
Previous Article
Quasiinvariant attractors of piecewise isometric systems
 DCDS Home
 This Issue

Next Article
Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure
Oscillations in a secondorder 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$x(th),\quad\alpha=$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 slowlyoscillating solutions are (orbitally) stable. Under extra conditions we show the uniqueness of a periodic slowlyoscillating 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) : 220230. doi: 10.3934/proc.1998.1998.220 
[2] 
Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differentialdifference equation. Conference Publications, 2009, 2009 (Special) : 385393. doi: 10.3934/proc.2009.2009.385 
[3] 
Urszula Foryś, Jan Poleszczuk. A delaydifferential equation model of HIV related cancerimmune system dynamics. Mathematical Biosciences & Engineering, 2011, 8 (2) : 627641. doi: 10.3934/mbe.2011.8.627 
[4] 
Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delaydifferential equations with large delay. Discrete & Continuous Dynamical Systems  A, 2015, 35 (1) : 537553. doi: 10.3934/dcds.2015.35.537 
[5] 
Eugen Stumpf. On a delay differential equation arising from a carfollowing model: Wavefront solutions with constantspeed and their stability. Discrete & Continuous Dynamical Systems  B, 2017, 22 (9) : 33173340. doi: 10.3934/dcdsb.2017139 
[6] 
Elimhan N. Mahmudov. Optimal control of second order delaydiscrete and delaydifferential inclusions with state constraints. Evolution Equations & Control Theory, 2018, 7 (3) : 501529. doi: 10.3934/eect.2018024 
[7] 
Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301313. doi: 10.3934/proc.1998.1998.301 
[8] 
Pasquale Palumbo, Simona Panunzi, Andrea De Gaetano. Qualitative behavior of a family of delaydifferential models of the GlucoseInsulin system. Discrete & Continuous Dynamical Systems  B, 2007, 7 (2) : 399424. doi: 10.3934/dcdsb.2007.7.399 
[9] 
Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems  A, 2013, 33 (6) : 23692387. doi: 10.3934/dcds.2013.33.2369 
[10] 
Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 20312068. doi: 10.3934/cpaa.2013.12.2031 
[11] 
Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 11051117. doi: 10.3934/cpaa.2014.13.1105 
[12] 
Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 15291542. doi: 10.3934/cpaa.2010.9.1529 
[13] 
Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems  A, 2018, 38 (7) : 36373661. doi: 10.3934/dcds.2018157 
[14] 
Fábio Natali, Ademir Pastor. Orbital stability of periodic waves for the KleinGordonSchrödinger system. Discrete & Continuous Dynamical Systems  A, 2011, 31 (1) : 221238. doi: 10.3934/dcds.2011.31.221 
[15] 
BaoZhu Guo, LiMing Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689694. doi: 10.3934/mbe.2011.8.689 
[16] 
Cemil Tunç. Stability, boundedness and uniform boundedness of solutions of nonlinear delay differential equations. Conference Publications, 2011, 2011 (Special) : 13951403. doi: 10.3934/proc.2011.2011.1395 
[17] 
Rafael Ortega. Stability and index of periodic solutions of a nonlinear telegraph equation. Communications on Pure & Applied Analysis, 2005, 4 (4) : 823837. doi: 10.3934/cpaa.2005.4.823 
[18] 
Sevdzhan Hakkaev. Orbital stability of solitary waves of the SchrödingerBoussinesq equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 10431050. doi: 10.3934/cpaa.2007.6.1043 
[19] 
Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with timevarying coefficients. Discrete & Continuous Dynamical Systems  A, 2017, 37 (12) : 61236138. doi: 10.3934/dcds.2017263 
[20] 
Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete & Continuous Dynamical Systems  A, 2013, 33 (7) : 31093134. doi: 10.3934/dcds.2013.33.3109 
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]