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June  2020, 19(6): 3257-3281. doi: 10.3934/cpaa.2020144

Stable periodic solutions for Nazarenko's equation

Szandra Beretka 1, and  Gabriella Vas 2,,

1. 

Bolyai Institute, University of Szeged, 1 Aradi vértanúk tere, Szeged, Hungary
2. 

MTA-SZTE Analysis and Stochastics Research Group, , Bolyai Institute, University of Szeged, 1 Aradi vértanúk tere, Szeged, Hungary

* Corresponding author

Received  July 2019 Revised  December 2019 Published  March 2020

Fund Project: This research was supported by the EU-funded Hungarian grant EFOP-3.6.1-16-2016-00008. Gabriella Vas was also supported by the National Research, Development and Innovation Office of Hungary, Grant No. K129322

In 1976 Nazarenko proposed studying the delay differential equation
$ \begin{equation*} \dot{y}(t) = -py(t)+\dfrac{qy(t)}{r+y^{n}(t-\tau)},\qquad t>0, \end{equation*} $
under the assumptions that
$ p,q,r,\tau\in\left(0,\infty\right) $
,
$ n\in\mathbb{N} = \left\{ 1,2,\ldots\right\} $
 and
$ q/p>r $
. We show that if
$ \tau $
 or
$ n $
 is large enough, then the positive periodic solution oscillating slowly about
$ K = \left(q/p-r\right)^{1/n} $
 is unique, and the corresponding periodic orbit is asymptotically stable. We also determine the asymptotic shape of the periodic solution as
$ n\rightarrow\infty $
.
Keywords: Delay differential equation, periodic solution, slow oscillation, uniqueness, stability.
Mathematics Subject Classification: Primary: 34K13, 92D25; Secondary: 34K27.
Citation: Szandra Beretka, Gabriella Vas. Stable periodic solutions for Nazarenko's equation. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3257-3281. doi: 10.3934/cpaa.2020144
References:
[1]

Y. Cao, Multiexistence of slowly oscillating periodic solutions for differential delay equations, SIAM J. Math. Anal., 26 (1995), 436-445.  doi: 10.1137/0526022.  Google Scholar

[2]

Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Differ. Equ., 128 (1996), 46-57.  doi: 10.1006/jdeq.1996.0088.  Google Scholar

[3]

J. L. Kaplan and J. A. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal., 6 (1975), 268-282.  doi: 10.1137/0506028.  Google Scholar

[4]

B. Kennedy and E. Stumpf, Multiple slowly oscillating periodic solutions for $x' (t) = f(x(t- 1))$ with negative feedback, Ann. Polon. Math., 118 (2016), 113-140.  doi: 10.4064/ap3899-10-2016.  Google Scholar

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I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Model., 35 (2002), 295-301.  doi: 10.1016/S0895-7177(01)00166-2.  Google Scholar

[6] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.   Google Scholar
[7]

E. Liz and G. Röst, Dichotomy results for delay differential equations with negative Schwarzian derivative, Nonlinear Anal. Real World Appl., 11 (2010), 1422-1430.  doi: 10.1016/j.nonrwa.2009.02.030.  Google Scholar

[8]

J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equ., 125 (1996), 441-489.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

[9]

V. G. Nazarenko, Influence of delay on auto-oscillations in cell populations, Biofisika, 21 (1976), 352-356.   Google Scholar

[10]

R. D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal., 19 (1975), 319-338.  doi: 10.1016/0022-1236(75)90061-0.  Google Scholar

[11]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 101 (1974), 263-306.  doi: 10.1007/BF02417109.  Google Scholar

[12]

R. D. Nussbaum, The range of periods of periodic solutions of $x' (t) = - \alpha f(x(t- 1))$, J. Math. Anal. Appl., 58 (1977), 280-292.  doi: 10.1016/0022-247X(77)90206-2.  Google Scholar

[13]

