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Regions of stability for a linear differential equation with two rationally dependent delays
1. | Department of Mathematics and Statistics, Nonlinear Dynamical Systems Group, Computational Sciences Research Center, San Diego State University, San Diego, CA 92182-7720, United States |
2. | Department of Mathematics, Grossmont College, El Cajon, CA 92020, United States |
References:
[1] |
J. Bélair, Stability of a differential-delay equation with two time lags,, in Oscillations, (1987), 305.
|
[2] |
J. Bélair and M. Mackey, A model for the regulation of mammalian platelet production,, Ann. N. Y. Acad. Sci., 504 (1987), 280.
doi: 10.1111/j.1749-6632.1987.tb48740.x. |
[3] |
J. Bélair and M. Mackey, Consumer memory and price fluctuations in commodity markets: An integrodifferential model,, J. Dyn. and Diff. Eqns., 1 (1989), 299.
doi: 10.1007/BF01053930. |
[4] |
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. |
[5] |
J. Bélair and S. A. Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation,, SIAM J. Appl. Math., 54 (1994), 1402.
doi: 10.1137/S0036139993248853. |
[6] |
J. Bélair, S. A. Campbell and P. v. d. Driessche, Frustration, stability, and delay-induced oscillations in a neural network model,, SIAM Journal on Applied Mathematics, 56 (1996), 245.
doi: 10.1137/S0036139994274526. |
[7] |
R. Bellman and K. L. Cooke, Differential-Difference Equations,, Lectures in Applied Mathematics, (1963).
|
[8] |
F. G. Boese, The delay-independent stability behaviour of a first order differential-difference equation with two constant lags,, preprint, (1993). Google Scholar |
[9] |
F. G. Boese, A new representation of a stability result of N. D. Hayes,, Z. Angew. Math. Mech., 73 (1993), 117.
doi: 10.1002/zamm.19930730215. |
[10] |
F. G. Boese, Stability in a special class of retarded difference-differential equations with interval-valued parameters,, Journal of Mathematical Analysis and Applications, 181 (1994), 227.
doi: 10.1006/jmaa.1994.1017. |
[11] |
D. M. Bortz, Eigenvalues for two-lag linear delay differential equations,, submitted, (2012). Google Scholar |
[12] |
R. D. Braddock and P. van den Driessche, A population model with two time delays,, in Quantitative Population Dynamics (eds. D. G. Chapman and V. F. Gallucci), (1981). Google Scholar |
[13] |
T. C. Busken, On the Asymptotic Stability of the Zero Solution for a Linear Differential Equation with Two Delays,, Master's Thesis, (2012). Google Scholar |
[14] |
S. A. Campbell and J. Bélair, Analytical and symbolically-assisted investigation of Hopf bifurcations in delay-differential equations,, Proceedings of the G. J. Butler Workshop in Mathematical Biology (Waterloo, 3 (1995), 137.
|
[15] |
K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics,, Math. Biosci., 16 (1973), 75.
doi: 10.1016/0025-5564(73)90046-1. |
[16] |
L. E. El'sgol'ts and S. Norkin, Introduction to the Theory of Differential Equations with Deviating Arguments,, Academic Press, (1977). Google Scholar |
[17] |
T. Elsken, The region of (in)stability of a 2-delay equation is connected,, J. Math. Anal. Appl., 261 (2001), 497.
doi: 10.1006/jmaa.2001.7536. |
[18] |
C. Guzelis and L. O. Chua, Stability analysis of generalized cellular neural networks,, International Journal of Circuit Theory and Applications, 21 (1993), 1.
doi: 10.1002/cta.4490210102. |
[19] |
J. Hale, E. Infante and P. Tsen, Stability in linear delay equations,, J. Math. Anal. Appl., 105 (1985), 533.
doi: 10.1016/0022-247X(85)90068-X. |
[20] |
J. K. Hale, Nonlinear oscillations in equations with delays,, in Nonlinear Oscillations in Biology (Proc. Tenth Summer Sem. Appl. Math., (1978), 157.
|
[21] |
J. K. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations,, J. Math. Anal. Appl., 178 (1993), 344.
doi: 10.1006/jmaa.1993.1312. |
[22] |
J. K. Hale and S. M. Tanaka, Square and pulse waves with two delays,, Journal of Dynamics and Differential Equations, 12 (2000), 1.
