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

Multiple periodic solutions of state-dependent threshold delay equations

Benjamin B. Kennedy 1,

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 $x'(t) = h(x(t - d(x_t)))$.
Keywords: nullhomotopic map., threshold-type delay, State-dependent delay.
Mathematics Subject Classification: 34K1.
Citation: Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Granas and J. Dugundji, "Fixed Point Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Benjamin Kennedy, Multiple periodic solutions of an equation with state-dependent delay, Journal of Dynamics and Differential Equations, 23 (2011), 283-313. Google Scholar

[9]

Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays, Differential and Integral Equations, 22 (2009), 679-724.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

John Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315.  Google Scholar

[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.  Google Scholar

[16]

H. Peters, Chaotic behavior of nonlinear differential-delay equations, Nonlinear Analysis: Theory, Methods & Applications, 7 (1983), 1315-1334.  Google Scholar

[17]

H.-W. Siegberg, Chaotic behavior of a class of differential-delay equations, Annali di Matematica Pura ed Applicata (4), 138 (1984), 15-33.  Google Scholar

[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.  Google Scholar

[19]

D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, Journal of Dynamics and Differential Equations, 20 (2008), 201-238.  Google Scholar

[20]

H.-O. Walther, A periodic solution of a differential equation with state-dependent delay, Journal of Differential Equations, 244 (2008), 1910-1945.  Google Scholar

[21]

H.-O. Walther, Stable periodic motion of a system with state-dependent delay, Differential and Integral Equations, 15 (2002), 923-944.  Google Scholar

[22]

P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics," Lecture Notes in Biomathematics, Vol. 1, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Granas and J. Dugundji, "Fixed Point Theory," Springer Monographs in Mathematics, Springer-Verlag, New York, 2003.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Benjamin Kennedy, Multiple periodic solutions of an equation with state-dependent delay, Journal of Dynamics and Differential Equations, 23 (2011), 283-313. Google Scholar

[9]

Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays, Differential and Integral Equations, 22 (2009), 679-724.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

John Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315.  Google Scholar

[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.  Google Scholar

[16]

H. Peters, Chaotic behavior of nonlinear differential-delay equations, Nonlinear Analysis: Theory, Methods & Applications, 7 (1983), 1315-1334.  Google Scholar

[17]

H.-W. Siegberg, Chaotic behavior of a class of differential-delay equations, Annali di Matematica Pura ed Applicata (4), 138 (1984), 15-33.  Google Scholar

[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.  Google Scholar

[19]

D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, Journal of Dynamics and Differential Equations, 20 (2008), 201-238.  Google Scholar

[20]

H.-O. Walther, A periodic solution of a differential equation with state-dependent delay, Journal of Differential Equations, 244 (2008), 1910-1945.  Google Scholar

[21]

H.-O. Walther, Stable periodic motion of a system with state-dependent delay, Differential and Integral Equations, 15 (2002), 923-944.  Google Scholar

[22]

P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics," Lecture Notes in Biomathematics, Vol. 1, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

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