# American Institute of Mathematical Sciences

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

## Stable periodic solutions for Nazarenko's equation

 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
 $n\rightarrow\infty$
.
Citation: Szandra Beretka, Gabriella Vas. Stable periodic solutions for Nazarenko's equation. Communications on Pure and 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. [2] Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Differ. Equ., 128 (1996), 46-57.  doi: 10.1006/jdeq.1996.0088. [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. [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. [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. [6] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [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. [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. [9] V. G. Nazarenko, Influence of delay on auto-oscillations in cell populations, Biofisika, 21 (1976), 352-356. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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##### 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. [2] Y. Cao, Uniqueness of periodic solution for differential delay equations, J. Differ. Equ., 128 (1996), 46-57.  doi: 10.1006/jdeq.1996.0088. [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. [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. [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. [6] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. [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. [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. [9] V. G. Nazarenko, Influence of delay on auto-oscillations in cell populations, Biofisika, 21 (1976), 352-356. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
The plot of $f$ for $p = 1$, $q = 4,$ $r = 1.5$ and $n = 10$
An element of $\mathcal{N}(A,B,\beta,\varepsilon)$
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 $t\in[0,\bar{\omega}]$
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
 $p=$ $q=$ $r=$ $n=$ $\tau\geq$ 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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