American Institute of Mathematical Sciences

Advanced
June  2020, 40(6): 3789-3812. doi: 10.3934/dcds.2020044

Asymptotic homogenization for delay-differential equations and a question of analyticity

John Mallet-Paret 1, and  Roger D. Nussbaum 2,

1. 

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
2. 

Department of Mathematics, Rutgers University, Piscataway, NJ 08854, USA

Received  March 2019 Revised  June 2019 Published  October 2019

Fund Project: The first author was partially supported by The Center for Nonlinear Analysis at Rutgers University. The second author was partially supported by NSF Grant DMS-1201328 and by The Lefschetz Center for Dynamical Systems at Brown University

We consider a class of nonautonomous delay-differential equations in which the time-varying coefficients have an oscillatory character, with zero mean value, and whose frequency approaches
$ +\infty $
 as
$ t\to\pm\infty $
. Typical simple examples are
$ x'(t) = \sin (t^q)x(t-1) \qquad\text{and}\qquad x'(t) = e^{it^q}x(t-1), \;\;\;\;\;\;\;\;{(*)} $
where
$ q\ge 2 $
 is an integer. Under various conditions, we show the existence of a unique solution with any prescribed finite limit
$ \lim\limits_{t\to-\infty}x(t) = x_- $
 at
$ -\infty $
. We also show, under appropriate conditions, that any solution of an initial value problem has a finite limit
$ \lim\limits_{t\to+\infty}x(t) = x_+ $
 at
$ +\infty $
, and thus we establish the existence of a class of heteroclinic solutions. We term this limiting phenomenon, and thus the existence of such solutions, "asymptotic homogenization." Note that in general, proving the existence of a bounded solution of a given delay-differential equation on a semi-infinite interval
$ (-\infty,-T] $
 is often highly nontrivial.
Our original interest in such solutions stems from questions concerning their smoothness. In particular, any solution
$ x: \mathbb{R}\to \mathbb{C} $
 of one of the equations in
$ (*) $
 with limits
$ x_\pm $
 at
$ \pm\infty $
 is
$ C^\infty $
, but it is unknown if such solutions are analytic. Nevertheless, one does know that any such solution of the second equation in
$ (*) $
 can be extended to the lower half plane
$ \{z\in \mathbb{C}\:|\: \mathop{{{\rm{Im}}}} z<0\} $
 as an analytic function.
Keywords: Delay-differential equation, heteroclinic solution, asymptotic behavior, analytic solution, rapid oscillation, averaging, homogenization.
Mathematics Subject Classification: Primary: 34K12, 34K25, 37C29; Secondary: 26E05, 26E10, 37C60.
Citation: John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044
References:
[1]

N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, Inc., New York, 1981.  Google Scholar

[2] J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969.   Google Scholar
[3]

J. Mallet-Paret and R. D. Nussbaum, Analyticity and nonanalyticity of solutions of delay-differential equations, SIAM J. Math. Anal., 46 (2014), 2468-2500.  doi: 10.1137/13091943X.  Google Scholar

[4]

J. Mallet-Paret and R. D. Nussbaum, Intricate structure of the analyticity set for solutions of a class of integral equations, J. Dynam. Differential Equations, 31 (2019), 1045-1077.  doi: 10.1007/s10884-019-09746-1.  Google Scholar

[5]

J. Mallet-Paret and R. D. Nussbaum, Analytic solutions of delay-differential equations, in preparation. Google Scholar

[6]

R. D. Nussbaum, Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20 (1973), 249-255.  doi: 10.1307/mmj/1029001104.  Google Scholar

[7]

A. Zygmund, Trigonometric Series, Vols. I, II, 2nd edition, Cambridge Univ. Press, New York, 1959.  Google Scholar

show all references

References:
[1]

N. G. de Bruijn, Asymptotic Methods in Analysis, Dover Publications, Inc., New York, 1981.  Google Scholar

[2] J. Dieudonné, Foundations of Modern Analysis, Pure and Applied Mathematics, Vol. 10-I, Academic Press, New York-London, 1969.   Google Scholar
[3]

J. Mallet-Paret and R. D. Nussbaum, Analyticity and nonanalyticity of solutions of delay-differential equations, SIAM J. Math. Anal., 46 (2014), 2468-2500.  doi: 10.1137/13091943X.  Google Scholar

[4]

J. Mallet-Paret and R. D. Nussbaum, Intricate structure of the analyticity set for solutions of a class of integral equations, J. Dynam. Differential Equations, 31 (2019), 1045-1077.  doi: 10.1007/s10884-019-09746-1.  Google Scholar

[5]

J. Mallet-Paret and R. D. Nussbaum, Analytic solutions of delay-differential equations, in preparation. Google Scholar

[6]

R. D. Nussbaum, Periodic solutions of analytic functional differential equations are analytic, Michigan Math. J., 20 (1973), 249-255.  doi: 10.1307/mmj/1029001104.  Google Scholar

[7]

A. Zygmund, Trigonometric Series, Vols. I, II, 2nd edition, Cambridge Univ. Press, New York, 1959.  Google Scholar

Table 1.  In every summation we assume that $ j,k\in \mathbb{Z}\setminus\{0\} $.
$\begin{array}{lcl} \Gamma_1(t) & = & {\frac{B(t^2)}{2t},}\\ \\ \Omega_1(t) & = & {\frac{B(t^2)}{2t^2},}\\ \\ \Gamma_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg) +\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t}\bigg),}\\ \\ \Omega_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{ia_ja_kk}{2 \omega j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{2 \omega^2 j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^3}\bigg)}\\ \\ & & {+\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t^2}\bigg),}\\ \\ \Gamma_3(t) & = & {\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg),}\\ \\ \Omega_3(t) & = & {-\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)A((t-2)^2)}\\ \\ & & {+\sum\limits_{j,k}\frac{a_ja_{-j}a_k(j-2k)}{4 \omega^2j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{4 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^3}\bigg),}\\ \\ \Omega_4(t) & = & {-\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)A((t-3)^2),} \end{array}$
Table Options

Download as excel

$\begin{array}{lcl} \Gamma_1(t) & = & {\frac{B(t^2)}{2t},}\\ \\ \Omega_1(t) & = & {\frac{B(t^2)}{2t^2},}\\ \\ \Gamma_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg) +\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t}\bigg),}\\ \\ \Omega_2(t) & = & {\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{ia_ja_kk}{2 \omega j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{2 \omega^2 j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^3}\bigg)}\\ \\ & & {+\sum\limits_j\frac{a_ja_{-j}}{4 \omega^2j^2}\bigg(\frac{e^{i \omega j(2t-1)}}{t^2}\bigg),}\\ \\ \Gamma_3(t) & = & {\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg),}\\ \\ \Omega_3(t) & = & {-\sum\limits_{{j,k}\atop{j+k\ne 0}}\frac{a_ja_k}{4 \omega^2j(j+k)}\bigg(\frac{e^{i \omega(jt^2+k(t-1)^2)}}{t^2}\bigg)A((t-2)^2)}\\ \\ & & {+\sum\limits_{j,k}\frac{a_ja_{-j}a_k(j-2k)}{4 \omega^2j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)}\\ \\ & & {+\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{4 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^3}\bigg),}\\ \\ \Omega_4(t) & = & {-\sum\limits_{j,k}\frac{ia_ja_{-j}a_k}{8 \omega^3j^2k}\bigg(\frac{e^{i \omega (j(2t-1)+k(t-2)^2)}}{t^2}\bigg)A((t-3)^2),} \end{array}$
[1]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[2]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[3]

Ran Zhang, Shengqiang Liu. On the asymptotic behaviour of traveling wave solution for a discrete diffusive epidemic model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1197-1204. doi: 10.3934/dcdsb.2020159
[4]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[5]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016
[7]

Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160
[8]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091
[9]

Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115
[10]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298
[11]

Feimin Zhong, Jinxing Xie, Yuwei Shen. Bargaining in a multi-echelon supply chain with power structure: KS solution vs. Nash solution. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020172
[12]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256
[13]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119
[14]

Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control & Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033
[15]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018
[16]

Yoichi Enatsu, Emiko Ishiwata, Takeo Ushijima. Traveling wave solution for a diffusive simple epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 835-850. doi: 10.3934/dcdss.2020387
[17]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[18]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[19]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[20]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (151)
  • HTML views (307)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences