# American Institute of Mathematical Sciences

## A cyclic system with delay and its characteristic equation

 1 Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB, Canada T2N 1N4 2 Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic 3 Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, USA 4 Instituto de Matematica y Fisica, Universidad de Talca, Casilla 747, Talca, Chile

S. I. Trofimchuk is the corresponding author, e-mail: trofimch@inst-mat.utalca.cl

Received  March 2017 Revised  July 2017 Published  January 2019

A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the existence of monotone solutions are derived in terms of roots of the characteristic equation.

Citation: Elena Braverman, Karel Hasik, Anatoli F. Ivanov, Sergei I. Trofimchuk. A cyclic system with delay and its characteristic equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020001
##### References:
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Conway, Functions of One Complex Variable, $2^{nd}$ edition, Springer, 1978. Google Scholar [8] B. C. Goodwin, Oscillatory behaviour in enzymatic control process, Adv. Enzime Regul., 3 (1965), 425-438. Google Scholar [9] I. Györy and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. Google Scholar [10] K. P. Hadeler, Delay equations in biology, in Lecture Notes in Mathematics, Springer, 730 (1979), 139-156. Google Scholar [11] K. P. Hadeler and J. Tomiuk, Periodic solutions of difference differential equations, Arch. Rat. Mech. Anal., 65 (1977), 87-95. doi: 10.1007/BF00289359. Google Scholar [12] J. K. Hale and A. F. Ivanov, On a high order differential delay equation, J. Math. Anal. Appl., 173 (1993), 505-514. doi: 10.1006/jmaa.1993.1083. Google Scholar [13] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [14] J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554. Google Scholar [15] A. F. Ivanov and B. Lani-Wayda, Periodic solutions for three-dimensional non-monotone cyclic systems with time delays, Discrete and Continuous Dynam. Systems- A, 11 (2004), 667-692. doi: 10.3934/dcds.2004.11.667. Google Scholar [16] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. Google Scholar [17] P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002. Google Scholar [18] B. Li, Oscillations of delay differential equations with variable coefficients, J. Math. Anal. Appl., 192 (1995), 312-321. doi: 10.1006/jmaa.1995.1173. Google Scholar [19] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar [20] J. Mahaffy, Periodic solutions of certain protein synthesis models, J. Math. Anal. Appl., 74 (1980), 72-105. doi: 10.1016/0022-247X(80)90115-8. Google Scholar [21] J. Mallet-Paret, Morse decompositions for delay differential equations, J. Differential Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X. Google Scholar [22] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-47. doi: 10.1023/A:1021889401235. Google Scholar [23] J. Mallet-Paret and R. D. Nussbaum, A differential delay equation arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249-292. doi: 10.1137/0520019. Google Scholar [24] J. Mallet-Paret and G. Sell, Systems of delay differential equations Ⅰ: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. Google Scholar [25] J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. Google Scholar [26] M. Pituk, Asymptotic behavior and oscillation of functional differential equations, J. Math. Anal. Appl., 322 (2006), 1140-1158. doi: 10.1016/j.jmaa.2005.09.081. Google Scholar [27] T. Scheper, D. Klinkenberg, C. Pennartz and J. van Pelt, A Mathematical model for the intracellular circadian rhythm generator, Journal of Neuroscience, 19 (1999), 40-47. doi: 10.1523/JNEUROSCI.19-01-00040.1999. Google Scholar [28] A. N. Sharkovsy, Yu. L. Maistrenko and E. Yu. Romanenko, Difference Equations and Their Perturbations, Kluwer Academic Publishers, 1993. doi: 10.1007/978-94-011-1763-0. Google Scholar [29] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer-Verlag, 2011. doi: 10.1007/978-1-4419-7646-8. Google Scholar [30] M. Wazewska-Czyzewska and A. Lasota, Matematyczne problemy dynamiki układu krwinek czerwonych, (Polish), [Mathematical models of the red cell system], Matematyka Stosowana, 6 (1976), 25-40.Google Scholar [31] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110879971. Google Scholar

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##### References:
 [1] U.an der Heiden, Periodic solutions of a nonlinear second order differential equation with delay, J. Math. Anal. Appl., 70 (1979), 599-609. doi: 10.1016/0022-247X(79)90068-4. Google Scholar [2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, London, 1963. Google Scholar [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. Google Scholar [4] O. Diekmann, S. van Gils, S. Verduyn Lunel and H.-O. Walther, Delay Equations: Complex, Functional, and Nonlinear Analysis, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. Google Scholar [5] Á. Elbert and I. P. Stavroulakis, Oscillation and nonoscillation criteria for delay differential equations, Proc. Amer. Math. Soc., 123 (1995), 1503-1510. doi: 10.1090/S0002-9939-1995-1242082-1. Google Scholar [6] T. Erneux, Applied Delay Differential Equations, Springer-Verlag, New York, 2009. Google Scholar [7] J. B. Conway, Functions of One Complex Variable, $2^{nd}$ edition, Springer, 1978. Google Scholar [8] B. C. Goodwin, Oscillatory behaviour in enzymatic control process, Adv. Enzime Regul., 3 (1965), 425-438. Google Scholar [9] I. Györy and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. Google Scholar [10] K. P. Hadeler, Delay equations in biology, in Lecture Notes in Mathematics, Springer, 730 (1979), 139-156. Google Scholar [11] K. P. Hadeler and J. Tomiuk, Periodic solutions of difference differential equations, Arch. Rat. Mech. Anal., 65 (1977), 87-95. doi: 10.1007/BF00289359. Google Scholar [12] J. K. Hale and A. F. Ivanov, On a high order differential delay equation, J. Math. Anal. Appl., 173 (1993), 505-514. doi: 10.1006/jmaa.1993.1083. Google Scholar [13] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [14] J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Natl. Acad. Sci. USA, 79 (1982), 2554-2558. doi: 10.1073/pnas.79.8.2554. Google Scholar [15] A. F. Ivanov and B. Lani-Wayda, Periodic solutions for three-dimensional non-monotone cyclic systems with time delays, Discrete and Continuous Dynam. Systems- A, 11 (2004), 667-692. doi: 10.3934/dcds.2004.11.667. Google Scholar [16] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. Google Scholar [17] P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002. Google Scholar [18] B. Li, Oscillations of delay differential equations with variable coefficients, J. Math. Anal. Appl., 192 (1995), 312-321. doi: 10.1006/jmaa.1995.1173. Google Scholar [19] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar [20] J. Mahaffy, Periodic solutions of certain protein synthesis models, J. Math. Anal. Appl., 74 (1980), 72-105. doi: 10.1016/0022-247X(80)90115-8. Google Scholar [21] J. Mallet-Paret, Morse decompositions for delay differential equations, J. Differential Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X. Google Scholar [22] J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-47. doi: 10.1023/A:1021889401235. Google Scholar [23] J. Mallet-Paret and R. D. Nussbaum, A differential delay equation arising in optics and physiology, SIAM J. Math. Anal., 20 (1989), 249-292. doi: 10.1137/0520019. Google Scholar [24] J. Mallet-Paret and G. Sell, Systems of delay differential equations Ⅰ: Floquet multipliers and discrete Lyapunov functions, J. Differential Equations, 125 (1996), 385-440. doi: 10.1006/jdeq.1996.0036. Google Scholar [25] J. Mallet-Paret and G. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037. Google Scholar [26] M. Pituk, Asymptotic behavior and oscillation of functional differential equations, J. Math. Anal. Appl., 322 (2006), 1140-1158. doi: 10.1016/j.jmaa.2005.09.081. Google Scholar [27] T. Scheper, D. Klinkenberg, C. Pennartz and J. van Pelt, A Mathematical model for the intracellular circadian rhythm generator, Journal of Neuroscience, 19 (1999), 40-47. doi: 10.1523/JNEUROSCI.19-01-00040.1999. Google Scholar [28] A. N. Sharkovsy, Yu. L. Maistrenko and E. Yu. Romanenko, Difference Equations and Their Perturbations, Kluwer Academic Publishers, 1993. doi: 10.1007/978-94-011-1763-0. Google Scholar [29] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer-Verlag, 2011. doi: 10.1007/978-1-4419-7646-8. Google Scholar [30] M. Wazewska-Czyzewska and A. Lasota, Matematyczne problemy dynamiki układu krwinek czerwonych, (Polish), [Mathematical models of the red cell system], Matematyka Stosowana, 6 (1976), 25-40.Google Scholar [31] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110879971. Google Scholar
Graphs of $\Theta_0(\omega) = \sum_{j = 1}^n\theta_j$ and $y = -\omega\tau+\pi(2k-1)$, $k\in\mathbb N$ for $n = 6$ (upper) and $n = 10$ (lower).
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