October  2015, 35(10): 4955-4986. doi: 10.3934/dcds.2015.35.4955

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

Received  July 2013 Revised  January 2015 Published  April 2015

Stability analysis is performed for a linear differential equation with two delays. Geometric arguments show that when the two delays are rationally dependent, then the region of stability increases. When the ratio has the form $1/n$, this study finds the asymptotic shape and size of the stability region. For example, a delay ratio of $1/3$ asymptotically produces a stability region about 44.3% larger than any nearby delay ratios, showing extreme sensitivity in the delays. The study provides a systematic and geometric approach to finding the eigenvalues on the boundary of stability for this delay differential equation. A nonlinear model with two delays illustrates how our methods can be applied.
Citation: Joseph M. Mahaffy, Timothy C. Busken. Regions of stability for a linear differential equation with two rationally dependent delays. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4955-4986. doi: 10.3934/dcds.2015.35.4955
References:
[1]

J. Bélair, Stability of a differential-delay equation with two time lags, in Oscillations, Bifurcations, and Chaos (eds. F. V. Atkinson, W. F. Langford and A. B. Mingarelli), CMS Conf. Proc., 8, AMS, Providence, RI, 1987, 305-315.

[2]

J. Bélair and M. Mackey, A model for the regulation of mammalian platelet production, Ann. N. Y. Acad. Sci., 504 (1987), 280-282. 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), 299-325. 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), 317-346. 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), 1402-1424. 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), 245-255. doi: 10.1137/S0036139994274526.

[7]

R. Bellman and K. L. Cooke, Differential-Difference Equations, Lectures in Applied Mathematics, Vol. 17, Academic Press, New York, N.Y., 1963.

[8]

F. G. Boese, The delay-independent stability behaviour of a first order differential-difference 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), 117-120. 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), 227-247. doi: 10.1006/jmaa.1994.1017.

[11]

D. M. Bortz, Eigenvalues for two-lag linear delay differential equations, submitted, arXiv:1206.6364, 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), Statistical Ecology Series, 13, International Cooperative Publishing House, Fairland, MD, 1981.

[13]

T. C. Busken, On the Asymptotic Stability of the Zero Solution for a Linear Differential Equation with Two Delays, Master's Thesis, San Diego State University, 2012.

[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, ON, 1993), Canad. Appl. Math. Quart., 3 (1995), 137-154.

[15]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci., 16 (1973), 75-101. 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, New York, NY, 1977.

[17]

T. Elsken, The region of (in)stability of a 2-delay equation is connected, J. Math. Anal. Appl., 261 (2001), 497-526. 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), 1-33. doi: 10.1002/cta.4490210102.

[19]

J. Hale, E. Infante and P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl., 105 (1985), 533-555. 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., Univ. Utah, Salt Lake City, Utah, 1978), Lectures in Applied Mathematics, 17, American Math. Soc., Providence, R. I., 1979, 157-185.

[21]

J. K. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl., 178 (1993), 344-362. 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), 1-30. 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, Algorithms, Applications (Würzburg, 1990), Internat. Ser. Numer. Math., 97, Birkhäuser, Basel, 1991, 155-159.

[24]

N. Hayes, Roots of the transcendental equation associated with a certain differential difference equation, J. London Math. Soc., 25 (1950), 226-232.

[25]

T. D. Howroyd and A. M. Russell, Cournot oligopoly models with time lags, J. Math. Econ., 13 (1984), 97-103. doi: 10.1016/0304-4068(84)90009-0.

[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), 153-159. 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), 254-280. 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), North Holland, Amsterdam, 1979, 287-295.

[30]

N. MacDonald, An activation-inhibition model of cyclic granulopoiesis in chronic granulocytic leukemia, Math. Biosci., 54 (1980), 61-70. 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), 497-509. 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, Department of Mathematical Sciences, San Diego State University, San Diego, CA, 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), 779-796. 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), 327-342. 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), 17-38. 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), 263-282. doi: 10.1086/284634.

[37]

R. D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations, Trans. Amer. Math. Soc., 238 (1978), 139-164. 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), 664-676. 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), 617-650. doi: 10.1007/BF01048262.

[40]

J. Ruiz-Claeyssen, Effects of delays on functional differential equations, J. Diff. Eq., 20 (1976), 404-440. 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), 435-449.

[42]

R. T. Wilsterman, An Analytic and Geometric Approach for Examining the Stability of Linear Differential Equations with Two Delays, Master's Thesis, San Diego State University, 2013.

[43]

T. Yoneyama and J. Sugie, On the stability region of differential equations with two delays, Funkcial. Ekvac., 31 (1988), 233-240.

[44]

E. Zaron, The Delay Differential Equation: $x'(t) = -ax(t) + bx(t-\tau_1) + cx(t-\tau_2)$, Technical report, Harvey Mudd College, Claremont, CA, 1987.

show all references

References:
[1]

J. Bélair, Stability of a differential-delay equation with two time lags, in Oscillations, Bifurcations, and Chaos (eds. F. V. Atkinson, W. F. Langford and A. B. Mingarelli), CMS Conf. Proc., 8, AMS, Providence, RI, 1987, 305-315.

[2]

J. Bélair and M. Mackey, A model for the regulation of mammalian platelet production, Ann. N. Y. Acad. Sci., 504 (1987), 280-282. 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), 299-325. 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), 317-346. 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), 1402-1424. 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), 245-255. doi: 10.1137/S0036139994274526.

[7]

R. Bellman and K. L. Cooke, Differential-Difference Equations, Lectures in Applied Mathematics, Vol. 17, Academic Press, New York, N.Y., 1963.

[8]

F. G. Boese, The delay-independent stability behaviour of a first order differential-difference 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), 117-120. 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), 227-247. doi: 10.1006/jmaa.1994.1017.

[11]

D. M. Bortz, Eigenvalues for two-lag linear delay differential equations, submitted, arXiv:1206.6364, 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), Statistical Ecology Series, 13, International Cooperative Publishing House, Fairland, MD, 1981.

[13]

T. C. Busken, On the Asymptotic Stability of the Zero Solution for a Linear Differential Equation with Two Delays, Master's Thesis, San Diego State University, 2012.

[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, ON, 1993), Canad. Appl. Math. Quart., 3 (1995), 137-154.

[15]

K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci., 16 (1973), 75-101. 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, New York, NY, 1977.

[17]

T. Elsken, The region of (in)stability of a 2-delay equation is connected, J. Math. Anal. Appl., 261 (2001), 497-526. 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), 1-33. doi: 10.1002/cta.4490210102.

[19]

J. Hale, E. Infante and P. Tsen, Stability in linear delay equations, J. Math. Anal. Appl., 105 (1985), 533-555. 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., Univ. Utah, Salt Lake City, Utah, 1978), Lectures in Applied Mathematics, 17, American Math. Soc., Providence, R. I., 1979, 157-185.

[21]

J. K. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations, J. Math. Anal. Appl., 178 (1993), 344-362. 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), 1-30. 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, Algorithms, Applications (Würzburg, 1990), Internat. Ser. Numer. Math., 97, Birkhäuser, Basel, 1991, 155-159.

[24]

N. Hayes, Roots of the transcendental equation associated with a certain differential difference equation, J. London Math. Soc., 25 (1950), 226-232.

[25]

T. D. Howroyd and A. M. Russell, Cournot oligopoly models with time lags, J. Math. Econ., 13 (1984), 97-103. doi: 10.1016/0304-4068(84)90009-0.

[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), 153-159. 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), 254-280. 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), North Holland, Amsterdam, 1979, 287-295.

[30]

N. MacDonald, An activation-inhibition model of cyclic granulopoiesis in chronic granulocytic leukemia, Math. Biosci., 54 (1980), 61-70. 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), 497-509. 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, Department of Mathematical Sciences, San Diego State University, San Diego, CA, 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), 779-796. 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), 327-342. 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), 17-38. 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), 263-282. doi: 10.1086/284634.

[37]

R. D. Nussbaum, A Hopf global bifurcation theorem for retarded functional differential equations, Trans. Amer. Math. Soc., 238 (1978), 139-164. 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), 664-676. 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), 617-650. doi: 10.1007/BF01048262.

[40]

J. Ruiz-Claeyssen, Effects of delays on functional differential equations, J. Diff. Eq., 20 (1976), 404-440. 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), 435-449.

[42]

R. T. Wilsterman, An Analytic and Geometric Approach for Examining the Stability of Linear Differential Equations with Two Delays, Master's Thesis, San Diego State University, 2013.

[43]

T. Yoneyama and J. Sugie, On the stability region of differential equations with two delays, Funkcial. Ekvac., 31 (1988), 233-240.

[44]

E. Zaron, The Delay Differential Equation: $x'(t) = -ax(t) + bx(t-\tau_1) + cx(t-\tau_2)$, Technical report, Harvey Mudd College, Claremont, CA, 1987.

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