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 & 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, (1972). Google Scholar

[2]

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

[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. doi: 10.1115/1.1568121. Google Scholar

[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). Google Scholar

[5]

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

[6]

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

[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. doi: 10.1016/S0025-5564(02)00218-3. Google Scholar

[8]

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

[9]

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

[10]

R. Bellman and K. L. Cooke, Differential-Difference Equations,, Mathematis in Science and Engineering, (1963). Google Scholar

[11]

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

[12]

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

[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. doi: 10.1017/S0334270000002939. Google Scholar

[14]

J. W. Brown and R. V. Churchill, Complex Variables and Applications,, 6th edition, (1996). Google Scholar

[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. doi: 10.3934/dcds.2015.35.4955. Google Scholar

[16]

C. Carathéodory, Theory of Functions of a Complex Variable, vol. 97 of AMS Chelsea Publishing Series,, 2nd edition, (2001). Google Scholar

[17]

L. Comtet, Advanced Combinatorics,, D. Reidel Publishing Company, (1974). Google Scholar

[18]

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

[19]

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

[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. doi: 10.1007/BF02124750. Google Scholar

[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, (1997), 197. doi: 10.1145/258726.258783. Google Scholar

[22]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, vol. 20 of Lecture Notes in Biomathematics,, Springer-Verlag, (1977). Google Scholar

[23]

N. D. de Bruijn, Asymptotic Methods in Analysis,, North Holland, (1958). Google Scholar

[24]

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

[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, (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar

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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, (1973). Google Scholar

[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. doi: 10.1016/j.dam.2005.02.009. Google Scholar

[28]

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

[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. doi: 10.1016/j.jmaa.2004.02.063. Google Scholar

[30]

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

[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. doi: 10.1115/1.1569950. Google Scholar

[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. doi: 10.1006/jmaa.1993.1312. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[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. doi: 10.1016/j.cnsns.2012.11.030. Google Scholar

[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. doi: 10.1016/j.jtbi.2014.06.005. Google Scholar

[39]

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

[40]

Y. Kuang, Delay Differential Equations With Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering,, Academic Press, (1993). Google Scholar

[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. doi: 10.1137/050634001. Google Scholar

[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. doi: 10.1006/jmaa.1999.6418. Google Scholar

[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, (2009). doi: 10.1007/978-3-642-02897-7. Google Scholar

[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, (2009), 131. doi: 10.1007/978-0-387-85595-0_5. Google Scholar

[45]

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

[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. doi: 10.1142/S0218127495000570. Google Scholar

[47]

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

[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. doi: 10.1155/S0161171200002970. Google Scholar

[49]

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

[50]

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

[51]

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

[52]

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

[53]

S. Ruan, Chapter 11: Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications (eds. O. Arino, (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[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. Google Scholar

[55]

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

[56]

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

[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. doi: 10.1049/iet-cta.2010.0162. Google Scholar

[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, (2012). doi: 10.1007/978-3-642-25221-1. Google Scholar

[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, (2011). doi: 10.1007/978-1-4419-7646-8. Google Scholar

[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. doi: 10.1007/s12591-010-0074-y. Google Scholar

[61]

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

[62]

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

[63]

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

[64]

F. S. Wheeler, Bell polynomials,, ACM SIGSAM Bull., 21 (1987), 44. doi: 10.1145/29309.29317. Google Scholar

[65]

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

[66]

Y. Yuan and J. Bélair, Stability and hopf bifurcation analysis for functional differential equation with distributed delay,, SIAM J. Appl. Dyn. Syst., 10 (2011), 551. doi: 10.1137/100794493. Google Scholar

[67]

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. doi: 10.1115/1.2428282. Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions,, Dover, (1972). Google Scholar

[2]

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

[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. doi: 10.1115/1.1568121. Google Scholar

[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). Google Scholar

[5]

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

[6]

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

[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. doi: 10.1016/S0025-5564(02)00218-3. Google Scholar

[8]

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

[9]

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

[10]

R. Bellman and K. L. Cooke, Differential-Difference Equations,, Mathematis in Science and Engineering, (1963). Google Scholar

[11]

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

[12]

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

[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. doi: 10.1017/S0334270000002939. Google Scholar

[14]

J. W. Brown and R. V. Churchill, Complex Variables and Applications,, 6th edition, (1996). Google Scholar

[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. doi: 10.3934/dcds.2015.35.4955. Google Scholar

[16]

C. Carathéodory, Theory of Functions of a Complex Variable, vol. 97 of AMS Chelsea Publishing Series,, 2nd edition, (2001). Google Scholar

[17]

L. Comtet, Advanced Combinatorics,, D. Reidel Publishing Company, (1974). Google Scholar

[18]

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

[19]

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

[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. doi: 10.1007/BF02124750. Google Scholar

[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, (1997), 197. doi: 10.1145/258726.258783. Google Scholar

[22]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics, vol. 20 of Lecture Notes in Biomathematics,, Springer-Verlag, (1977). Google Scholar

[23]

N. D. de Bruijn, Asymptotic Methods in Analysis,, North Holland, (1958). Google Scholar

[24]

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

[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, (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar

[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, (1973). Google Scholar

[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. doi: 10.1016/j.dam.2005.02.009. Google Scholar

[28]

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

[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. doi: 10.1016/j.jmaa.2004.02.063. Google Scholar

[30]

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

[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. doi: 10.1115/1.1569950. Google Scholar

[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. doi: 10.1006/jmaa.1993.1312. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[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. doi: 10.1016/j.cnsns.2012.11.030. Google Scholar

[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. doi: 10.1016/j.jtbi.2014.06.005. Google Scholar

[39]

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

[40]

Y. Kuang, Delay Differential Equations With Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering,, Academic Press, (1993). Google Scholar

[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. doi: 10.1137/050634001. Google Scholar

[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. doi: 10.1006/jmaa.1999.6418. Google Scholar

[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, (2009). doi: 10.1007/978-3-642-02897-7. Google Scholar

[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, (2009), 131. doi: 10.1007/978-0-387-85595-0_5. Google Scholar

[45]

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

[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. doi: 10.1142/S0218127495000570. Google Scholar

[47]

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

[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. doi: 10.1155/S0161171200002970. Google Scholar

[49]

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

[50]

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

[51]

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

[52]

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

[53]

S. Ruan, Chapter 11: Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications (eds. O. Arino, (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[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. Google Scholar

[55]

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

[56]

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

[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. doi: 10.1049/iet-cta.2010.0162. Google Scholar

[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, (2012). doi: 10.1007/978-3-642-25221-1. Google Scholar

[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, (2011). doi: 10.1007/978-1-4419-7646-8. Google Scholar

[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. doi: 10.1007/s12591-010-0074-y. Google Scholar

[61]

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

[62]

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

[63]

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

[64]

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