October  2016, 21(8): 2409-2422. doi: 10.3934/dcdsb.2016053

Characteristic roots for two-lag linear delay differential equations

1. 

Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, United States

Received  October 2015 Revised  February 2016 Published  September 2016

We consider the class of two-lag linear delay differential equations and develop a series expansion to solve for the roots of the nonlinear characteristic equation. The expansion draws on results from complex analysis, combinatorics, special functions, and classical analysis for differential equations. Supporting numerical results are presented along with application of our method to study the stability of a two-lag model from ecology.
Citation: David M. Bortz. Characteristic roots for two-lag linear delay differential equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2409-2422. doi: 10.3934/dcdsb.2016053
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, 1972.

[2]

R. Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant, and Bell Matrices, World Scientific, Singapore, 2001. doi: 10.1142/9789812799838.

[3]

F. M. Asl and A. G. Ulsoy, Analysis of a system of linear delay differential equations, J. Dyn. Syst. Meas. Control, 125 (2003), 215-223. doi: 10.1115/1.1568121.

[4]

F. M. Asl and A. G. Ulsoy, Closure to "Discussion of 'Analysis of a system of linear delay differential equations' '' (2007, ASME J. Dyn. Syst., Meas., Control, 129, pp. 121-122), J. Dyn. Syst. Meas. Control, 129 (2007), 123.

[5]

C. E. Avellar and J. K. Hale, On the zeros of exponential polynomials, J. Math. Anal. Appl., 73 (1980), 434-452. doi: 10.1016/0022-247X(80)90289-9.

[6]

B. Balachandran, T. Kalmár-Nagy and D. E. Gilsinn (eds.), Delay Differential Equations: Recent Advances and New Directions, Springer US, Boston, MA, 2009.

[7]

H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Math. Biosci., 183 (2003), 63-91. doi: 10.1016/S0025-5564(02)00218-3.

[8]

E. T. Bell, Partition polynomials, Ann. Math., 29 (1927), 38-46. doi: 10.2307/1967979.

[9]

E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277. doi: 10.2307/1968431.

[10]

R. Bellman and K. L. Cooke, Differential-Difference Equations, Mathematis in Science and Engineering, Academic Press, New York, NY, 1963.

[11]

F. G. Boese, Stability Criteria for Second-Order Dynamical Systems Involving Several Time Delays, SIAM J. Math. Anal., 26 (1995), 1306-1330. doi: 10.1137/S0036141091200848.

[12]

D. M. Bortz, Eigenvalues for a two-lag linear delay differential equation, IFAC-PapersOnLine, 48 (2015), 13-16.

[13]

R. D. Braddock and P. van den Driessche, On a two lag differential delay equation, J. Aust. Math. Soc. Ser. B Appl. Math., 24 (1983), 292-317. doi: 10.1017/S0334270000002939.

[14]

J. W. Brown and R. V. Churchill, Complex Variables and Applications, 6th edition, McGraw-Hill Book Company, Inc., New York, NY, 1996.

[15]

T. C. Busken and J. M. Mahaffy, Regions of stability for a linear differential equation with two rationally dependent delays, Discrete Contin. Dyn. Syst., 35 (2015), 4955-4986. doi: 10.3934/dcds.2015.35.4955.

[16]

C. Carathéodory, Theory of Functions of a Complex Variable, vol. 97 of AMS Chelsea Publishing Series, 2nd edition, AMS Chelsea Pub., Rhode Island, 2001.

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L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht, Holland, 1974.

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K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260. doi: 10.1007/s002850050051.

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K. L. Cooke and P. van den Driessche, On zeros of some transcendental functions, Eunkcialaj Ekvacioj, 29 (1986), 77-90.

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R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the LambertW function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750.

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W. Deng, C. Li and J. Lü, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 48 (2007), 409-416. doi: 10.1007/s11071-006-9094-0.

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O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, vol. 110 of Applied Mathematical Sciences, Springer-Verlag, New York, NY, 1995. doi: 10.1007/978-1-4612-4206-2.

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L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theoryand Application of Differential Equations with Deviating Arguments, vol. 105 of Mathematics in Science and Engineering, Academic Press, New York, NY, 1973.

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L. H. Encinas, A. M. del Rey and J. M. Masqué, Faà di Bruno's formula, lattices, and partitions, Discrete Appl. Math., 148 (2005), 246-255. doi: 10.1016/j.dam.2005.02.009.

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T. Erneux, Applied Delay Differential Equations, vol. 3 of Surveys and Tutorials in Applied Mathematical Sciences, Springer New York, New York, NY, 2009.

[29]

J. Forde and P. Nelson, Applications of Sturm sequences to bifurcation analysis of delay differential equation models, J. Math. Anal. Appl., 300 (2004), 273-284. doi: 10.1016/j.jmaa.2004.02.063.

[30]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Springer, New York, NY, 1992. doi: 10.1007/978-94-015-7920-9.

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K. Gu and S.-I. Niculescu, Survey on recent results in the stability and control of time-delay systems, J. Dyn. Syst. Meas. Control, 125 (2003), 158-165. doi: 10.1115/1.1569950.

[32]

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.

[33]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, NY, 1993. doi: 10.1007/978-1-4612-4342-7.

[34]

E. Jarlebring, Critical delays and polynomial eigenvalue problems, J. Comput. Appl. Math., 224 (2009), 296-306. doi: 10.1016/j.cam.2008.05.004.

[35]

E. Jarlebring and T. Damm, The Lambert W function and the spectrum of some multidimensional time-delay systems, Automatica, 43 (2007), 2124-2128. doi: 10.1016/j.automatica.2007.04.001.

[36]

W. P. Johnson, The curious history of Faà di Bruno's formula, Am. Math. Mon., 109 (2002), 217-234. doi: 10.2307/2695352.

[37]

F. A. Khasawneh and B. P. Mann, A spectral element approach for the stability analysis of time-periodic delay equations with multiple delays, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2129-2141. doi: 10.1016/j.cnsns.2012.11.030.

[38]

S. M. Kissler, C. Cichowitz, S. Sankaranarayanan and D. M. Bortz, Determination of personalized diabetes treatment plans using a two-delay model, J. Theor. Biol., 359 (2014), 101-111. doi: 10.1016/j.jtbi.2014.06.005.

[39]

D. E. Knuth, Two notes on notation, Am. Math. Mon., 99 (1992), 403-422. doi: 10.2307/2325085.

[40]

Y. Kuang, Delay Differential Equations With Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, New York, NY, 1993.

[41]

J. Li and Y. Kuang, Analysis of a Model of the Glucose-Insulin Regulatory System with Two Delays, SIAM J. Appl. Math., 67 (2007), 757-776. doi: 10.1137/050634001.

[42]

X. Li, S. Ruan and J. Wei, Stability and Bifurcation in Delay-Differential Equations with Two Delays, J. Math. Anal. Appl., 236 (1999), 254-280. doi: 10.1006/jmaa.1999.6418.

[43]

J. J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi, Topics in Time Delay Systems: Analysis, Algorithms and Control, Lecture Notes in Control and Information Sciences, 388. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02897-7.

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X. Long, T. Insperger and B. Balachandran, Systems with periodic coefficients and periodically varying delays: Semidiscretization-based stability analysis, in Delay Differential Equations (eds. D. E. Gilsinn, T. Kalmár-Nagy and B. Balachandran), Springer US, Boston, MA, 2009, 131-153. doi: 10.1007/978-0-387-85595-0_5.

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N. MacDonald, Biological Delay Systems: Linear Stability Theory, vol. 8 of Cambridge Studies in Math. Biology, Cambridge University Press, Cambridge, UK, 1989.

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J. M. Mahaffy, K. M. Joiner and P. J. Zak, A geometric analysis of stability regions for a linear differential equation with two delays, Int. J. Bifurc. Chaos, 5 (1995), 779-796. doi: 10.1142/S0218127495000570.

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D. Michie, "Memo'' functions and machine learning, Nature, 218 (1968), p306. doi: 10.1038/218306c0.

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R. L. Mishkov, Generalization of the formula of Faa di Bruno for a composite function with a vector argument, Int. J. Math. Math. Sci., 24 (2000), 481-491. doi: 10.1155/S0161171200002970.

[49]

C. J. Moreno, The zeros of exponential polynomials (I), Compsitio Math., 26 (1973), 69-78.

[50]

L. Olien and J. Bélair, Bifurcations, stability, and monotonicity properties of a delayed neural network model, Phys. Nonlinear Phenom., 102 (1997), 349-363. doi: 10.1016/S0167-2789(96)00215-1.

[51]

J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5.

[52]

J. F. Ritt, On the zeros of exponential polynomials, Trans. Am. Math. Soc., 31 (1929), 680-686. doi: 10.1090/S0002-9947-1929-1501506-6.

[53]

S. Ruan, Chapter 11: Delay differential equations in single species dynamics, in Delay Differential Equations and Applications (eds. O. Arino, M. Hbid and E. A. Dads), vol. 205 of NATO Science Series, Springer Netherlands, Dordrecht, Holland, 2006, 477-517. doi: 10.1007/1-4020-3647-7_11.

[54]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 10 (2003), 863-874.

[55]

G. Samaey and B. Sandstede, Determining stability of pulses for partial differential equations with time delays, Dyn. Syst., 20 (2005), 201-222. doi: 10.1080/14689360500035693.

[56]

Y. Sasaki, On zeros of exponential polynomials and quantum algorithms, Quantum Inf. Process., 9 (2010), 419-427. doi: 10.1007/s11128-009-0148-3.

[57]

R. Sipahi and N. Olgac, Stability intricacies of two-delay linear systems in the presence of delay cross-talk, IET Control Theory Appl., 5 (2011), 990-998. doi: 10.1049/iet-cta.2010.0162.

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Rifat Sipahi, Tomáš Vyhlídal, Silviu-Iulian Niculescu and Pierdomenico Pepe (eds.), Time Delay Systems: Methods, Applications and New Trends, vol. 423 of Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-25221-1.

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H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57 of Texts in Applied Mathematics, Springer New York, New York, NY, 2011. doi: 10.1007/978-1-4419-7646-8.

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P. K. Srivastava, M. Banerjee and P. Chandra, A primary infection model for hiv and immune response with two discrete time delays, Differ. Equ. Dyn. Syst., 18 (2010), 385-399. doi: 10.1007/s12591-010-0074-y.

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G. Stépán, Retarded Dynamical Systems: Stability and Characteristic Functions, vol. 210 of Pitman research notes in mathematics series, Longman Scientific andTechnical, Harlow, 1989.

[62]

J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Phys. Nonlinear Phenom., 130 (1999), 255-272. doi: 10.1016/S0167-2789(99)00009-3.

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J. Wei and Y. Yuan, Synchronized Hopf bifurcation analysis in a neural network model with delays, J. Math. Anal. Appl., 312 (2005), 205-229. doi: 10.1016/j.jmaa.2005.03.049.

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S. Yi, P. W. Nelson and A. G. Ulsoy, Time-Delay Systems: Analysis and Control Using the Lambert W Function, World Scientific Press, London, UK, 2010. doi: 10.1142/9789814307406.

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N. Zafer, Discussion: "Analysis of a system of linear delay differential equations'' (Asl, F. M., and Ulsoy, A. G., 2003, ASME J. Dyn. Syst., Meas., Control, 125, pp. 215-223), J. Dyn. Syst. Meas. Control, 129 (2007), 121-122. doi: 10.1115/1.2428282.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, 1972.

[2]

R. Aldrovandi, Special Matrices of Mathematical Physics: Stochastic, Circulant, and Bell Matrices, World Scientific, Singapore, 2001. doi: 10.1142/9789812799838.

[3]

F. M. Asl and A. G. Ulsoy, Analysis of a system of linear delay differential equations, J. Dyn. Syst. Meas. Control, 125 (2003), 215-223. doi: 10.1115/1.1568121.

[4]

F. M. Asl and A. G. Ulsoy, Closure to "Discussion of 'Analysis of a system of linear delay differential equations' '' (2007, ASME J. Dyn. Syst., Meas., Control, 129, pp. 121-122), J. Dyn. Syst. Meas. Control, 129 (2007), 123.

[5]

C. E. Avellar and J. K. Hale, On the zeros of exponential polynomials, J. Math. Anal. Appl., 73 (1980), 434-452. doi: 10.1016/0022-247X(80)90289-9.

[6]

B. Balachandran, T. Kalmár-Nagy and D. E. Gilsinn (eds.), Delay Differential Equations: Recent Advances and New Directions, Springer US, Boston, MA, 2009.

[7]

H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the modeling of viral delays in HIV infection dynamics, Math. Biosci., 183 (2003), 63-91. doi: 10.1016/S0025-5564(02)00218-3.

[8]

E. T. Bell, Partition polynomials, Ann. Math., 29 (1927), 38-46. doi: 10.2307/1967979.

[9]

E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277. doi: 10.2307/1968431.

[10]

R. Bellman and K. L. Cooke, Differential-Difference Equations, Mathematis in Science and Engineering, Academic Press, New York, NY, 1963.

[11]

F. G. Boese, Stability Criteria for Second-Order Dynamical Systems Involving Several Time Delays, SIAM J. Math. Anal., 26 (1995), 1306-1330. doi: 10.1137/S0036141091200848.

[12]

D. M. Bortz, Eigenvalues for a two-lag linear delay differential equation, IFAC-PapersOnLine, 48 (2015), 13-16.

[13]

R. D. Braddock and P. van den Driessche, On a two lag differential delay equation, J. Aust. Math. Soc. Ser. B Appl. Math., 24 (1983), 292-317. doi: 10.1017/S0334270000002939.

