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]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023
[2]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[3]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[4]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[5]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109
[6]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[7]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[8]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[9]

Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006
[10]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190
[11]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055
[12]

V. Vijayakumar, R. Udhayakumar, K. Kavitha. On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay. Evolution Equations & Control Theory, 2021, 10 (2) : 271-296. doi: 10.3934/eect.2020066
[13]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[14]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008
[15]

Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[16]

Simone Calogero, Juan Calvo, Óscar Sánchez, Juan Soler. Dispersive behavior in galactic dynamics. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 1-16. doi: 10.3934/dcdsb.2010.14.1
[17]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[18]

Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067
[19]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[20]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (159)
  • HTML views (326)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences