AVERAGING PRINCIPLE FOR NONLINEAR DIFFERENTIAL SYSTEMS WITH JORDAN BLOCKS

Shuyuan Xiao; Zhicheng Tong

doi:10.11948/20230355

2024 Volume 14 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2024 > 14(2): 1097-1110

Next Article Previous Article

Shuyuan Xiao, Zhicheng Tong. AVERAGING PRINCIPLE FOR NONLINEAR DIFFERENTIAL SYSTEMS WITH JORDAN BLOCKS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 1097-1110. doi: 10.11948/20230355

Citation:

Shuyuan Xiao, Zhicheng Tong. AVERAGING PRINCIPLE FOR NONLINEAR DIFFERENTIAL SYSTEMS WITH JORDAN BLOCKS[J]. Journal of Applied Analysis & Computation, 2024, 14(2): 1097-1110. doi: 10.11948/20230355

AVERAGING PRINCIPLE FOR NONLINEAR DIFFERENTIAL SYSTEMS WITH JORDAN BLOCKS

Shuyuan Xiao^1,,
Zhicheng Tong^2, ,

1.
School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, China
2.
School of Mathematics, Jilin University, Changchun 130012, China

Author Bio: Email: xiaosy716@nenu.edu.cn(S. Xiao)
Corresponding author: Email: tongzc20@mails.jlu.edu.cn(Z. Tong)

Abstract

Abstract

This paper studies a perturbed differential system
$\begin{equation*} \frac{\partial v}{\partial t}=Av+\varepsilon H(v), \ v(0)=v_0, \ \varepsilon\in(0, 1], \end{equation*}$
where $A$ is a linear operator having purely imaginary eigenvalues with Jordan blocks, and $H$ is an analytic perturbation satisfying $H(v)=\mathcal{O}(|v|^2)$ as $|v|\rightarrow 0$. Such a case cannot be dealt with straightforwardly by the averaging principle due to the difficulties presenting by $ A $. To this end, by employing the Poincaré normal form with nilpotent term for nonlinear quasiperiodic system to simplify the above differential system, we extend the classical Krylov-Bogoliubov averaging method to nonlinear systems admitting Jordan blocks.
- Krylov-Bogoliubov averaging method /
- Jordan blocks /
- Poincaré normal form /
- nilpotent term
MSC: 34C29

FullText(HTML)

References

[1]	P. Fatou, Sur le mouvement d'un systéme soumis à des forces à courte période, Bull. Soc. Math. Fr., 1928, 56, 98–139. Google Scholar
[2]	P. Gao, Averaging principle for stochastic Kuramoto-Sivashinsky equation with a fast oscillation, Discrete Contin. Dyn. Syst., 2018, 38(11), 5649–5684. DOI: 10.3934/dcds.2018247. CrossRef Google Scholar
[3]	P. Gao, Averaging principles for stochastic 2D Navier-Stokes equations, J. Stat. Phys., 2022, 186(2), 28, 29 pp. DOI: 10.1007/s10955-022-02876-9. Google Scholar
[4]	P. Gao and Y. Li, Averaging principle for the Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 2017, 22(6), 2147–2168. DOI: 10.3934/dcdsb.2017089. CrossRef Google Scholar
[5]	I. M. Gel'fand, Lectures on Linear Algebra, Interscience Tracts in Pure and Applied Mathematics, Vol. 9., Interscience Publishers, New York-London, 1961. Google Scholar
[6]	G. Huang, An averaging theorem for nonlinear Schrödinger equations with small nonlinearities, Discrete Contin. Dyn. Sys., 2014, 34(9), 3555–3574. DOI: 10.3934/dcds.2014.34.3555. CrossRef Google Scholar
[7]	G. Huang, S. B. Kuksin and A. M. Maiocchi, Time-averaging for weakly nonlinear CGL equations with arbitrary potentials, Hamiltonian Partial Differential Equations and Applications, Vol. 75, Springer, New York, 2015, 323–349. DOI: 10.1007/978-1-4939-2950-4_11. Google Scholar
[8]	W. W. Jian, S. B. Kuksin and Y. Wu, Krylov-Bogolyubov averaging, Uspekhi Mat. Nauk, 2020, 75(3), 37–54. DOI: 10.4213/rm9933. CrossRef Google Scholar
[9]	R. Z. Khas'minskii, The behavior of a conservative system under the action of slight friction and slight random noise, Prikl. Mat. Meh., 1964, 28(5), 931–935. DOI: 10.1016/0021-8928(64)90017-6. CrossRef Google Scholar
[10]	V. G. Kolomiets, A. I. Mel'nikov and T. M. Muhitdinov, Averaging of stochastic systems of integral-differential equations with Poisson noise, Ukrain. Mat. Zh., 1991, 43(2), 273–277. DOI: 10.1007/BF01060515. CrossRef Google Scholar
[11]	N. M. Krylov and N. N. Bogoliubov, Introduction to Nonlinear Mechanics, Princeton University Press, Princeton, 1947. Google Scholar
[12]	L. N'Goran and M. N'zi, Averaging principle for multivalued stochastic differential equations, Random Oper. Stoch. Equ., 2001, 9(4), 399–407. DOI: 10.1515/rose.2001.9.4.399. CrossRef Google Scholar
[13]	J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, Vol. 59, Springer-Verlag, New York, 1985. DOI: 10.1007/978-1-4757-4575-7. Google Scholar
[14]	R. L. Stratnovich, Izbrannye Voprosy Teorii Fluktuatsii v Radiotekhnike, 1961. Google Scholar
[15]	I. Vrkoč, Extension of the averaging method to stochastic equations, Czechoslovak Math. J., 1966, 16(91), 518–544. Google Scholar
[16]	S. Y. Xiao and Y. Li, Normal form for nonlinear quasiperiodic systems, submitted. Google Scholar
[17]	J. M. Xing, X. Yang and Y. Li, Affine-periodic solutions by averaging methods, Sci. China Math., 2018, 61(3), 439–452. DOI: 10.1007/s11425-016-0455-1. CrossRef Google Scholar
[18]	Y. Xu, J. Q. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Phys. D, 2011, 240, 1395–1401. DOI: 10.1016/j.physd.2011.06.001. CrossRef Google Scholar
[19]	N. T. Zung, Convergence versus integrability in Birkhoff normal form, Ann. Math., 2005, 161(1), 141–156. DOI: 10.4007/annals.2005.161.141. CrossRef Google Scholar