-
Previous Article
Energy cascades for NLS on the torus
- DCDS Home
- This Issue
-
Next Article
Density of orbits in laminations and the space of critical portraits
Absolute and delay-dependent stability of equations with a distributed delay
1. | Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4 |
2. | Department of Mathematics, Moscow PF University, Miklukho-Maklaya str. 6, Moscow 117198, Russian Federation |
References:
[1] |
L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients, Comput. Math. Appl., 51 (2006), 1-16.
doi: 10.1016/j.camwa.2005.09.001. |
[2] |
L. Berezansky and E. Braverman, On existence and attractivity of periodic solutions for the hematopoiesis equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13B (2006), suppl., 103-116. |
[3] |
L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Math. Comput. Modelling, 48 (2008), 287-304.
doi: 10.1016/j.mcm.2007.10.003. |
[4] |
L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[5] |
N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dyn. Nat. Soc., 2007, Art. ID 92959, 25 pp. |
[6] |
E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with a distributed delay, Can. Appl. Math. Quart., 14 (2006), 107-128. |
[7] |
W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.
doi: 10.1017/S030500410002990X. |
[8] |
C. Corduneanu, "Functional Equations with Causal Operators," Stability and Control: Theory, Methods and Applications, 16, Taylor & Francis, London, 2002. |
[9] |
K. Gopalsamy, N. Bantsur and S. Trofimchuk, A note on global attractivity in models of hematopoiesis, Ukrainian Math. J., 50 (1998), 3-12.
doi: 10.1007/BF02514684. |
[10] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[11] |
I. Győri and G. Ladas, "Oscillation Theory of Delay Differential Equations. With Applications," Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. |
[12] |
I. Győri and S. Trofimchuk, Global attractivity in $x^{\'}(t)= -\delta x(t) +p f(x(t-h))$, Dynam. Syst. Appl., 8 (1999), 197-210. |
[13] |
K. P. Hadeler and J. Tomiuk, Periodic solutions of difference-differential equations, Arch. Rat. Mech. Anal., 65 (1977), 87-95.
doi: 10.1007/BF00289359. |
[14] |
A. F. Ivanov, On global stability in a nonlinear discrete model, Nonlinear Anal., 23 (1994), 1383-1389.
doi: 10.1016/0362-546X(94)90133-3. |
[15] |
A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in "Dynamics Reported: Expositions in Dynamical Systems," Dynam. Rep. Expositions Dynam. Systems (New Series), 1, Springer, Berlin, (1992), 164-224. |
[16] |
G. Karakostas, Ch. Philos and Y. Sficas, Stable steady state of some population models, J. Dynam. Diff. Eq., 4 (1992), 161-190. |
[17] |
T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[18] |
T. Krisztin, H.-O. Walther and J. Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," Fields Inst. Monogr., 11, AMS, Providence, RI, 1999. |
[19] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. |
[20] |
I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Modelling, 35 (2002), 295-301. |
[21] |
M. R. S. Kulenović, G. Ladas and Y. Sficas, Global attractivity in Nicholson's blowflies, Appl. Anal., 43 (1992), 109-124. |
[22] |
B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Am. Math. Soc., 351 (1999), 901-945.
doi: 10.1090/S0002-9947-99-02351-X. |
[23] |
E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199.
doi: 10.3934/dcdsb.2007.7.191. |
[24] |
E. Liz, C. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential Integral Equations, 15 (2002), 875-896. |
[25] |
E. Liz, M. Pinto, V. Tkachenko and S. Tromichuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math, 63 (2005), 56-70. |
[26] |
E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete and Continuous Dynamical Systems, 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[27] |
E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622.
doi: 10.1137/S0036141001399222. |
[28] |
E. Liz, E. Trofimchuk and S. Trofimchuk, Mackey-Glass type delay differential equations near the boundary of absolute stability, J. Math. Anal. Appl., 275 (2002), 747-760.
doi: 10.1016/S0022-247X(02)00416-X. |
[29] |
J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first-order nonlinear differential delay equations, Chaos, 3 (1993), 167-176.
doi: 10.1063/1.165982. |
[30] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[31] |
J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl. (4), 145 (1986), 33-128.
doi: 10.1007/BF01790539. |
[32] |
J. Mallet-Paret and R. Nussbaum, A differential-delay equation arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249-292.
doi: 10.1137/0520019. |
[33] |
J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[34] |
A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool., 2 (1954), 9-65.
doi: 10.1071/ZO9540009. |
[35] |
G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669. |
[36] |
S. H. Saker, Oscillation and global attractivity in hematopoiesis model with delay time, Appl. Math. Comput., 136 (2003), 241-250.
doi: 10.1016/S0096-3003(02)00035-8. |
[37] |
D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.
doi: 10.1137/0135020. |
show all references
References:
[1] |
L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients, Comput. Math. Appl., 51 (2006), 1-16.
doi: 10.1016/j.camwa.2005.09.001. |
[2] |
L. Berezansky and E. Braverman, On existence and attractivity of periodic solutions for the hematopoiesis equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13B (2006), suppl., 103-116. |
[3] |
L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Math. Comput. Modelling, 48 (2008), 287-304.
doi: 10.1016/j.mcm.2007.10.003. |
[4] |
L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[5] |
N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dyn. Nat. Soc., 2007, Art. ID 92959, 25 pp. |
[6] |
E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with a distributed delay, Can. Appl. Math. Quart., 14 (2006), 107-128. |
[7] |
W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43.
