American Institute of Mathematical Sciences

Advanced
June  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

Ábel Garab 1,

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.
Keywords: period function, delayed cellular neural networks, periodic orbit., Delay differential equation, discrete Lyapunov functional, neural networks.
Mathematics Subject Classification: Primary: 34K13, 92B2.
Citation: Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems, 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), 46-57. doi: 10.1006/jdeq.1996.0088.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. K. Hale, "Theory of Functional Differential Equations," Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[8]

T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback, Electron. J. Qual. Theory Differ. Equ., Szeged, (2000), 1-12.  Google Scholar

[9]

T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback, Fields Inst. Commun., 29 (2001), 267-296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Wu, "Introduction to Neural Dynamics and Signal Transmission," de Gruyter, Berlin, 2001. doi: 10.1515/9783110879971.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. K. Hale, "Theory of Functional Differential Equations," Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar

[8]

T. Krisztin, Periodic orbits and the global attractor for delayed monotone negative feedback, Electron. J. Qual. Theory Differ. Equ., Szeged, (2000), 1-12.  Google Scholar

[9]

T. Krisztin, Unstable sets of periodic orbits and the global attractor for delayed feedback, Fields Inst. Commun., 29 (2001), 267-296.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Wu, "Introduction to Neural Dynamics and Signal Transmission," de Gruyter, Berlin, 2001. doi: 10.1515/9783110879971.  Google Scholar

[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.  Google Scholar

[1]

Ying Sue Huang. Resynchronization of delayed neural networks. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 397-401. doi: 10.3934/dcds.2001.7.397
[2]

Cheng-Hsiung Hsu, Suh-Yuh Yang. Structure of a class of traveling waves in delayed cellular neural networks. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 339-359. doi: 10.3934/dcds.2005.13.339
[3]

Benedetta Lisena. Average criteria for periodic neural networks with delay. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 761-773. doi: 10.3934/dcdsb.2014.19.761
[4]

Xilin Fu, Zhang Chen. New discrete analogue of neural networks with nonlinear amplification function and its periodic dynamic analysis. Conference Publications, 2007, 2007 (Special) : 391-398. doi: 10.3934/proc.2007.2007.391
[5]

Larry Turyn. Cellular neural networks: asymmetric templates and spatial chaos. Conference Publications, 2003, 2003 (Special) : 864-871. doi: 10.3934/proc.2003.2003.864
[6]

Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6047-6056. doi: 10.3934/dcdsb.2021001
[7]

Chuangxia Huang, Hedi Yang, Jinde Cao. Weighted pseudo almost periodicity of multi-proportional delayed shunting inhibitory cellular neural networks with D operator. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1259-1272. doi: 10.3934/dcdss.2020372
[8]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2021, 8 (2) : 131-152. doi: 10.3934/jcd.2021006
[9]

Ricai Luo, Honglei Xu, Wu-Sheng Wang, Jie Sun, Wei Xu. A weak condition for global stability of delayed neural networks. Journal of Industrial & Management Optimization, 2016, 12 (2) : 505-514. doi: 10.3934/jimo.2016.12.505
[10]

Cheng-Hsiung Hsu, Suh-Yuh Yang. Traveling wave solutions in cellular neural networks with multiple time delays. Conference Publications, 2005, 2005 (Special) : 410-419. doi: 10.3934/proc.2005.2005.410
[11]

Zuowei Cai, Jianhua Huang, Lihong Huang. Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3591-3614. doi: 10.3934/dcdsb.2017181
[12]

Ivanka Stamova, Gani Stamov. On the stability of sets for reaction–diffusion Cohen–Grossberg delayed neural networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1429-1446. doi: 10.3934/dcdss.2020370
[13]

Sylvia Novo, Rafael Obaya, Ana M. Sanz. Exponential stability in non-autonomous delayed equations with applications to neural networks. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 517-536. doi: 10.3934/dcds.2007.18.517
[14]

Xiaochen Mao, Weijie Ding, Xiangyu Zhou, Song Wang, Xingyong Li. Complexity in time-delay networks of multiple interacting neural groups. Electronic Research Archive, 2021, 29 (5) : 2973-2985. doi: 10.3934/era.2021022
[15]

Meiyu Sui, Yejuan Wang, Peter E. Kloeden. Pullback attractors for stochastic recurrent neural networks with discrete and distributed delays. Electronic Research Archive, 2021, 29 (2) : 2187-2221. doi: 10.3934/era.2020112
[16]

Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457
[17]

Tatyana S. Turova. Structural phase transitions in neural networks. Mathematical Biosciences & Engineering, 2014, 11 (1) : 139-148. doi: 10.3934/mbe.2014.11.139
[18]

Suh-Yuh Yang, Cheng-Hsiung Hsu. Existence of monotonic traveling waves in modified RTD-based cellular neural networks. Conference Publications, 2005, 2005 (Special) : 930-939. doi: 10.3934/proc.2005.2005.930
[19]

Benedict Leimkuhler, Charles Matthews, Tiffany Vlaar. Partitioned integrators for thermodynamic parameterization of neural networks. Foundations of Data Science, 2019, 1 (4) : 457-489. doi: 10.3934/fods.2019019
[20]

Jianping Zhou, Yamin Liu, Ju H. Park, Qingkai Kong, Zhen Wang. Fault-tolerant anti-synchronization control for chaotic switched neural networks with time delay and reaction diffusion. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1569-1589. doi: 10.3934/dcdss.2020357

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy