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Unique periodic orbits of a delay differential equation with piecewise linear feedback function
1. | Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, H-6720, Hungary |
References:
[1] |
Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Diff. Eq., 128 (1996), 46-57.
doi: 10.1006/jdeq.1996.0088. |
[2] |
Y. Chen and J. Wu, Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential Integral Equations, 14 (2001), 1181-1236. |
[3] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex and Nonlinear Analysis," Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[4] |
Á. Garab and T. Krisztin, The period function of a delay differential equation and an application, Per. Math. Hungar., 63 (2011), 173-190.
doi: 10.1007/s10998-011-8173-2. |
[5] |
K. Gopalsamy and X.-Z. He, Stability in asymmetric Hopfield nets with transmission delays, Phys. D, 40 (1994), 344-358.
doi: 10.1016/0167-2789(94)90043-4. |
[6] |
I. Gyõri and F. Hartung, Stability analysis of a single neuron model with delay, J. Comput. Appl. Math., 157 (2003), 73-92.
doi: 10.1016/S0377-0427(03)00376-5. |
[7] |
J. K. Hale, "Theory of Functional Differential Equations," Springer-Verlag, New York-Heidelberg, 1977. |
[8] |
T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback, Electron. J. Qual. Theory Differ. Equ., Szeged, (2000), 1-12. |
[9] |
T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback, Fields Inst. Commun., 29 (2001), 267-296. |
[10] |
T. Krisztin and G. Vas, Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, J. Dynam. Differential Equations, 23 (2011), 727-790.
doi: 10.1007/s10884-011-9225-2. |
[11] |
T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[12] |
T. Krisztin, H.-O. Walther and J. Wu, "Shape, Smoothness, and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," Fields Institute Monograph Series, 11, AMS, Providence, 1999. |
[13] |
J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Diff. Eq., 125 (1996), 385-440.
doi: 10.1006/jdeq.1996.0036. |
[14] |
J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Diff. Eq., 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[15] |
G. Vas, Asymptotic constancy and periodicity for a single neuron model with delay, Nonlinear Anal., 71 (2009), 2268-2277.
doi: 10.1016/j.na.2009.01.078. |
[16] |
J. Wu, "Introduction to Neural Dynamics and Signal Transmission," de Gruyter, Berlin, 2001.
doi: 10.1515/9783110879971. |
[17] |
T. Yi, Y. Chen and J. Wu, Periodic solutions and the global attractor in a system of delay differential equations, SIAM J. Math. Anal., 42 (2010), 24-63.
doi: 10.1137/080725283. |
show all references
References:
[1] |
Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Diff. Eq., 128 (1996), 46-57.
doi: 10.1006/jdeq.1996.0088. |
[2] |
Y. Chen and J. Wu, Existence and attraction of a phase-locked oscillation in a delayed network of two neurons, Differential Integral Equations, 14 (2001), 1181-1236. |
[3] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex and Nonlinear Analysis," Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[4] |
Á. Garab and T. Krisztin, The period function of a delay differential equation and an application, Per. Math. Hungar., 63 (2011), 173-190.
doi: 10.1007/s10998-011-8173-2. |
[5] |
K. Gopalsamy and X.-Z. He, Stability in asymmetric Hopfield nets with transmission delays, Phys. D, 40 (1994), 344-358.
doi: 10.1016/0167-2789(94)90043-4. |
[6] |
I. Gyõri and F. Hartung, Stability analysis of a single neuron model with delay, J. Comput. Appl. Math., 157 (2003), 73-92.
doi: 10.1016/S0377-0427(03)00376-5. |
[7] |
J. K. Hale, "Theory of Functional Differential Equations," Springer-Verlag, New York-Heidelberg, 1977. |
[8] |
T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback, Electron. J. Qual. Theory Differ. Equ., Szeged, (2000), 1-12. |
[9] |
T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback, Fields Inst. Commun., 29 (2001), 267-296. |
[10] |
T. Krisztin and G. Vas, Large-amplitude periodic solutions for differential equations with delayed monotone positive feedback, J. Dynam. Differential Equations, 23 (2011), 727-790.
doi: 10.1007/s10884-011-9225-2. |
[11] |
T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[12] |
T. Krisztin, H.-O. Walther and J. Wu, "Shape, Smoothness, and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," Fields Institute Monograph Series, 11, AMS, Providence, 1999. |
[13] |
J. Mallet-Paret and G. Sell, Systems of differential delay equations: Floquet multipliers and discrete Lyapunov functions, J. Diff. Eq., 125 (1996), 385-440.
doi: 10.1006/jdeq.1996.0036. |
[14] |
J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Diff. Eq., 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[15] |
G. Vas, Asymptotic constancy and periodicity for a single neuron model with delay, Nonlinear Anal., 71 (2009), 2268-2277.
doi: 10.1016/j.na.2009.01.078. |
[16] |
J. Wu, "Introduction to Neural Dynamics and Signal Transmission," de Gruyter, Berlin, 2001.
doi: 10.1515/9783110879971. |
[17] |
T. Yi, Y. Chen and J. Wu, Periodic solutions and the global attractor in a system of delay differential equations, SIAM J. Math. Anal., 42 (2010), 24-63.
doi: 10.1137/080725283. |
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