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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 921827720, United States 
2.  Department of Mathematics, Grossmont College, El Cajon, CA 92020, United States 
References:
[1] 
J. Bélair, Stability of a differentialdelay 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.17496632.1987.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, Agestructured and two delay models for erythropoiesis,, Math. Biosci., 128 (1995), 317. doi: 10.1016/00255564(94)00078E. 
[5] 
J. Bélair and S. A. Campbell, Stability and bifurcations of equilibria in a multipledelayed 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 delayinduced 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, DifferentialDifference Equations,, Lectures in Applied Mathematics, (1963). 
[8] 
F. G. Boese, The delayindependent stability behaviour of a first order differentialdifference equation with two constant lags,, preprint, (1993). 
[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 differencedifferential equations with intervalvalued parameters,, Journal of Mathematical Analysis and Applications, 181 (1994), 227. doi: 10.1006/jmaa.1994.1017. 
[11] 
D. M. Bortz, Eigenvalues for twolag linear delay differential equations,, submitted, (2012). 
[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). 
[13] 
T. C. Busken, On the Asymptotic Stability of the Zero Solution for a Linear Differential Equation with Two Delays,, Master's Thesis, (2012). 
[14] 
S. A. Campbell and J. Bélair, Analytical and symbolicallyassisted investigation of Hopf bifurcations in delaydifferential 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: 10.1016/00255564(73)900461. 
[16] 
L. E. El'sgol'ts and S. Norkin, Introduction to the Theory of Differential Equations with Deviating Arguments,, Academic Press, (1977). 
[17] 
T. Elsken, The region of (in)stability of a 2delay 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: 10.1016/0022247X(85)90068X. 
[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: 10.1016/03044068(84)900090. 
[26] 
, E. F. Infante,, Personal Communication, (1975). 
[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 delaydifferential 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. 
[30] 
N. MacDonald, An activationinhibition model of cyclic granulopoiesis in chronic granulocytic leukemia,, Math. Biosci., 54 (1980), 61. doi: 10.1016/00255564(81)900766. 
[31] 
M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors,, J. Econ. Theory, 48 (1989), 497. doi: 10.1016/00220531(89)900392. 
[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). 
[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: 10.1016/01672789(89)900882. 
[35] 
S. Mohamad and K. Gopalsamy, Exponential stability of continuoustime and discretetime cellular neural networks with delays,, Applied Mathematics and Computation, 135 (2003), 17. doi: 10.1016/S00963003(01)002995. 
[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 delaydifferential parasitoidhost 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/S00029947197804829130. 
[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. RuizClaeyssen, Effects of delays on functional differential equations,, J. Diff. Eq., 20 (1976), 404. doi: 10.1016/00220396(76)901170. 
[41] 
S. Sakata, Asymptotic stability for a linear system of differentialdifference 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). 
[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). 
show all references
References:
[1] 
J. Bélair, Stability of a differentialdelay 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.17496632.1987.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, Agestructured and two delay models for erythropoiesis,, Math. Biosci., 128 (1995), 317. doi: 10.1016/00255564(94)00078E. 
[5] 
J. Bélair and S. A. Campbell, Stability and bifurcations of equilibria in a multipledelayed 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 delayinduced 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, DifferentialDifference Equations,, Lectures in Applied Mathematics, (1963). 
[8] 
F. G. Boese, The delayindependent stability behaviour of a first order differentialdifference equation with two constant lags,, preprint, (1993). 
[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 differencedifferential equations with intervalvalued parameters,, Journal of Mathematical Analysis and Applications, 181 (1994), 227. doi: 10.1006/jmaa.1994.1017. 
[11] 
D. M. Bortz, Eigenvalues for twolag linear delay differential equations,, submitted, (2012). 
[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). 
[13] 
T. C. Busken, On the Asymptotic Stability of the Zero Solution for a Linear Differential Equation with Two Delays,, Master's Thesis, (2012). 
[14] 
S. A. Campbell and J. Bélair, Analytical and symbolicallyassisted investigation of Hopf bifurcations in delaydifferential 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: 10.1016/00255564(73)900461. 
[16] 
L. E. El'sgol'ts and S. Norkin, Introduction to the Theory of Differential Equations with Deviating Arguments,, Academic Press, (1977). 
[17] 
T. Elsken, The region of (in)stability of a 2delay 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: 10.1016/0022247X(85)90068X. 
[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: 10.1016/03044068(84)900090. 
[26] 
, E. F. Infante,, Personal Communication, (1975). 
[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 delaydifferential 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. 
[30] 
N. MacDonald, An activationinhibition model of cyclic granulopoiesis in chronic granulocytic leukemia,, Math. Biosci., 54 (1980), 61. doi: 10.1016/00255564(81)900766. 
[31] 
M. C. Mackey, Commodity price fluctuations: Price dependent delays and nonlinearities as explanatory factors,, J. Econ. Theory, 48 (1989), 497. doi: 10.1016/00220531(89)900392. 
[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). 
[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: 10.1016/01672789(89)900882. 
[35] 
S. Mohamad and K. Gopalsamy, Exponential stability of continuoustime and discretetime cellular neural networks with delays,, Applied Mathematics and Computation, 135 (2003), 17. doi: 10.1016/S00963003(01)002995. 
[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 delaydifferential parasitoidhost 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/S00029947197804829130. 
[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. RuizClaeyssen, Effects of delays on functional differential equations,, J. Diff. Eq., 20 (1976), 404. doi: 10.1016/00220396(76)901170. 
[41] 
S. Sakata, Asymptotic stability for a linear system of differentialdifference 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). 
[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). 
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