-
Previous Article
Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure
- DCDS Home
- This Issue
-
Next Article
Quasi-invariant attractors of piecewise isometric systems
Oscillations in a second-order discontinuous system with delay
| 1. | School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel |
| 2. | Department of Electrical Engineering & Systems, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel |
| 3. | Division of Postgraduate and Investigation, Chihuahua Institute of Technology, Chihuahua, Chi, C.P. 31160, Mexico |
$\alpha x''(t)=-x'(t)+F(x(t),t)-$sign$x(t-h),\quad\alpha=$const$>0,\ $ $h=$const$>0,$
which is a model for a scalar system with a discontinuous negative delayed feedback, and study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. Our main conclusion is that, in the autonomous case $F(x,t)\equiv F(x)$, for $|F(x)|<1$, there are periodic solutions with different frequencies of oscillations, though only slowly-oscillating solutions are (orbitally) stable. Under extra conditions we show the uniqueness of a periodic slowly-oscillating solution. We also give a criterion for the existence of bounded oscillations in the case of unbounded function $F(x,t)$. Our approach consists basically in reducing the problem to the study of dynamics of some discrete scalar system.
| [1] |
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 |
| [2] |
Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385 |
| [3] |
Urszula Foryś, Jan Poleszczuk. A delay-differential equation model of HIV related cancer--immune system dynamics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 627-641. doi: 10.3934/mbe.2011.8.627 |
| [4] |
Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090 |
| [5] |
Byungsoo Moon. Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4409-4437. doi: 10.3934/dcdss.2021123 |
| [6] |
Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537 |
| [7] |
John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044 |
| [8] |
Nguyen Thi Van Anh. On periodic solutions to a class of delay differential variational inequalities. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021045 |
| [9] |
Giovana Alves, Fábio Natali. Periodic waves for the cubic-quintic nonlinear Schrodinger equation: Existence and orbital stability. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022101 |
| [10] |
Xiaoxiao Zheng, Hui Wu. Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components. Mathematical Foundations of Computing, 2020, 3 (1) : 11-24. doi: 10.3934/mfc.2020002 |
| [11] |
Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations and Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024 |
| [12] |
Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139 |
| [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] |
José Manuel Palacios. Orbital and asymptotic stability of a train of peakons for the Novikov equation. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2475-2518. doi: 10.3934/dcds.2020372 |
| [15] |
Pasquale Palumbo, Simona Panunzi, Andrea De Gaetano. Qualitative behavior of a family of delay-differential models of the Glucose-Insulin system. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 399-424. doi: 10.3934/dcdsb.2007.7.399 |
| [16] |
Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369 |
| [17] |
Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031 |
| [18] |
Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105 |
| [19] |
Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529 |
| [20] |
Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]