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 & Continuous Dynamical Systems - A, 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, (1987), 305.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Bellman and K. L. Cooke, Differential-Difference Equations,, Lectures in Applied Mathematics, (1963).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. K. Hale, Nonlinear oscillations in equations with delays,, in Nonlinear Oscillations in Biology (Proc. Tenth Summer Sem. Appl. Math., (1978), 157.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Haller and G. Stépán, Codimension two bifurcation in an approximate model for delayed robot control,, in Bifurcation and Chaos: Analysis, (1990), 155.   Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

S. Sakata, Asymptotic stability for a linear system of differential-difference equations,, Funkcial. Ekvac., 41 (1998), 435.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

R. Bellman and K. L. Cooke, Differential-Difference Equations,, Lectures in Applied Mathematics, (1963).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

J. K. Hale, Nonlinear oscillations in equations with delays,, in Nonlinear Oscillations in Biology (Proc. Tenth Summer Sem. Appl. Math., (1978), 157.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Haller and G. Stépán, Codimension two bifurcation in an approximate model for delayed robot control,, in Bifurcation and Chaos: Analysis, (1990), 155.   Google Scholar

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

S. Sakata, Asymptotic stability for a linear system of differential-difference equations,, Funkcial. Ekvac., 41 (1998), 435.   Google Scholar

[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.   Google Scholar

[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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