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Breather continuation from infinity in nonlinear oscillator chains
Multiple periodic solutions of statedependent threshold delay equations
1.  Department of Mathematics, Gettysburg College, Gettysburg, PA 173251484, United States 
We also describe part of the global dynamics of the model equation $x'(t) = h(x(t  d(x_t)))$.
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), 1631. 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), 273295. 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, SpringerVerlag, New York, 1995. Google Scholar 
[4] 
L. M. Fridman, È. M. Fridman and E. I. Shustin, Steadystate regimes in an autonomous system with a discontinuity and delay, Differential Equations, 29 (1993), 11611166. Google Scholar 
[5] 
A. Granas and J. Dugundji, "Fixed Point Theory," Springer Monographs in Mathematics, SpringerVerlag, New York, 2003. Google Scholar 
[6] 
Ferenc Hartung, Tibor Krisztin, HansOtto Walther and Jianhong Wu, Functional differential equations with statedependent 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/NorthHolland, Amsterdam, (2006), 435545. 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), 811832. Google Scholar 
[8] 
Benjamin Kennedy, Multiple periodic solutions of an equation with statedependent delay, Journal of Dynamics and Differential Equations, 23 (2011), 283313. Google Scholar 
[9] 
Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays, Differential and Integral Equations, 22 (2009), 679724. Google Scholar 
[10] 
Tibor Krisztin and Ovide Arino, The twodimensional attractor of a differential equation with statedependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453522. Google Scholar 
[11] 
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous statedependent delay equations, Nonlinear Analysis: Theory, Methods, & Applications, 19 (1992), 855872. 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), 6195. Google Scholar 
[13] 
John MalletParet, Roger D. Nussbaum and Panagiotis Paraskevopoulous, Periodic solutions for functional differential equations with multiple statedependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101162. Google Scholar 
[14] 
John MalletParet, Morse decompositions for delaydifferential equations, Journal of Differential Equations, 72 (1988), 270315. 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), 114135. Google Scholar 
[16] 
H. Peters, Chaotic behavior of nonlinear differentialdelay equations, Nonlinear Analysis: Theory, Methods & Applications, 7 (1983), 13151334. Google Scholar 
[17] 
H.W. Siegberg, Chaotic behavior of a class of differentialdelay equations, Annali di Matematica Pura ed Applicata (4), 138 (1984), 1533. Google Scholar 
[18] 
H. L. Smith and Y. Kuang, Periodic solutions of differential delay equations with thresholdtype delays, in "Oscillation and Dynamics in Delay Equations" (San Francisco, CA, 1991), Contemporary Mathematics, 129, Amer. Math. Soc., Providence, RI, (1992), 153176. Google Scholar 
[19] 
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, Journal of Dynamics and Differential Equations, 20 (2008), 201238. Google Scholar 
[20] 
H.O. Walther, A periodic solution of a differential equation with statedependent delay, Journal of Differential Equations, 244 (2008), 19101945. Google Scholar 
[21] 
H.O. Walther, Stable periodic motion of a system with statedependent delay, Differential and Integral Equations, 15 (2002), 923944. Google Scholar 
[22] 
P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics," Lecture Notes in Biomathematics, Vol. 1, SpringerVerlag, BerlinNew 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), 1631. 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), 273295. 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, SpringerVerlag, New York, 1995. Google Scholar 
[4] 
L. M. Fridman, È. M. Fridman and E. I. Shustin, Steadystate regimes in an autonomous system with a discontinuity and delay, Differential Equations, 29 (1993), 11611166. Google Scholar 
[5] 
A. Granas and J. Dugundji, "Fixed Point Theory," Springer Monographs in Mathematics, SpringerVerlag, New York, 2003. Google Scholar 
[6] 
Ferenc Hartung, Tibor Krisztin, HansOtto Walther and Jianhong Wu, Functional differential equations with statedependent 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/NorthHolland, Amsterdam, (2006), 435545. 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), 811832. Google Scholar 
[8] 
Benjamin Kennedy, Multiple periodic solutions of an equation with statedependent delay, Journal of Dynamics and Differential Equations, 23 (2011), 283313. Google Scholar 
[9] 
Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays, Differential and Integral Equations, 22 (2009), 679724. Google Scholar 
[10] 
Tibor Krisztin and Ovide Arino, The twodimensional attractor of a differential equation with statedependent delay, Journal of Dynamics and Differential Equations, 13 (2001), 453522. Google Scholar 
[11] 
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous statedependent delay equations, Nonlinear Analysis: Theory, Methods, & Applications, 19 (1992), 855872. 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), 6195. Google Scholar 
[13] 
John MalletParet, Roger D. Nussbaum and Panagiotis Paraskevopoulous, Periodic solutions for functional differential equations with multiple statedependent time lags, Topological Methods in Nonlinear Analysis, 3 (1994), 101162. Google Scholar 
[14] 
John MalletParet, Morse decompositions for delaydifferential equations, Journal of Differential Equations, 72 (1988), 270315. 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), 114135. Google Scholar 
[16] 
H. Peters, Chaotic behavior of nonlinear differentialdelay equations, Nonlinear Analysis: Theory, Methods & Applications, 7 (1983), 13151334. Google Scholar 
[17] 
H.W. Siegberg, Chaotic behavior of a class of differentialdelay equations, Annali di Matematica Pura ed Applicata (4), 138 (1984), 1533. Google Scholar 
[18] 
H. L. Smith and Y. Kuang, Periodic solutions of differential delay equations with thresholdtype delays, in "Oscillation and Dynamics in Delay Equations" (San Francisco, CA, 1991), Contemporary Mathematics, 129, Amer. Math. Soc., Providence, RI, (1992), 153176. Google Scholar 
[19] 
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, Journal of Dynamics and Differential Equations, 20 (2008), 201238. Google Scholar 
[20] 
H.O. Walther, A periodic solution of a differential equation with statedependent delay, Journal of Differential Equations, 244 (2008), 19101945. Google Scholar 
[21] 
H.O. Walther, Stable periodic motion of a system with statedependent delay, Differential and Integral Equations, 15 (2002), 923944. Google Scholar 
[22] 
P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics," Lecture Notes in Biomathematics, Vol. 1, SpringerVerlag, BerlinNew York, 1974. Google Scholar 
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