doi: 10.1023/A:1009052718531. |
[23] |
G. Haller and G. Stépán, Codimension two bifurcation in an approximate model for delayed robot control,, in Bifurcation and Chaos: Analysis, (1990), 155.
|
[24] |
N. Hayes, Roots of the transcendental equation associated with a certain differential difference equation,, J. London Math. Soc., 25 (1950), 226.
|
[25] |
T. D. Howroyd and A. M. Russell, Cournot oligopoly models with time lags,, J. Math. Econ., 13 (1984), 97.
doi: 10.1016/0304-4068(84)90009-0. |
[26] |
, E. F. Infante,, Personal Communication, (1975). Google Scholar |
[27] |
I. S. Levitskaya, Stability domain of a linear differential equation with two delays,, Comput. Math. Appl., 51 (2006), 153.
doi: 10.1016/j.camwa.2005.05.011. |
[28] |
X. Li, S. Ruan and J. Wei, Stability and bifurcation in delay-differential equations with two delays,, Journal of Mathematical Analysis and Applications, 236 (1999), 254.
doi: 10.1006/jmaa.1999.6418. |
[29] |
N. MacDonald, Cyclical neutropenia; Models with two cell types and two time lags,, in Biomathematics and Cell Kinetics (eds. A. J. Valleron and P. D. M. Macdonald), (1979), 287. Google Scholar |
[30] |
N. MacDonald, An activation-inhibition model of cyclic granulopoiesis in chronic granulocytic leukemia,, Math. Biosci., 54 (1980), 61.
doi: 10.1016/0025-5564(81)90076-6. |
[31] |
M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors,, J. Econ. Theory, 48 (1989), 497.
doi: 10.1016/0022-0531(89)90039-2. |
[32] |
J. M. Mahaffy, P. J. Zak and K. M. Joiner, A Three Parameter Stability Analysis for a Linear Differential Equation with Two Delays,, Technical report, (1993). Google Scholar |
[33] |
J. M. Mahaffy, P. J. Zak and K. M. Joiner, A geometric analysis of stability regions for a linear differential equation with two delays,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 779.
doi: 10.1142/S0218127495000570. |
[34] |
M. Mizuno and K. Ikeda, An unstable mode selection rule: Frustrated optical instability due to two competing boundary conditions,, Physica D, 36 (1989), 327.
doi: 10.1016/0167-2789(89)90088-2. |
[35] |
S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays,, Applied Mathematics and Computation, 135 (2003), 17.
doi: 10.1016/S0096-3003(01)00299-5. |
[36] |
W. W. Murdoch, R. M. Nisbet, S. P. Blythe, W. S. C. Gurney and J. D. Reeve, An invulnerable age class and stability in delay-differential parasitoid-host models,, American Naturalist, 129 (1987), 263.
doi: 10.1086/284634. |
[37] |
R. D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations,, Trans. Amer. Math. Soc., 238 (1978), 139.
doi: 10.1090/S0002-9947-1978-0482913-0. |
[38] |
M. Piotrowska, A remark on the ode with two discrete delays,, Journal of Mathematical Analysis and Applications, 329 (2007), 664.
doi: 10.1016/j.jmaa.2006.06.078. |
[39] |
C. G. Ragazzo and C. P. Malta, Singularity structure of the Hopf bifurcation surface of a differential equation with two delays,, Journal of Dynamics and Differential Equations, 4 (1992), 617.
doi: 10.1007/BF01048262. |
[40] |
J. Ruiz-Claeyssen, Effects of delays on functional differential equations,, J. Diff. Eq., 20 (1976), 404.
doi: 10.1016/0022-0396(76)90117-0. |
[41] |
S. Sakata, Asymptotic stability for a linear system of differential-difference equations,, Funkcial. Ekvac., 41 (1998), 435.
|
[42] |
R. T. Wilsterman, An Analytic and Geometric Approach for Examining the Stability of Linear Differential Equations with Two Delays,, Master's Thesis, (2013). Google Scholar |
[43] |
T. Yoneyama and J. Sugie, On the stability region of differential equations with two delays,, Funkcial. Ekvac., 31 (1988), 233.
|
[44] |
E. Zaron, The Delay Differential Equation: $x'(t) = -ax(t) + bx(t-\tau_1) + cx(t-\tau_2)$,, Technical report, (1987). Google Scholar |
show all references
References:
[1] |
J. Bélair, Stability of a differential-delay equation with two time lags,, in Oscillations, (1987), 305.
|
[2] |
J. Bélair and M. Mackey, A model for the regulation of mammalian platelet production,, Ann. N. Y. Acad. Sci., 504 (1987), 280.
doi: 10.1111/j.1749-6632.1987.tb48740.x. |
[3] |
J. Bélair and M. Mackey, Consumer memory and price fluctuations in commodity markets: An integrodifferential model,, J. Dyn. and Diff. Eqns., 1 (1989), 299.
doi: 10.1007/BF01053930. |
[4] |
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. |
[5] |
J. Bélair and S. A. Campbell, Stability and bifurcations of equilibria in a multiple-delayed differential equation,, SIAM J. Appl. Math., 54 (1994), 1402.
doi: 10.1137/S0036139993248853. |
[6] |
J. Bélair, S. A. Campbell and P. v. d. Driessche, Frustration, stability, and delay-induced oscillations in a neural network model,, SIAM Journal on Applied Mathematics, 56 (1996), 245.
doi: 10.1137/S0036139994274526. |
[7] |
R. Bellman and K. L. Cooke, Differential-Difference Equations,, Lectures in Applied Mathematics, (1963).
|
[8] |
F. G. Boese, The delay-independent stability behaviour of a first order differential-difference equation with two constant lags,, preprint, (1993). Google Scholar |
[9] |
F. G. Boese, A new representation of a stability result of N. D. Hayes,, Z. Angew. Math. Mech., 73 (1993), 117.
doi: 10.1002/zamm.19930730215. |
[10] |
F. G. Boese, Stability in a special class of retarded difference-differential equations with interval-valued parameters,, Journal of Mathematical Analysis and Applications, 181 (1994), 227.
doi: 10.1006/jmaa.1994.1017. |
[11] |
D. M. Bortz, Eigenvalues for two-lag linear delay differential equations,, submitted, (2012). Google Scholar |
[12] |
R. D. Braddock and P. van den Driessche, A population model with two time delays,, in Quantitative Population Dynamics (eds. D. G. Chapman and V. F. Gallucci), (1981). Google Scholar |
[13] |
T. C. Busken, On the Asymptotic Stability of the Zero Solution for a Linear Differential Equation with Two Delays,, Master's Thesis, (2012). Google Scholar |
[14] |
S. A. Campbell and J. Bélair, Analytical and symbolically-assisted investigation of Hopf bifurcations in delay-differential equations,, Proceedings of the G. J. Butler Workshop in Mathematical Biology (Waterloo, 3 (1995), 137.
|
[15] |
K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics,, Math. Biosci., 16 (1973), 75.
doi: 10.1016/0025-5564(73)90046-1. |
[16] |
L. E. El'sgol'ts and S. Norkin, Introduction to the Theory of Differential Equations with Deviating Arguments,, Academic Press, (1977). Google Scholar |
[17] |
T. Elsken, The region of (in)stability of a 2-delay equation is connected,, J. Math. Anal. Appl., 261 (2001), 497.
doi: 10.1006/jmaa.2001.7536. |
[18] |
C. Guzelis and L. O. Chua, Stability analysis of generalized cellular neural networks,, International Journal of Circuit Theory and Applications, 21 (1993), 1.
doi: 10.1002/cta.4490210102. |
[19] |
J. Hale, E. Infante and P. Tsen, Stability in linear delay equations,, J. Math. Anal. Appl., 105 (1985), 533.
doi: 10.1016/0022-247X(85)90068-X. |
[20] |
J. K. Hale, Nonlinear oscillations in equations with delays,, in Nonlinear Oscillations in Biology (Proc. Tenth Summer Sem. Appl. Math., (1978), 157.
|
[21] |
J. K. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations,, J. Math. Anal. Appl., 178 (1993), 344.
doi: 10.1006/jmaa.1993.1312. |
[22] |
J. K. Hale and S. M. Tanaka, Square and pulse waves with two delays,, Journal of Dynamics and Differential Equations, 12 (2000), 1.
doi: 10.1023/A:1009052718531. |
[23] |
G. Haller and G. Stépán, Codimension two bifurcation in an approximate model for delayed robot control,, in Bifurcation and Chaos: Analysis, (1990), 155.
