# American Institute of Mathematical Sciences

2013, 33(6): 2369-2387. doi: 10.3934/dcds.2013.33.2369

## 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

Received  January 2012 Revised  April 2012 Published  December 2012

In this paper we study the scalar delay differential equation \linebreak $\dot{x}(t)=-ax(t) + bf(x(t-\tau))$ with feedback function $f(\xi)=\frac{1}{2}(|\xi+1|-|\xi-1|)$ and with real parameters $a>0,\ \tau>0$ and $b\neq 0$, which can model a single neuron or a group of synchronized neurons. We give necessary and sufficient conditions for existence and uniqueness of periodic orbits with prescribed oscillation frequencies. We also investigate the period of the slowly oscillating periodic solution as a function of the delay. Based on the obtained results we state an analogous theorem concerning existence and uniqueness of periodic orbits of a certain type of system of delay differential equations. The proofs are based among others on theory of monotone systems and discrete Lyapunov functionals.
Citation: Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
##### References:
 [1] Y. Cao, Uniqueness of periodic solution for differential delay equations,, J. Diff. Eq., 128 (1996), 46. 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. [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex and Nonlinear Analysis,", Springer-Verlag, (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. 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. 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. doi: 10.1016/S0377-0427(03)00376-5. [7] J. K. Hale, "Theory of Functional Differential Equations,", Springer-Verlag, (1977). [8] T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback,, Electron. J. Qual. Theory Differ. Equ., (2000), 1. [9] T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback,, Fields Inst. Commun., 29 (2001), 267. [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. 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. 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 (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. 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. 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. doi: 10.1016/j.na.2009.01.078. [16] J. Wu, "Introduction to Neural Dynamics and Signal Transmission,", de Gruyter, (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. doi: 10.1137/080725283.

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##### References:
 [1] Y. Cao, Uniqueness of periodic solution for differential delay equations,, J. Diff. Eq., 128 (1996), 46. 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. [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex and Nonlinear Analysis,", Springer-Verlag, (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. 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. 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. doi: 10.1016/S0377-0427(03)00376-5. [7] J. K. Hale, "Theory of Functional Differential Equations,", Springer-Verlag, (1977). [8] T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback,, Electron. J. Qual. Theory Differ. Equ., (2000), 1. [9] T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback,, Fields Inst. Commun., 29 (2001), 267. [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. 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. 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 (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. 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. 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. doi: 10.1016/j.na.2009.01.078. [16] J. Wu, "Introduction to Neural Dynamics and Signal Transmission,", de Gruyter, (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. doi: 10.1137/080725283.
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