# American Institute of Mathematical Sciences

May  2012, 32(5): 1801-1833. doi: 10.3934/dcds.2012.32.1801

## Multiple periodic solutions of state-dependent threshold delay equations

 1 Department of Mathematics, Gettysburg College, Gettysburg, PA 17325-1484, United States

Received  November 2010 Revised  September 2011 Published  January 2012

We prove the existence of multiple periodic solutions for scalar-valued state-dependent delay equations of the form $x'(t) = f(x(t - d(x_t)))$, where $d(x_t)$ is given by a threshold condition and $f$ is close, in a suitable sense, to the step function $h(x) = -\mbox{sign}(x)$. We construct maps whose fixed points correspond to periodic solutions and show that these maps have nontrivial fixed points via homotopy to constant maps.
We also describe part of the global dynamics of the model equation $x'(t) = h(x(t - d(x_t)))$.
Citation: Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801
##### References:
 [1] W. Alt, Periodic solutions of some autonomous differential equations with variable time delay, in "Functional Differential Equations and Approximation of Fixed Points" (eds. H. O. Peitgen and H.-O. Walther, Proc. Summer School and Conf., Univ. Bonn, Bonn, 1978), Lecture Notes in Math., 730, Springer, Berlin, (1979), 16-31. [2] U. an der Heiden and H.-O. Walther, Existence of chaos in control systems with delayed feedback, Journal of Differential Equations, 47 (1983), 273-295. [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Funtional, Complex, and Nonlinear Analysis," Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995. [4] L. M. Fridman, È. M. Fridman and E. I. Shustin, Steady-state regimes in an autonomous system with a discontinuity and delay, Differential Equations, 29 (1993), 1161-1166. [5] A. Granas and J. Dugundji, "Fixed Point Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. [6] Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. III (eds. A. Cañada, P. Dràbek and A. Fonda), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545. [7] Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback, Differential and Integral Equations, 12 (1999), 811-832. [8] Benjamin Kennedy, Multiple periodic solutions of an equation with state-dependent delay, Journal of Dynamics and Differential Equations, 23 (2011), 283-313. [9] Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays, Differential and Integral Equations, 22 (2009), 679-724. [10] Tibor Krisztin and Ovide Arino, The two-dimensional attractor of a differential equation with state-dependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453-522. [11] Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Analysis: Theory, Methods, & Applications, 19 (1992), 855-872. [12] P. Magal and O. Arino, Existence of periodic solutions for a state dependent delay differential equation, Journal of Differential Equations, 165 (2000), 61-95. [13] John Mallet-Paret, Roger D. Nussbaum and Panagiotis Paraskevopoulous, Periodic solutions for functional differential equations with multiple state-dependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101-162. [14] John Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315. [15] R. Nisbet and W. S. C. Gurney, The systematic formulation of population models with dynamically varying instar duration, Theoretical Population Biology, 23 (1983), 114-135. [16] H. Peters, Chaotic behavior of nonlinear differential-delay equations, Nonlinear Analysis: Theory, Methods & Applications, 7 (1983), 1315-1334. [17] H.-W. Siegberg, Chaotic behavior of a class of differential-delay equations, Annali di Matematica Pura ed Applicata (4), 138 (1984), 15-33. [18] H. L. Smith and Y. Kuang, Periodic solutions of differential delay equations with threshold-type delays, in "Oscillation and Dynamics in Delay Equations" (San Francisco, CA, 1991), Contemporary Mathematics, 129, Amer. Math. Soc., Providence, RI, (1992), 153-176. [19] D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, Journal of Dynamics and Differential Equations, 20 (2008), 201-238. [20] H.-O. Walther, A periodic solution of a differential equation with state-dependent delay, Journal of Differential Equations, 244 (2008), 1910-1945. [21] H.-O. Walther, Stable periodic motion of a system with state-dependent delay, Differential and Integral Equations, 15 (2002), 923-944. [22] P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics," Lecture Notes in Biomathematics, Vol. 1, Springer-Verlag, Berlin-New York, 1974.

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##### References:
 [1] W. Alt, Periodic solutions of some autonomous differential equations with variable time delay, in "Functional Differential Equations and Approximation of Fixed Points" (eds. H. O. Peitgen and H.-O. Walther, Proc. Summer School and Conf., Univ. Bonn, Bonn, 1978), Lecture Notes in Math., 730, Springer, Berlin, (1979), 16-31. [2] U. an der Heiden and H.-O. Walther, Existence of chaos in control systems with delayed feedback, Journal of Differential Equations, 47 (1983), 273-295. [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Funtional, Complex, and Nonlinear Analysis," Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995. [4] L. M. Fridman, È. M. Fridman and E. I. Shustin, Steady-state regimes in an autonomous system with a discontinuity and delay, Differential Equations, 29 (1993), 1161-1166. [5] A. Granas and J. Dugundji, "Fixed Point Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. [6] Ferenc Hartung, Tibor Krisztin, Hans-Otto Walther and Jianhong Wu, Functional differential equations with state-dependent delays: Theory and applications, in "Handbook of Differential Equations: Ordinary Differential Equations," Vol. III (eds. A. Cañada, P. Dràbek and A. Fonda), Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2006), 435-545. [7] Anatoli F. Ivanov and Jérôme Losson, Stable rapidly oscillating solutions in delay equations with negative feedback, Differential and Integral Equations, 12 (1999), 811-832. [8] Benjamin Kennedy, Multiple periodic solutions of an equation with state-dependent delay, Journal of Dynamics and Differential Equations, 23 (2011), 283-313. [9] Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays, Differential and Integral Equations, 22 (2009), 679-724. [10] Tibor Krisztin and Ovide Arino, The two-dimensional attractor of a differential equation with state-dependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453-522. [11] Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous state-dependent delay equations, Nonlinear Analysis: Theory, Methods, & Applications, 19 (1992), 855-872. [12] P. Magal and O. Arino, Existence of periodic solutions for a state dependent delay differential equation, Journal of Differential Equations, 165 (2000), 61-95. [13] John Mallet-Paret, Roger D. Nussbaum and Panagiotis Paraskevopoulous, Periodic solutions for functional differential equations with multiple state-dependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101-162. [14] John Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315. [15] R. Nisbet and W. S. C. Gurney, The systematic formulation of population models with dynamically varying instar duration, Theoretical Population Biology, 23 (1983), 114-135. [16] H. Peters, Chaotic behavior of nonlinear differential-delay equations, Nonlinear Analysis: Theory, Methods & Applications, 7 (1983), 1315-1334. [17] H.-W. Siegberg, Chaotic behavior of a class of differential-delay equations, Annali di Matematica Pura ed Applicata (4), 138 (1984), 15-33. [18] H. L. Smith and Y. Kuang, Periodic solutions of differential delay equations with threshold-type delays, in "Oscillation and Dynamics in Delay Equations" (San Francisco, CA, 1991), Contemporary Mathematics, 129, Amer. Math. Soc., Providence, RI, (1992), 153-176. [19] D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, Journal of Dynamics and Differential Equations, 20 (2008), 201-238. [20] H.-O. Walther, A periodic solution of a differential equation with state-dependent delay, Journal of Differential Equations, 244 (2008), 1910-1945. [21] H.-O. Walther, Stable periodic motion of a system with state-dependent delay, Differential and Integral Equations, 15 (2002), 923-944. [22] P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics," Lecture Notes in Biomathematics, Vol. 1, Springer-Verlag, Berlin-New York, 1974.
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