doi: 10.1017/S030500410002990X. |
[8] |
C. Corduneanu, "Functional Equations with Causal Operators," Stability and Control: Theory, Methods and Applications, 16, Taylor & Francis, London, 2002. |
[9] |
K. Gopalsamy, N. Bantsur and S. Trofimchuk, A note on global attractivity in models of hematopoiesis, Ukrainian Math. J., 50 (1998), 3-12.
doi: 10.1007/BF02514684. |
[10] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[11] |
I. Győri and G. Ladas, "Oscillation Theory of Delay Differential Equations. With Applications," Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991. |
[12] |
I. Győri and S. Trofimchuk, Global attractivity in $x^{\'}(t)= -\delta x(t) +p f(x(t-h))$, Dynam. Syst. Appl., 8 (1999), 197-210. |
[13] |
K. P. Hadeler and J. Tomiuk, Periodic solutions of difference-differential equations, Arch. Rat. Mech. Anal., 65 (1977), 87-95.
doi: 10.1007/BF00289359. |
[14] |
A. F. Ivanov, On global stability in a nonlinear discrete model, Nonlinear Anal., 23 (1994), 1383-1389.
doi: 10.1016/0362-546X(94)90133-3. |
[15] |
A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in "Dynamics Reported: Expositions in Dynamical Systems," Dynam. Rep. Expositions Dynam. Systems (New Series), 1, Springer, Berlin, (1992), 164-224. |
[16] |
G. Karakostas, Ch. Philos and Y. Sficas, Stable steady state of some population models, J. Dynam. Diff. Eq., 4 (1992), 161-190. |
[17] |
T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[18] |
T. Krisztin, H.-O. Walther and J. Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," Fields Inst. Monogr., 11, AMS, Providence, RI, 1999. |
[19] |
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. |
[20] |
I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Modelling, 35 (2002), 295-301. |
[21] |
M. R. S. Kulenović, G. Ladas and Y. Sficas, Global attractivity in Nicholson's blowflies, Appl. Anal., 43 (1992), 109-124. |
[22] |
B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Am. Math. Soc., 351 (1999), 901-945.
doi: 10.1090/S0002-9947-99-02351-X. |
[23] |
E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199.
doi: 10.3934/dcdsb.2007.7.191. |
[24] |
E. Liz, C. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential Integral Equations, 15 (2002), 875-896. |
[25] |
E. Liz, M. Pinto, V. Tkachenko and S. Tromichuk, A global stability criterion for a family of delayed population models, Quart. Appl. Math, 63 (2005), 56-70. |
[26] |
E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete and Continuous Dynamical Systems, 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[27] |
E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622.
doi: 10.1137/S0036141001399222. |
[28] |
E. Liz, E. Trofimchuk and S. Trofimchuk, Mackey-Glass type delay differential equations near the boundary of absolute stability, J. Math. Anal. Appl., 275 (2002), 747-760.
doi: 10.1016/S0022-247X(02)00416-X. |
[29] |
J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first-order nonlinear differential delay equations, Chaos, 3 (1993), 167-176.
doi: 10.1063/1.165982. |
[30] |
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[31] |
J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl. (4), 145 (1986), 33-128.
doi: 10.1007/BF01790539. |
[32] |
J. Mallet-Paret and R. Nussbaum, A differential-delay equation arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249-292.
doi: 10.1137/0520019. |
[33] |
J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[34] |
A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool., 2 (1954), 9-65.
doi: 10.1071/ZO9540009. |
[35] |
G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669. |
[36] |
S. H. Saker, Oscillation and global attractivity in hematopoiesis model with delay time, Appl. Math. Comput., 136 (2003), 241-250.
doi: 10.1016/S0096-3003(02)00035-8. |
[37] |
D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267.
doi: 10.1137/0135020. |
[1] |
Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191 |
[2] |
Valentin Duruisseaux, Antony R. Humphries. Bistability, bifurcations and chaos in the Mackey-Glass equation. Journal of Computational Dynamics, 2022 doi: 10.3934/jcd.2022009 |
[3] |
Ahmed Elhassanein. Complex dynamics of a forced discretized version of the Mackey-Glass delay differential equation. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 93-105. doi: 10.3934/dcdsb.2015.20.93 |
[4] |
Xiaoyuan Chang, Junping Shi. Bistable and oscillatory dynamics of Nicholson's blowflies equation with Allee effect. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021242 |
[5] |
Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150 |
[6] |
Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control and Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 |
[7] |
Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 |
[8] |
Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305 |
[9] |
Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268 |
[10] |
Sze-Bi Hsu, Ming-Chia Li, Weishi Liu, Mikhail Malkin. Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1465-1492. doi: 10.3934/dcds.2003.9.1465 |
[11] |
Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014 |
[12] |
Tarik Mohammed Touaoula. Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2473-2490. doi: 10.3934/cpaa.2020108 |
[13] |
G. A. Enciso, E. D. Sontag. Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 549-578. doi: 10.3934/dcds.2006.14.549 |
[14] |
Chadi Nour. Construction of solutions to a global Eikonal equation. Conference Publications, 2007, 2007 (Special) : 779-783. doi: 10.3934/proc.2007.2007.779 |
[15] |
Kazuki Himoto, Hideaki Matsunaga. The limits of solutions of a linear delay integral equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3033-3048. doi: 10.3934/dcdsb.2020050 |
[16] |
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 |
[17] |
Szandra Beretka, Gabriella Vas. Stable periodic solutions for Nazarenko's equation. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3257-3281. doi: 10.3934/cpaa.2020144 |
[18] |
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 |
[19] |
Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361 |
[20] |
Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]