[14]

J. W. Brown and R. V. Churchill, Complex Variables and Applications, 6th edition, McGraw-Hill Book Company, Inc., New York, NY, 1996.

[15]

T. C. Busken and J. M. Mahaffy, Regions of stability for a linear differential equation with two rationally dependent delays, Discrete Contin. Dyn. Syst., 35 (2015), 4955-4986. doi: 10.3934/dcds.2015.35.4955.

[16]

C. Carathéodory, Theory of Functions of a Complex Variable, vol. 97 of AMS Chelsea Publishing Series, 2nd edition, AMS Chelsea Pub., Rhode Island, 2001.

[17]

L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company, Dordrecht, Holland, 1974.

[18]

K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35 (1996), 240-260. doi: 10.1007/s002850050051.

[19]

K. L. Cooke and P. van den Driessche, On zeros of some transcendental functions, Eunkcialaj Ekvacioj, 29 (1986), 77-90.

[20]

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the LambertW function, Adv. Comput. Math., 5 (1996), 329-359. doi: 10.1007/BF02124750.

[21]

R. M. Corles, D. J. Jeffrey and D. E. Knuth, A sequence of series for the Lambert w function, in Proceedings of the 1997 international symposium on Symbolic and algebraic computation, ACM Press, 1997, 197-204. doi: 10.1145/258726.258783.

[22]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, vol. 20 of Lecture Notes in Biomathematics, Springer-Verlag, New York, NY, 1977.

[23]

N. D. de Bruijn, Asymptotic Methods in Analysis, North Holland, The Netherlands, 1958.

[24]

W. Deng, C. Li and J. Lü, Stability analysis of linear fractional differential system with multiple time delays, Nonlinear Dyn., 48 (2007), 409-416. doi: 10.1007/s11071-006-9094-0.

[25]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, vol. 110 of Applied Mathematical Sciences, Springer-Verlag, New York, NY, 1995. doi: 10.1007/978-1-4612-4206-2.

[26]

L. E. El'sgol'ts and S. B. Norkin, Introduction to the Theoryand Application of Differential Equations with Deviating Arguments, vol. 105 of Mathematics in Science and Engineering, Academic Press, New York, NY, 1973.

[27]

L. H. Encinas, A. M. del Rey and J. M. Masqué, Faà di Bruno's formula, lattices, and partitions, Discrete Appl. Math., 148 (2005), 246-255. doi: 10.1016/j.dam.2005.02.009.

[28]

T. Erneux, Applied Delay Differential Equations, vol. 3 of Surveys and Tutorials in Applied Mathematical Sciences, Springer New York, New York, NY, 2009.

[29]

J. Forde and P. Nelson, Applications of Sturm sequences to bifurcation analysis of delay differential equation models, J. Math. Anal. Appl., 300 (2004), 273-284. doi: 10.1016/j.jmaa.2004.02.063.

[30]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Springer, New York, NY, 1992. doi: 10.1007/978-94-015-7920-9.

[31]

K. Gu and S.-I. Niculescu, Survey on recent results in the stability and control of time-delay systems, J. Dyn. Syst. Meas. Control, 125 (2003), 158-165. doi: 10.1115/1.1569950.

[32]

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.

[33]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer-Verlag, New York, NY, 1993. doi: 10.1007/978-1-4612-4342-7.

[34]

E. Jarlebring, Critical delays and polynomial eigenvalue problems, J. Comput. Appl. Math., 224 (2009), 296-306. doi: 10.1016/j.cam.2008.05.004.

[35]

E. Jarlebring and T. Damm, The Lambert W function and the spectrum of some multidimensional time-delay systems, Automatica, 43 (2007), 2124-2128. doi: 10.1016/j.automatica.2007.04.001.

[36]

W. P. Johnson, The curious history of Faà di Bruno's formula, Am. Math. Mon., 109 (2002), 217-234. doi: 10.2307/2695352.

[37]

F. A. Khasawneh and B. P. Mann, A spectral element approach for the stability analysis of time-periodic delay equations with multiple delays, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2129-2141. doi: 10.1016/j.cnsns.2012.11.030.

[38]

S. M. Kissler, C. Cichowitz, S. Sankaranarayanan and D. M. Bortz, Determination of personalized diabetes treatment plans using a two-delay model, J. Theor. Biol., 359 (2014), 101-111. doi: 10.1016/j.jtbi.2014.06.005.

[39]

D. E. Knuth, Two notes on notation, Am. Math. Mon., 99 (1992), 403-422. doi: 10.2307/2325085.

[40]

Y. Kuang, Delay Differential Equations With Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, New York, NY, 1993.

[41]

J. Li and Y. Kuang, Analysis of a Model of the Glucose-Insulin Regulatory System with Two Delays, SIAM J. Appl. Math., 67 (2007), 757-776. doi: 10.1137/050634001.

[42]

X. Li, S. Ruan and J. Wei, Stability and Bifurcation in Delay-Differential Equations with Two Delays, J. Math. Anal. Appl., 236 (1999), 254-280. doi: 10.1006/jmaa.1999.6418.

[43]

J. J. Loiseau, W. Michiels, S.-I. Niculescu and R. Sipahi, Topics in Time Delay Systems: Analysis, Algorithms and Control, Lecture Notes in Control and Information Sciences, 388. Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-02897-7.

[44]

X. Long, T. Insperger and B. Balachandran, Systems with periodic coefficients and periodically varying delays: Semidiscretization-based stability analysis, in Delay Differential Equations (eds. D. E. Gilsinn, T. Kalmár-Nagy and B. Balachandran), Springer US, Boston, MA, 2009, 131-153. doi: 10.1007/978-0-387-85595-0_5.

[45]

N. MacDonald, Biological Delay Systems: Linear Stability Theory, vol. 8 of Cambridge Studies in Math. Biology, Cambridge University Press, Cambridge, UK, 1989.

[46]

J. M. Mahaffy, K. M. Joiner and P. J. Zak, A geometric analysis of stability regions for a linear differential equation with two delays, Int. J. Bifurc. Chaos, 5 (1995), 779-796. doi: 10.1142/S0218127495000570.

[47]

D. Michie, "Memo'' functions and machine learning, Nature, 218 (1968), p306. doi: 10.1038/218306c0.

[48]

R. L. Mishkov, Generalization of the formula of Faa di Bruno for a composite function with a vector argument, Int. J. Math. Math. Sci., 24 (2000), 481-491. doi: 10.1155/S0161171200002970.

[49]

C. J. Moreno, The zeros of exponential polynomials (I), Compsitio Math., 26 (1973), 69-78.

[50]

L. Olien and J. Bélair, Bifurcations, stability, and monotonicity properties of a delayed neural network model, Phys. Nonlinear Phenom., 102 (1997), 349-363. doi: 10.1016/S0167-2789(96)00215-1.

[51]

J.-P. Richard, Time-delay systems: An overview of some recent advances and open problems, Automatica, 39 (2003), 1667-1694. doi: 10.1016/S0005-1098(03)00167-5.

[52]

J. F. Ritt, On the zeros of exponential polynomials, Trans. Am. Math. Soc., 31 (1929), 680-686. doi: 10.1090/S0002-9947-1929-1501506-6.

[53]

S. Ruan, Chapter 11: Delay differential equations in single species dynamics, in Delay Differential Equations and Applications (eds. O. Arino, M. Hbid and E. A. Dads), vol. 205 of NATO Science Series, Springer Netherlands, Dordrecht, Holland, 2006, 477-517. doi: 10.1007/1-4020-3647-7_11.

[54]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 10 (2003), 863-874.

[55]

G. Samaey and B. Sandstede, Determining stability of pulses for partial differential equations with time delays, Dyn. Syst., 20 (2005), 201-222. doi: 10.1080/14689360500035693.

[56]

Y. Sasaki, On zeros of exponential polynomials and quantum algorithms, Quantum Inf. Process., 9 (2010), 419-427. doi: 10.1007/s11128-009-0148-3.

[57]

R. Sipahi and N. Olgac, Stability intricacies of two-delay linear systems in the presence of delay cross-talk, IET Control Theory Appl., 5 (2011), 990-998. doi: 10.1049/iet-cta.2010.0162.

[58]

Rifat Sipahi, Tomáš Vyhlídal, Silviu-Iulian Niculescu and Pierdomenico Pepe (eds.), Time Delay Systems: Methods, Applications and New Trends, vol. 423 of Lecture Notes in Control and Information Sciences, Springer Berlin Heidelberg, Berlin, Heidelberg, 2012. doi: 10.1007/978-3-642-25221-1.

[59]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, vol. 57 of Texts in Applied Mathematics, Springer New York, New York, NY, 2011. doi: 10.1007/978-1-4419-7646-8.

[60]

P. K. Srivastava, M. Banerjee and P. Chandra, A primary infection model for hiv and immune response with two discrete time delays, Differ. Equ. Dyn. Syst., 18 (2010), 385-399. doi: 10.1007/s12591-010-0074-y.

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