
Previous Article
Blowup phenomena for the 3D compressible MHD equations
 DCDS Home
 This Issue

Next Article
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, 730 (1979), 16. 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. 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 (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), 1161. Google Scholar 
[5] 
A. Granas and J. Dugundji, "Fixed Point Theory,", Springer Monographs in Mathematics, (2003). Google Scholar 
[6] 
Ferenc Hartung, Tibor Krisztin, HansOtto Walther and Jianhong Wu, Functional differential equations with statedependent delays: Theory and applications,, in, (2006), 435. 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. Google Scholar 
[8] 
Benjamin Kennedy, Multiple periodic solutions of an equation with statedependent delay,, Journal of Dynamics and Differential Equations, 23 (2011), 283. Google Scholar 
[9] 
Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays,, Differential and Integral Equations, 22 (2009), 679. 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), 453. Google Scholar 
[11] 
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous statedependent delay equations,, Nonlinear Analysis: Theory, 19 (1992), 855. 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. 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), 101. Google Scholar 
[14] 
John MalletParet, Morse decompositions for delaydifferential equations,, Journal of Differential Equations, 72 (1988), 270. 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. Google Scholar 
[16] 
H. Peters, Chaotic behavior of nonlinear differentialdelay equations,, Nonlinear Analysis: Theory, 7 (1983), 1315. Google Scholar 
[17] 
H.W. Siegberg, Chaotic behavior of a class of differentialdelay equations,, Annali di Matematica Pura ed Applicata (4), 138 (1984), 15. Google Scholar 
[18] 
H. L. Smith and Y. Kuang, Periodic solutions of differential delay equations with thresholdtype delays,, in, 129 (1992), 153. Google Scholar 
[19] 
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions,, Journal of Dynamics and Differential Equations, 20 (2008), 201. Google Scholar 
[20] 
H.O. Walther, A periodic solution of a differential equation with statedependent delay,, Journal of Differential Equations, 244 (2008), 1910. Google Scholar 
[21] 
H.O. Walther, Stable periodic motion of a system with statedependent delay,, Differential and Integral Equations, 15 (2002), 923. Google Scholar 
[22] 
P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics,", Lecture Notes in Biomathematics, (1974). Google Scholar 
show all references
References:
[1] 
W. Alt, Periodic solutions of some autonomous differential equations with variable time delay,, in, 730 (1979), 16. 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. 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 (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), 1161. Google Scholar 
[5] 
A. Granas and J. Dugundji, "Fixed Point Theory,", Springer Monographs in Mathematics, (2003). Google Scholar 
[6] 
Ferenc Hartung, Tibor Krisztin, HansOtto Walther and Jianhong Wu, Functional differential equations with statedependent delays: Theory and applications,, in, (2006), 435. 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. Google Scholar 
[8] 
Benjamin Kennedy, Multiple periodic solutions of an equation with statedependent delay,, Journal of Dynamics and Differential Equations, 23 (2011), 283. Google Scholar 
[9] 
Benjamin Kennedy, Periodic solutions of delay equations with several fixed delays,, Differential and Integral Equations, 22 (2009), 679. 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), 453. Google Scholar 
[11] 
Y. Kuang and H. L. Smith, Slowly oscillating periodic solutions of autonomous statedependent delay equations,, Nonlinear Analysis: Theory, 19 (1992), 855. 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. 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), 101. Google Scholar 
[14] 
John MalletParet, Morse decompositions for delaydifferential equations,, Journal of Differential Equations, 72 (1988), 270. 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. Google Scholar 
[16] 
H. Peters, Chaotic behavior of nonlinear differentialdelay equations,, Nonlinear Analysis: Theory, 7 (1983), 1315. Google Scholar 
[17] 
H.W. Siegberg, Chaotic behavior of a class of differentialdelay equations,, Annali di Matematica Pura ed Applicata (4), 138 (1984), 15. Google Scholar 
[18] 
