American Institute of Mathematical Sciences

Advanced
March  2010, 13(2): 327-345. doi: 10.3934/dcdsb.2010.13.327

Stability implications of delay distribution for first-order and second-order systems

Gábor Kiss 1, and  Bernd Krauskopf 2,

1. 

Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom
2. 

Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR

Received  February 2009 Revised  August 2009 Published  December 2009

In application areas, such as biology, physics and engineering, delays arise naturally because of the time it takes for the system to react to internal or external events. Often the associated mathematical model features more than one delay that are then weighted by some distribution function. This paper considers the effect of delay distribution on the asymptotic stability of the zero solution of functional differential equations - the corresponding mathematical models. We first show that the asymptotic stability of the zero solution of a first-order scalar equation with symmetrically distributed delays follows from the stability of the corresponding equation where the delay is fixed and given by the mean of the distribution. This result completes a proof of a stability condition in [Bernard, S., Bélair, J. and Mackey, M. C. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete Contin. Dyn. Syst. Ser. B, 1(2):233-256, 2001], which was motivated in turn by an application from biology. We also discuss the corresponding case of second-order scalar delay differential equations, because they arise in physical systems that involve oscillating components. An example shows that it is not possible to give a general result for the second-order case. Namely, the boundaries of the stability regions of the distributed-delay equation and of the mean-delay equation may intersect, even if the distribution is symmetric.
Keywords: Hybrid testing., Differential Equations, Distributed Delay.
Mathematics Subject Classification: Primary: 34K20, 34K06; Secondary: 92C37, 62P3.
Citation: Gábor Kiss, Bernd Krauskopf. Stability implications of delay distribution for first-order and second-order systems. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 327-345. doi: 10.3934/dcdsb.2010.13.327
[1]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[2]

Zhenyu Lu, Junhao Hu, Xuerong Mao. Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4099-4116. doi: 10.3934/dcdsb.2019052
[3]

Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062
[4]

Wei Mao, Yanan Jiang, Liangjian Hu, Xuerong Mao. Stabilization by intermittent control for hybrid stochastic differential delay equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021055
[5]

Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[6]

Samuel Bernard, Jacques Bélair, Michael C Mackey. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 233-256. doi: 10.3934/dcdsb.2001.1.233
[7]

Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35
[8]

Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041
[9]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268
[10]

Michael Dellnitz, Mirko Hessel-Von Molo, Adrian Ziessler. On the computation of attractors for delay differential equations. Journal of Computational Dynamics, 2016, 3 (1) : 93-112. doi: 10.3934/jcd.2016005
[11]

Hermann Brunner, Stefano Maset. Time transformations for delay differential equations. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 751-775. doi: 10.3934/dcds.2009.25.751
[12]

Klaudiusz Wójcik, Piotr Zgliczyński. Topological horseshoes and delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 827-852. doi: 10.3934/dcds.2005.12.827
[13]

Serhiy Yanchuk, Leonhard Lücken, Matthias Wolfrum, Alexander Mielke. Spectrum and amplitude equations for scalar delay-differential equations with large delay. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 537-553. doi: 10.3934/dcds.2015.35.537
[14]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[15]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215
[16]

Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633
[17]

Sana Netchaoui, Mohamed Ali Hammami, Tomás Caraballo. Pullback exponential attractors for differential equations with delay. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1345-1358. doi: 10.3934/dcdss.2020367
[18]

Eduardo Liz, Manuel Pinto, Gonzalo Robledo, Sergei Trofimchuk, Victor Tkachenko. Wright type delay differential equations with negative Schwarzian. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 309-321. doi: 10.3934/dcds.2003.9.309
[19]

Igor Chueshov, Michael Scheutzow. Invariance and monotonicity for stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1533-1554. doi: 10.3934/dcdsb.2013.18.1533
[20]

C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (188)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences