2015, 20(7): 1855-1876. doi: 10.3934/dcdsb.2015.20.1855

Optimal linear stability condition for scalar differential equations with distributed delay

1. 

Université de Lyon; CNRS UMR 5208, Université Lyon 1; Institut Camille Jordan, INRIA Team Dracula, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France

Received  April 2014 Revised  February 2015 Published  July 2015

Linear scalar differential equations with distributed delays appear in the study of the local stability of nonlinear differential equations with feedback, which are common in biology and physics. Negative feedback loops tend to promote oscillations around steady states, and their stability depends on the particular shape of the delay distribution. Since in applications the mean delay is often the only reliable information available about the distribution, it is desirable to find conditions for stability that are independent from the shape of the distribution. We show here that for a given mean delay, the linear equation with distributed delay is asymptotically stable if the associated differential equation with a discrete delay is asymptotically stable. We illustrate this criterion on a compartment model of hematopoietic cell dynamics to obtain sufficient conditions for stability.
Citation: 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
References:
[1]

M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia,, SIAM J. Appl. Math., 65 (2005), 1328. doi: 10.1137/040604698.

[2]

R. Anderson, Geometric and probabilistic stability criteria for delay systems,, Math. Biosci., 105 (1991), 81. doi: 10.1016/0025-5564(91)90049-O.

[3]

R. Anderson, Intrinsic parameters and stability of differential-delay equations,, J. Math. Anal. Appl., 163 (1992), 184. doi: 10.1016/0022-247X(92)90287-N.

[4]

R. Apostu and M. Mackey, Understanding cyclical thrombocytopenia: A mathematical modeling approach,, J. Theor. Biol., 251 (2008), 297. doi: 10.1016/j.jtbi.2007.11.029.

[5]

F. Atay, Distributed delays facilitate amplitude death of coupled oscillators,, Phys. Rev. Lett., 91 (2003). doi: 10.1103/PhysRevLett.91.094101.

[6]

F. Atay, Delayed feedback control near Hopf bifurcation,, Discrete Contin. Dynam. Systems Ser. S, 1 (2008), 197. doi: 10.3934/dcdss.2008.1.197.

[7]

S. Basu, A. Dunn and A. Ward, G-CSF: Function and modes of action,, Int. J. Mol. Med., 10 (2002), 3.

[8]

J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis,, Math. Biosci., 128 (1995), 317. doi: 10.1016/0025-5564(94)00078-E.

[9]

R. Bellman and K. Cooke, Differential-Difference Equations,, Academic press, (1963).

[10]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144. doi: 10.1137/S0036141000376086.

[11]

L. Berezansky and E. Braverman, Stability of linear differential equations with a distributed delay,, Comm. Pure Appl. Math., 10 (2011), 1361. doi: 10.3934/cpaa.2011.10.1361.

[12]

L. Berezansky and E. Braverman, Stability of equations with a distributed delay, monotone production and nonlinear mortality,, Nonlinearity, 26 (2013), 2833. doi: 10.1088/0951-7715/26/10/2833.

[13]

S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay,, Discrete Contin. Dynam. Systems Ser. B, 1 (2001), 233. doi: 10.3934/dcdsb.2001.1.233.

[14]

S. Bernard, J. Belair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling,, J. Theor. Biol., 223 (2003), 283. doi: 10.1016/S0022-5193(03)00090-0.

[15]

S. Bernard, B. Čajavec, L. Pujo-Menjouet, M. Mackey and H. Herzel, Modelling transcriptional feedback loops: The role of Gro/TLE1 in Hes1 oscillations,, Philos. Trans. R. Soc. London, 364 (2006), 1155. doi: 10.1098/rsta.2006.1761.

[16]

F. Boese, The stability chart for the linearized cushing equation with a discrete delay and gamma-distributed delays,, J. Math. Anal. Appl., 140 (1989), 510. doi: 10.1016/0022-247X(89)90081-4.

[17]

S. Campbell, Time delays in neural systems,, in Handbook of Brain Connectivity, (2007), 65. doi: 10.1007/978-3-540-71512-2_2.

[18]

S. Campbell and R. Jessop, Approximating the stability region for a differential equation with a distributed delay,, Math. Mod. Nat. Phenom., 4 (2009), 1. doi: 10.1051/mmnp/20094201.

[19]

C. Colijn and M. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia,, J. Theor. Biol., 237 (2005), 117. doi: 10.1016/j.jtbi.2005.03.033.

[20]

C. Colijn and M. Mackey, A mathematical model of hematopoiesis - II. Cyclical neutropenia,, J. Theor. Biol., 237 (2005), 133. doi: 10.1016/j.jtbi.2005.03.034.

[21]

C. Colijn and M. Mackey, Bifurcation and bistability in a model of hematopoietic regulation,, SIAM J. App. Dynam. Sys., 6 (2007), 378. doi: 10.1137/050640072.

[22]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches,, J. Math. Anal. Appl., 86 (1982), 592. doi: 10.1016/0022-247X(82)90243-8.

[23]

F. Crauste, Stability and Hopf bifurcation for a first-order delay differential equation with distributed delay,, in Complex Time-Delay Systems, (2010), 263.

[24]

T. Erneux, Applied Delay Differential Equations,, Springer Verlag, (2009).

[25]

C. Eurich, A. Thiel and L. Fahse, Distributed delays stabilize ecological feedback systems,, Phys. Rev. Lett., 94 (2005). doi: 10.1103/PhysRevLett.94.158104.

[26]

J. Hale, Functional differential equations with infinite delays,, J. Math. Anal. Appl., 48 (1974), 276. doi: 10.1016/0022-247X(74)90233-9.

[27]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac, 21 (1978), 11.

[28]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations,, Berlin: Springer, (1993). doi: 10.1007/978-1-4612-4342-7.

[29]

N. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation,, J. Lond. Math. Soc., 25 (1950), 226.

[30]

C. Huang and S. Vandewalle, An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays,, SIAM J. Sci. Comput., 25 (2004), 1608. doi: 10.1137/S1064827502409717.

[31]

G. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[32]

K. Kaushansky, The molecular mechanisms that control thrombopoiesis,, J Clin Invest, 115 (2005), 3339. doi: 10.1172/JCI26674.

[33]

G. Kiss and B. Krauskopf, Stability implications of delay distribution for first-order and second-order systems,, Discrete Contin. Dynam. Systems Ser. B, 13 (2010), 327. doi: 10.3934/dcdsb.2010.13.327.

[34]

M. Koury and M. Bondurant, Erythropoietin retards DNA breakdown and prevents programmed death in erythroid progenitor cells,, Science, 248 (1990), 378. doi: 10.1126/science.2326648.

[35]

T. Krisztin, Stability for functional differential equations and some variational problems,, Tohoku Math. J, 42 (1990), 407. doi: 10.2748/tmj/1178227618.

[36]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Pr, (1993).

[37]

Y. Kuang, Nonoccurrence of stability switching in systems of differential equations with distributed delays,, Quart. Appl. Math., 52 (1994), 569.

[38]

J. Lei and M. Mackey, Multistability in an age-structured model of hematopoeisis: Cyclical neutropenia,, J. Theor. Biol., 270 (2011), 143. doi: 10.1016/j.jtbi.2010.11.024.

[39]

N. MacDonald, Biological Delay Systems: Linear Stability Theory,, Cambridge Studies in Mathematical Biology, (1989).

[40]

M. C. Mackey, Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis,, Blood, 51 (1978), 941.

[41]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326.

[42]

U. Meyer, J. Shao, S. Chakrabarty, S. Brandt, H. Luksch and R. Wessel, Distributed delays stabilize neural feedback systems,, Biol. Cybern., 99 (2008), 79. doi: 10.1007/s00422-008-0239-8.

[43]

R. Miyazaki, Characteristic equation and asymptotic behavior of delay-differential equation,, Funkcial. Ekvac., 40 (1997), 471.

[44]

N. Monk, Oscillatory expression of Hes1, p53, and NF-$\kappa$B driven by transcriptional time delays,, Curr. Biol., 13 (2003), 1409. doi: 10.1016/S0960-9822(03)00494-9.

[45]

H. Ozbay, C. Bonnet and J. Clairambault, Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics,, in Decision and Control, (2008), 2050. doi: 10.1109/CDC.2008.4738654.

[46]

K. Rateitschak and O. Wolkenhauer, Intracellular delay limits cyclic changes in gene expression,, Math. Biosci., 205 (2007), 163. doi: 10.1016/j.mbs.2006.08.010.

[47]

O. Solomon and E. Fridman, New stability conditions for systems with distributed delays,, Automatica J. IFAC, 49 (2013), 3467. doi: 10.1016/j.automatica.2013.08.025.

[48]

G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions,, Longman Scientific & Technical New York, (1989).

[49]

T. Stiehl and A. Marciniak-Czochra, Characterization of stem cells using mathematical models of multistage cell lineages,, Math. Comp. Models., 53 (2011), 1505. doi: 10.1016/j.mcm.2010.03.057.

[50]

X. Tang, Asymptotic behavior of a differential equation with distributed delays,, J. Math. Anal. Appl., 301 (2005), 313. doi: 10.1016/j.jmaa.2004.07.023.

show all references

References:
[1]

M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia,, SIAM J. Appl. Math., 65 (2005), 1328. doi: 10.1137/040604698.

[2]

R. Anderson, Geometric and probabilistic stability criteria for delay systems,, Math. Biosci., 105 (1991), 81. doi: 10.1016/0025-5564(91)90049-O.

[3]

R. Anderson, Intrinsic parameters and stability of differential-delay equations,, J. Math. Anal. Appl., 163 (1992), 184. doi: 10.1016/0022-247X(92)90287-N.

[4]

R. Apostu and M. Mackey, Understanding cyclical thrombocytopenia: A mathematical modeling approach,, J. Theor. Biol., 251 (2008), 297. doi: 10.1016/j.jtbi.2007.11.029.

[5]

F. Atay, Distributed delays facilitate amplitude death of coupled oscillators,, Phys. Rev. Lett., 91 (2003). doi: 10.1103/PhysRevLett.91.094101.

[6]

F. Atay, Delayed feedback control near Hopf bifurcation,, Discrete Contin. Dynam. Systems Ser. S, 1 (2008), 197. doi: 10.3934/dcdss.2008.1.197.

[7]

S. Basu, A. Dunn and A. Ward, G-CSF: Function and modes of action,, Int. J. Mol. Med., 10 (2002), 3.

[8]

J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis,, Math. Biosci., 128 (1995), 317. doi: 10.1016/0025-5564(94)00078-E.

[9]

R. Bellman and K. Cooke, Differential-Difference Equations,, Academic press, (1963).

[10]

E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters,, SIAM J. Math. Anal., 33 (2002), 1144. doi: 10.1137/S0036141000376086.

[11]

L. Berezansky and E. Braverman, Stability of linear differential equations with a distributed delay,, Comm. Pure Appl. Math., 10 (2011), 1361. doi: 10.3934/cpaa.2011.10.1361.

[12]

L. Berezansky and E. Braverman, Stability of equations with a distributed delay, monotone production and nonlinear mortality,, Nonlinearity, 26 (2013), 2833. doi: 10.1088/0951-7715/26/10/2833.

[13]

S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay,, Discrete Contin. Dynam. Systems Ser. B, 1 (2001), 233. doi: 10.3934/dcdsb.2001.1.233.

[14]

S. Bernard, J. Belair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling,, J. Theor. Biol., 223 (2003), 283. doi: 10.1016/S0022-5193(03)00090-0.

[15]

S. Bernard, B. Čajavec, L. Pujo-Menjouet, M. Mackey and H. Herzel, Modelling transcriptional feedback loops: The role of Gro/TLE1 in Hes1 oscillations,, Philos. Trans. R. Soc. London, 364 (2006), 1155. doi: 10.1098/rsta.2006.1761.

[16]

F. Boese, The stability chart for the linearized cushing equation with a discrete delay and gamma-distributed delays,, J. Math. Anal. Appl., 140 (1989), 510. doi: 10.1016/0022-247X(89)90081-4.

[17]

S. Campbell, Time delays in neural systems,, in Handbook of Brain Connectivity, (2007), 65. doi: 10.1007/978-3-540-71512-2_2.

[18]

S. Campbell and R. Jessop, Approximating the stability region for a differential equation with a distributed delay,, Math. Mod. Nat. Phenom., 4 (2009), 1. doi: 10.1051/mmnp/20094201.

[19]

C. Colijn and M. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia,, J. Theor. Biol., 237 (2005), 117. doi: 10.1016/j.jtbi.2005.03.033.

[20]

C. Colijn and M. Mackey, A mathematical model of hematopoiesis - II. Cyclical neutropenia,, J. Theor. Biol., 237 (2005), 133. doi: 10.1016/j.jtbi.2005.03.034.

[21]

C. Colijn and M. Mackey, Bifurcation and bistability in a model of hematopoietic regulation,, SIAM J. App. Dynam. Sys., 6 (2007), 378. doi: 10.1137/050640072.

[22]

K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches,, J. Math. Anal. Appl., 86 (1982), 592. doi: 10.1016/0022-247X(82)90243-8.

[23]

F. Crauste, Stability and Hopf bifurcation for a first-order delay differential equation with distributed delay,, in Complex Time-Delay Systems, (2010), 263.

[24]

T. Erneux, Applied Delay Differential Equations,, Springer Verlag, (2009).

[25]

C. Eurich, A. Thiel and L. Fahse, Distributed delays stabilize ecological feedback systems,, Phys. Rev. Lett., 94 (2005). doi: 10.1103/PhysRevLett.94.158104.

[26]

J. Hale, Functional differential equations with infinite delays,, J. Math. Anal. Appl., 48 (1974), 276. doi: 10.1016/0022-247X(74)90233-9.

[27]

J. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac, 21 (1978), 11.

[28]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations,, Berlin: Springer, (1993). doi: 10.1007/978-1-4612-4342-7.

[29]

N. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation,, J. Lond. Math. Soc., 25 (1950), 226.

[30]

C. Huang and S. Vandewalle, An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays,, SIAM J. Sci. Comput., 25 (2004), 1608. doi: 10.1137/S1064827502409717.

[31]

G. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[32]

K. Kaushansky, The molecular mechanisms that control thrombopoiesis,, J Clin Invest, 115 (2005), 3339. doi: 10.1172/JCI26674.

[33]

G. Kiss and B. Krauskopf, Stability implications of delay distribution for first-order and second-order systems,, Discrete Contin. Dynam. Systems Ser. B, 13 (2010), 327. doi: 10.3934/dcdsb.2010.13.327.

[34]

M. Koury and M. Bondurant, Erythropoietin retards DNA breakdown and prevents programmed death in erythroid progenitor cells,, Science, 248 (1990), 378. doi: 10.1126/science.2326648.

[35]

T. Krisztin, Stability for functional differential equations and some variational problems,, Tohoku Math. J, 42 (1990), 407. doi: 10.2748/tmj/1178227618.

[36]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics,, Academic Pr, (1993).

[37]

Y. Kuang, Nonoccurrence of stability switching in systems of differential equations with distributed delays,, Quart. Appl. Math., 52 (1994), 569.

[38]

J. Lei and M. Mackey, Multistability in an age-structured model of hematopoeisis: Cyclical neutropenia,, J. Theor. Biol., 270 (2011), 143. doi: 10.1016/j.jtbi.2010.11.024.

[39]

N. MacDonald, Biological Delay Systems: Linear Stability Theory,, Cambridge Studies in Mathematical Biology, (1989).

[40]

M. C. Mackey, Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis,, Blood, 51 (1978), 941.

[41]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326.

[42]

U. Meyer, J. Shao, S. Chakrabarty, S. Brandt, H. Luksch and R. Wessel, Distributed delays stabilize neural feedback systems,, Biol. Cybern., 99 (2008), 79. doi: 10.1007/s00422-008-0239-8.

[43]

R. Miyazaki, Characteristic equation and asymptotic behavior of delay-differential equation,, Funkcial. Ekvac., 40 (1997), 471.

[44]

N. Monk, Oscillatory expression of Hes1, p53, and NF-$\kappa$B driven by transcriptional time delays,, Curr. Biol., 13 (2003), 1409. doi: 10.1016/S0960-9822(03)00494-9.

[45]

H. Ozbay, C. Bonnet and J. Clairambault, Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics,, in Decision and Control, (2008), 2050. doi: 10.1109/CDC.2008.4738654.

[46]

K. Rateitschak and O. Wolkenhauer, Intracellular delay limits cyclic changes in gene expression,, Math. Biosci., 205 (2007), 163. doi: 10.1016/j.mbs.2006.08.010.

[47]

O. Solomon and E. Fridman, New stability conditions for systems with distributed delays,, Automatica J. IFAC, 49 (2013), 3467. doi: 10.1016/j.automatica.2013.08.025.

[48]

G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions,, Longman Scientific & Technical New York, (1989).

[49]

T. Stiehl and A. Marciniak-Czochra, Characterization of stem cells using mathematical models of multistage cell lineages,, Math. Comp. Models., 53 (2011), 1505. doi: 10.1016/j.mcm.2010.03.057.

[50]

X. Tang, Asymptotic behavior of a differential equation with distributed delays,, J. Math. Anal. Appl., 301 (2005), 313. doi: 10.1016/j.jmaa.2004.07.023.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[8]

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

[9]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

Tomás Caraballo, P.E. Kloeden, Pedro Marín-Rubio. Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 177-196. doi: 10.3934/dcds.2007.19.177

[16]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

[17]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[18]

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

[19]

Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23

[20]

Roman Srzednicki. A theorem on chaotic dynamics and its application to differential delay equations. Conference Publications, 2001, 2001 (Special) : 362-365. doi: 10.3934/proc.2001.2001.362

2016 Impact Factor: 0.994

Metrics

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

Other articles
by authors

[Back to Top]