American Institute of Mathematical Sciences

Advanced
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

Joseph M. Mahaffy 1, and  Timothy C. Busken 2,

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.
Keywords: exponential polynomial, stability analysis, eigenvalue., Delay differential equation, bifurcation.
Mathematics Subject Classification: Primary: 37C75, 37G15; Secondary: 39B8.
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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