January  2020, 13(1): 1-29. doi: 10.3934/dcdss.2020001

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 and Continuous Dynamical Systems - S, 2020, 13 (1) : 1-29. doi: 10.3934/dcdss.2020001
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.

[2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, London, 1963. 
[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[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.

[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.

[6]

T. Erneux, Applied Delay Differential Equations, Springer-Verlag, New York, 2009.

[7]

J. B. Conway, Functions of One Complex Variable, $2^{nd}$ edition, Springer, 1978.

[8]

B. C. Goodwin, Oscillatory behaviour in enzymatic control process, Adv. Enzime Regul., 3 (1965), 425-438. 

[9] I. Györy and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. 
[10]

K. P. Hadeler, Delay equations in biology, in Lecture Notes in Mathematics, Springer, 730 (1979), 139-156.

[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.

[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.

[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.

[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.

[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.

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.

[17]

P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002.

[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.

[19]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

T. ScheperD. KlinkenbergC. 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.

[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.

[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.

[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.

[31]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110879971.

show all references

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.

[2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, London, 1963. 
[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011.

[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.

[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.

[6]

T. Erneux, Applied Delay Differential Equations, Springer-Verlag, New York, 2009.

[7]

J. B. Conway, Functions of One Complex Variable, $2^{nd}$ edition, Springer, 1978.

[8]

B. C. Goodwin, Oscillatory behaviour in enzymatic control process, Adv. Enzime Regul., 3 (1965), 425-438. 

[9] I. Györy and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. 
[10]

K. P. Hadeler, Delay equations in biology, in Lecture Notes in Mathematics, Springer, 730 (1979), 139-156.

[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.

[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.

[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.

[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.

[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.

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.

[17]

P. D. Lax, Functional Analysis, Wiley-Interscience, New York, 2002.

[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.

[19]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.  doi: 10.1126/science.267326.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

T. ScheperD. KlinkenbergC. 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.

[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.

[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.

[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.

[31]

J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter & Co., Berlin, 2001. doi: 10.1515/9783110879971.

Figure 1.  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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