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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 and Continuous Dynamical Systems, 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.

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

[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.

[5]

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

[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.

[7]

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

show all references

References:
[1]

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

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

[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.

[5]

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

[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.

[7]

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

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