H. L. Smith and Y. Kuang, Periodic solutions of differential delay equations with thresholdtype delays,, in, 129 (1992), 153. Google Scholar 
[19] 
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions,, Journal of Dynamics and Differential Equations, 20 (2008), 201. Google Scholar 
[20] 
H.O. Walther, A periodic solution of a differential equation with statedependent delay,, Journal of Differential Equations, 244 (2008), 1910. Google Scholar 
[21] 
H.O. Walther, Stable periodic motion of a system with statedependent delay,, Differential and Integral Equations, 15 (2002), 923. Google Scholar 
[22] 
P. Waltman, "Deterministic Threshold Models in the Theory of Epidemics,", Lecture Notes in Biomathematics, (1974). Google Scholar 
[1] 
Qingwen Hu. A model of regulatory dynamics with thresholdtype statedependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863882. doi: 10.3934/mbe.2018039 
[2] 
HansOtto Walther. On Poisson's statedependent delay. Discrete & Continuous Dynamical Systems  A, 2013, 33 (1) : 365379. doi: 10.3934/dcds.2013.33.365 
[3] 
István Györi, Ferenc Hartung. Exponential stability of a statedependent delay system. Discrete & Continuous Dynamical Systems  A, 2007, 18 (4) : 773791. doi: 10.3934/dcds.2007.18.773 
[4] 
Ovide Arino, Eva Sánchez. A saddle point theorem for functional statedependent delay differential equations. Discrete & Continuous Dynamical Systems  A, 2005, 12 (4) : 687722. doi: 10.3934/dcds.2005.12.687 
[5] 
Tibor Krisztin. A local unstable manifold for differential equations with statedependent delay. Discrete & Continuous Dynamical Systems  A, 2003, 9 (4) : 9931028. doi: 10.3934/dcds.2003.9.993 
[6] 
Eugen Stumpf. Local stability analysis of differential equations with statedependent delay. Discrete & Continuous Dynamical Systems  A, 2016, 36 (6) : 34453461. doi: 10.3934/dcds.2016.36.3445 
[7] 
Odo Diekmann, Karolína Korvasová. Linearization of solution operators for statedependent delay equations: A simple example. Discrete & Continuous Dynamical Systems  A, 2016, 36 (1) : 137149. doi: 10.3934/dcds.2016.36.137 
[8] 
Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with statedependent delay. Discrete & Continuous Dynamical Systems  A, 2017, 37 (7) : 39393961. doi: 10.3934/dcds.2017167 
[9] 
A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple statedependent delays. Discrete & Continuous Dynamical Systems  A, 2012, 32 (8) : 27012727. doi: 10.3934/dcds.2012.32.2701 
[10] 
Hermann Brunner, Stefano Maset. Time transformations for statedependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 2345. doi: 10.3934/cpaa.2010.9.23 
[11] 
Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded statedependent delay equation. Conference Publications, 2001, 2001 (Special) : 5665. doi: 10.3934/proc.2001.2001.56 
[12] 
Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with statedependent delay. Discrete & Continuous Dynamical Systems  B, 2017, 22 (8) : 31673197. doi: 10.3934/dcdsb.2017169 
[13] 
Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with statedependent delay. Discrete & Continuous Dynamical Systems  B, 2018, 23 (10) : 43294360. doi: 10.3934/dcdsb.2018167 
[14] 
Qingwen Hu, Bernhard LaniWayda, Eugen Stumpf. Preface: Delay differential equations with statedependent delays and their applications. Discrete & Continuous Dynamical Systems  S, 2020, 13 (1) : ⅰⅰ. doi: 10.3934/dcdss.20201i 
[15] 
Shangzhi Li, Shangjiang Guo. Dynamics of a stagestructured population model with a statedependent delay. Discrete & Continuous Dynamical Systems  B, 2017, 22 (11) : 00. doi: 10.3934/dcdsb.2020071 
[16] 
Benjamin B. Kennedy. A statedependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems  B, 2013, 18 (6) : 16331650. doi: 10.3934/dcdsb.2013.18.1633 
[17] 
Igor Chueshov, Peter E. Kloeden, Meihua Yang. Long term dynamics of second orderintime stochastic evolution equations with statedependent delay. Discrete & Continuous Dynamical Systems  B, 2018, 23 (3) : 9911009. doi: 10.3934/dcdsb.2018139 
[18] 
Xianlong Fu. Approximate controllability of semilinear nonautonomous evolution systems with statedependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517534. doi: 10.3934/eect.2017026 
[19] 
Alexander Rezounenko. Stability of a viral infection model with statedependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems  B, 2017, 22 (4) : 15471563. doi: 10.3934/dcdsb.2017074 
[20] 
Jan Sieber. Finding periodic orbits in statedependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems  A, 2012, 32 (8) : 26072651. doi: 10.3934/dcds.2012.32.2607 
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]