R. D. Nussbaum, Uniqueness and nonuniqueness for periodic solutions of $x'(t) = -g(x(t -1))$, J. Differ. Equ., 34 (1979), 25-54.  doi: 10.1016/0022-0396(79)90016-0.  Google Scholar

[14]

S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications (eds. O. Arino, M. L. Hbid and E. Ait Dads), NATO Sci. Ser. Ⅱ Math. Phys. Chem., Vol. 205, Springer, Dordrecht, (2006), 477–517. doi: 10.1007/1-4020-3647-7_11.  Google Scholar

[15]

S. H. Saker and J. O. Alzabut, Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear Anal. Real World Appl., 8 (2007), 1029-1039.  doi: 10.1016/j.nonrwa.2006.06.001.  Google Scholar

[16]

Y. Song and Y. Peng, Periodic solutions of a nonautonomous periodic model of population with continuous and discrete time, J. Comput. Appl. Math., 188 (2006), 256-264.  doi: 10.1016/j.cam.2005.04.017.  Google Scholar

[17]

Y. Song, J. Wei and M. Han, Local and global Hopf bifurcation in a delayed hematopoiesis model, Int. J. Bifurcation Chaos Appl. Sci. Eng., 14 (2004), 3909-3919.  doi: 10.1142/S0218127404011697.  Google Scholar

[18]

H. O. Walther, Contracting return maps for some delay differential equations, in Topics in Functional Differential and Difference Equations (Lisbon, 1999), Fields Inst. Commun., Vol. 29, American Mathematical Society, Providence, RI, (2001), 349–360.  Google Scholar

[19]

Q. Wang, J. Wen, S. Qiu and C. Guo, Numerical oscillations for first-order nonlinear delay differential equations in a hematopoiesis model, Adv. Differ. Equ., (2013), 17. doi: 10.1186/1687-1847-2013-163.  Google Scholar

[20]

J. Wu, Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.  Google Scholar

show all references

References:
[1]

Y. Cao, Multiexistence of slowly oscillating periodic solutions for differential delay equations, SIAM J. Math. Anal., 26 (1995), 436-445.  doi: 10.1137/0526022.  Google Scholar

[2]

Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Differ. Equ., 128 (1996), 46-57.  doi: 10.1006/jdeq.1996.0088.  Google Scholar

[3]

J. L. Kaplan and J. A. Yorke, On the stability of a periodic solution of a differential delay equation, SIAM J. Math. Anal., 6 (1975), 268-282.  doi: 10.1137/0506028.  Google Scholar

[4]

B. Kennedy and E. Stumpf, Multiple slowly oscillating periodic solutions for $x' (t) = f(x(t- 1))$ with negative feedback, Ann. Polon. Math., 118 (2016), 113-140.  doi: 10.4064/ap3899-10-2016.  Google Scholar

[5]

I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Model., 35 (2002), 295-301.  doi: 10.1016/S0895-7177(01)00166-2.  Google Scholar

[6] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.   Google Scholar
[7]

E. Liz and G. Röst, Dichotomy results for delay differential equations with negative Schwarzian derivative, Nonlinear Anal. Real World Appl., 11 (2010), 1422-1430.  doi: 10.1016/j.nonrwa.2009.02.030.  Google Scholar

[8]

J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equ., 125 (1996), 441-489.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

[9]

V. G. Nazarenko, Influence of delay on auto-oscillations in cell populations, Biofisika, 21 (1976), 352-356.   Google Scholar

[10]

R. D. Nussbaum, A global bifurcation theorem with applications to functional differential equations, J. Funct. Anal., 19 (1975), 319-338.  doi: 10.1016/0022-1236(75)90061-0.  Google Scholar

[11]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 101 (1974), 263-306.  doi: 10.1007/BF02417109.  Google Scholar

[12]

R. D. Nussbaum, The range of periods of periodic solutions of $x' (t) = - \alpha f(x(t- 1))$, J. Math. Anal. Appl., 58 (1977), 280-292.  doi: 10.1016/0022-247X(77)90206-2.  Google Scholar

[13]

R. D. Nussbaum, Uniqueness and nonuniqueness for periodic solutions of $x'(t) = -g(x(t -1))$, J. Differ. Equ., 34 (1979), 25-54.  doi: 10.1016/0022-0396(79)90016-0.  Google Scholar

[14]

S. Ruan, Delay differential equations in single species dynamics, in Delay differential equations and applications (eds. O. Arino, M. L. Hbid and E. Ait Dads), NATO Sci. Ser. Ⅱ Math. Phys. Chem., Vol. 205, Springer, Dordrecht, (2006), 477–517. doi: 10.1007/1-4020-3647-7_11.  Google Scholar

[15]

S. H. Saker and J. O. Alzabut, Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear Anal. Real World Appl., 8 (2007), 1029-1039.  doi: 10.1016/j.nonrwa.2006.06.001.  Google Scholar

[16]

Y. Song and Y. Peng, Periodic solutions of a nonautonomous periodic model of population with continuous and discrete time, J. Comput. Appl. Math., 188 (2006), 256-264.  doi: 10.1016/j.cam.2005.04.017.  Google Scholar

[17]

Y. Song, J. Wei and M. Han, Local and global Hopf bifurcation in a delayed hematopoiesis model, Int. J. Bifurcation Chaos Appl. Sci. Eng., 14 (2004), 3909-3919.  doi: 10.1142/S0218127404011697.  Google Scholar

[18]

H. O. Walther, Contracting return maps for some delay differential equations, in Topics in Functional Differential and Difference Equations (Lisbon, 1999), Fields Inst. Commun., Vol. 29, American Mathematical Society, Providence, RI, (2001), 349–360.  Google Scholar

[19]

Q. Wang, J. Wen, S. Qiu and C. Guo, Numerical oscillations for first-order nonlinear delay differential equations in a hematopoiesis model, Adv. Differ. Equ., (2013), 17. doi: 10.1186/1687-1847-2013-163.  Google Scholar

[20]

J. Wu, Symmetric functional-differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.  Google Scholar

Figure 1.  The plot of $ f $ for $ p = 1 $, $ q = 4, $ $ r = 1.5 $ and $ n = 10 $
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Figure 2.  An element of $ \mathcal{N}(A,B,\beta,\varepsilon) $
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Figure 3.  Upper and lower estimates for the SOP solution $ \bar{x} $ of (1.4) if $ p = 2.8 $, $ q = 6, $ $ r = 1.3 $, $ \tau = 5 $ and $ n = 350 $. For these parameters, $ |\bar{x}(t)-v(t)|<0.54 $ for all $ t\in[0,\bar{\omega}] $
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Table 1.  A few parameters for which Theorem 1.1 holds
$ p= $ $ q= $ $ r= $ $ n= $ $ \tau\geq $
2.8 6 1.3 19 5.16
2.8 6.9 0.9 25 2.41
2.8 6.9 0.9 2 23.68
1.9 4.2 0.8 20 3.88
0.7 1.3 0.7 30 8.84
1.9 6.9 0.8 15 8.16
6.6 9.3 0.4 10 2.63
3 5.3 1.3 15 9.71
8.8 5.9 0.5 20 8.52
9 6.4 0.4 5 6.62
9 6.4 0.4 2 16.54
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$ p= $ $ q= $ $ r= $ $ n= $ $ \tau\geq $
2.8 6 1.3 19 5.16
2.8 6.9 0.9 25 2.41
2.8 6.9 0.9 2 23.68
1.9 4.2 0.8 20 3.88
0.7 1.3 0.7 30 8.84
1.9 6.9 0.8 15 8.16
6.6 9.3 0.4 10 2.63
3 5.3 1.3 15 9.71
8.8 5.9 0.5 20 8.52
9 6.4 0.4 5 6.62
9 6.4 0.4 2 16.54
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