February & March  2004, 11(2&3): 667-692. doi: 10.3934/dcds.2004.11.667

Periodic solutions for three-dimensional non-monotone cyclic systems with time delays

1. 

Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, United States

2. 

Mathematisches Institut der Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany

Received  January 2003 Revised  March 2004 Published  June 2004

We study a model for three cyclically coupled neurons with eventually negative delayed feedback, and without symmetry or monotonicity properties. Periodic solutions are obtained from the Schauder fixed point theorem. It turns out that, contrary to lower dimensional cases, instability at zero does not exclude monotonously decaying solutions.
Citation: Anatoli F. Ivanov, Bernhard Lani-Wayda. Periodic solutions for three-dimensional non-monotone cyclic systems with time delays. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 667-692. doi: 10.3934/dcds.2004.11.667
[1]

Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692

[2]

Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222

[3]

Elena Braverman, Karel Hasik, Anatoli F. Ivanov, Sergei I. Trofimchuk. A cyclic system with delay and its characteristic equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-29. doi: 10.3934/dcdss.2020001

[4]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220

[5]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

[6]

Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517

[7]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775

[8]

Weigao Ge, Li Zhang. Multiple periodic solutions of delay differential systems with $2k-1$ lags via variational approach. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4925-4943. doi: 10.3934/dcds.2016013

[9]

Nicholas Long. Fixed point shifts of inert involutions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1297-1317. doi: 10.3934/dcds.2009.25.1297

[10]

Tiphaine Jézéquel, Patrick Bernard, Eric Lombardi. Homoclinic orbits with many loops near a $0^2 i\omega$ resonant fixed point of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3153-3225. doi: 10.3934/dcds.2016.36.3153

[11]

Benjamin B. Kennedy. A state-dependent delay equation with negative feedback and "mildly unstable" rapidly oscillating periodic solutions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1633-1650. doi: 10.3934/dcdsb.2013.18.1633

[12]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807

[13]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301

[14]

D. Q. Cao, Y. R. Yang, Y. M. Ge. Characteristic equation approach to stability measures of linear neutral systems with multiple time delays. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 95-105. doi: 10.3934/dcds.2007.17.95

[15]

Yakov Krasnov, Alexander Kononovich, Grigory Osharovich. On a structure of the fixed point set of homogeneous maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1017-1027. doi: 10.3934/dcdss.2013.6.1017

[16]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[17]

Luis Hernández-Corbato, Francisco R. Ruiz del Portal. Fixed point indices of planar continuous maps. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2979-2995. doi: 10.3934/dcds.2015.35.2979

[18]

Antonio Garcia. Transition tori near an elliptic-fixed point. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 381-392. doi: 10.3934/dcds.2000.6.381

[19]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[20]

Thomas I. Seidman, Olaf Klein. Periodic solutions of isotone hybrid systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 483-493. doi: 10.3934/dcdsb.2013.18.483

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]