|
[24] |
N. Hayes, Roots of the transcendental equation associated with a certain differential difference equation,, J. London Math. Soc., 25 (1950), 226.
|
[25] |
T. D. Howroyd and A. M. Russell, Cournot oligopoly models with time lags,, J. Math. Econ., 13 (1984), 97.
doi: 10.1016/0304-4068(84)90009-0. |
[26] |
, E. F. Infante,, Personal Communication, (1975). Google Scholar |
[27] |
I. S. Levitskaya, Stability domain of a linear differential equation with two delays,, Comput. Math. Appl., 51 (2006), 153.
doi: 10.1016/j.camwa.2005.05.011. |
[28] |
X. Li, S. Ruan and J. Wei, Stability and bifurcation in delay-differential equations with two delays,, Journal of Mathematical Analysis and Applications, 236 (1999), 254.
doi: 10.1006/jmaa.1999.6418. |
[29] |
N. MacDonald, Cyclical neutropenia; Models with two cell types and two time lags,, in Biomathematics and Cell Kinetics (eds. A. J. Valleron and P. D. M. Macdonald), (1979), 287. Google Scholar |
[30] |
N. MacDonald, An activation-inhibition model of cyclic granulopoiesis in chronic granulocytic leukemia,, Math. Biosci., 54 (1980), 61.
doi: 10.1016/0025-5564(81)90076-6. |
[31] |
M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors,, J. Econ. Theory, 48 (1989), 497.
doi: 10.1016/0022-0531(89)90039-2. |
[32] |
J. M. Mahaffy, P. J. Zak and K. M. Joiner, A Three Parameter Stability Analysis for a Linear Differential Equation with Two Delays,, Technical report, (1993). Google Scholar |
[33] |
J. M. Mahaffy, P. J. Zak and K. M. Joiner, A geometric analysis of stability regions for a linear differential equation with two delays,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5 (1995), 779.
doi: 10.1142/S0218127495000570. |
[34] |
M. Mizuno and K. Ikeda, An unstable mode selection rule: Frustrated optical instability due to two competing boundary conditions,, Physica D, 36 (1989), 327.
doi: 10.1016/0167-2789(89)90088-2. |
[35] |
S. Mohamad and K. Gopalsamy, Exponential stability of continuous-time and discrete-time cellular neural networks with delays,, Applied Mathematics and Computation, 135 (2003), 17.
doi: 10.1016/S0096-3003(01)00299-5. |
[36] |
W. W. Murdoch, R. M. Nisbet, S. P. Blythe, W. S. C. Gurney and J. D. Reeve, An invulnerable age class and stability in delay-differential parasitoid-host models,, American Naturalist, 129 (1987), 263.
doi: 10.1086/284634. |
[37] |
R. D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations,, Trans. Amer. Math. Soc., 238 (1978), 139.
doi: 10.1090/S0002-9947-1978-0482913-0. |
[38] |
M. Piotrowska, A remark on the ode with two discrete delays,, Journal of Mathematical Analysis and Applications, 329 (2007), 664.
doi: 10.1016/j.jmaa.2006.06.078. |
[39] |
C. G. Ragazzo and C. P. Malta, Singularity structure of the Hopf bifurcation surface of a differential equation with two delays,, Journal of Dynamics and Differential Equations, 4 (1992), 617.
doi: 10.1007/BF01048262. |
[40] |
J. Ruiz-Claeyssen, Effects of delays on functional differential equations,, J. Diff. Eq., 20 (1976), 404.
doi: 10.1016/0022-0396(76)90117-0. |
[41] |
S. Sakata, Asymptotic stability for a linear system of differential-difference equations,, Funkcial. Ekvac., 41 (1998), 435.
|
[42] |
R. T. Wilsterman, An Analytic and Geometric Approach for Examining the Stability of Linear Differential Equations with Two Delays,, Master's Thesis, (2013). Google Scholar |
[43] |
T. Yoneyama and J. Sugie, On the stability region of differential equations with two delays,, Funkcial. Ekvac., 31 (1988), 233.
|
[44] |
E. Zaron, The Delay Differential Equation: $x'(t) = -ax(t) + bx(t-\tau_1) + cx(t-\tau_2)$,, Technical report, (1987). Google Scholar |
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