 11, Amer. Math. Soc., Providence, RI, 1999. · Zbl 1004.34002
[22] Krisztin, T.; Walther, H.-O.; Wu, J., The structure of an attracting set defined by delayed and monotone positive feedback, Cwi q., 12, 3&4, 315-327, (1999) · Zbl 1071.34517
[23] T. Krisztin, J. Wu, Smooth manifolds of connecting orbits for delayed monotone feedback, in preparation.
[24] Kuang, Y.; Smith, H.L., Periodic solutions of differential delay equations with threshold-type delays, (), 153-176 · Zbl 0762.34044
[25] Kuang, Y.; Smith, H.L., Slowly oscillating periodic solutions of autonomous state-dependent delay differential equations, Nonlinear anal., 19, 855-872, (1992) · Zbl 0774.34054
[26] Mackey, M.C.; Milton, J., Feedback delays and the origin of blood cell dynamics, Comm. theoret. biol., 1, 299-327, (1990)
[27] Mallet-Paret, J., Morse decompositions for delay-differential equations, J. differential equations, 72, 270-315, (1988) · Zbl 0648.34082
[28] Mallet-Paret, J.; Nussbaum, R.D., A differential-delay equation arising in optics and physiology, SIAM J. math. anal., 20, 249-292, (1989) · Zbl 0676.34043
[29] Mallet-Paret, J.; Nussbaum, R.D., Boundary layer phenomena for differential delay equations with state-dependent time lagsi, Arch. rational mech. anal., 120, 99-146, (1992) · Zbl 0763.34056
[30] Mallet-Paret, J.; Nussbaum, R.D.; Paraskevopoulos, P., Periodic solutions for functional differential equations with multiple state-dependent time lags, Topol. methods nonlinear anal., 3, 101-162, (1994) · Zbl 0808.34080
[31] Mallet-Paret, J.; Sell, G., Systems of differential delay equationsfloquet multipliers and discrete Lyapunov functions, J. differential equations, 125, 385-440, (1986) · Zbl 0849.34055
[32] H.-O. Walther, The solution manifold and C1-smoothness for differential equations with state dependent delay, preprint 2002.
[33] Walther, H.-O., Stable periodic motion of a system with state dependent delay, Differential and integral equations, 15, 923-944, (2002) · Zbl 1034.34085
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