On bifurcation of eigenvalues along convex symplectic paths

Chang, Yinshan; Long, Yiming; Wang, Jian

doi:10.1016/j.anihpc.2018.04.001

Chang, Yinshan ; Long, Yiming ; Wang, Jian

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 1, pp. 75-102.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We consider a continuously differentiable curve $t \mapsto γ (t)$ in the space of $2 n \times 2 n$ real symplectic matrices, which is the solution of the following ODE:

\frac{d γ}{d t} (t) = J_{2 n} A (t) γ (t), γ (0) \in Sp (2 n, R),

where

J = J_{2 n} \overset{def}{=} [\begin{matrix} 0 & {Id}_{n} \\ - {Id}_{n} & 0 \end{matrix}]

and

A : t \mapsto A (t)

is a continuous path in the space of

2 n \times 2 n

real matrices which are symmetric. Under a certain convexity assumption (which includes the particular case that

A (t)

is strictly positive definite for all

t \in R

), we investigate the dynamics of the eigenvalues of

γ (t)

when t varies, which are closely related to the stability of such Hamiltonian dynamical systems. We rigorously prove the qualitative behavior of the branching of eigenvalues and explicitly give the first order asymptotics of the eigenvalues. This generalizes classical Krein–Lyubarskii theorem on the analytic bifurcation of the Floquet multipliers under a linear perturbation of the Hamiltonian. As a corollary, we give a rigorous proof of the following statement of Ekeland:

{t \in R : γ (t) has a Krein indefinite eigenvalue of modulus 1}

is a discrete set.

MR Zbl

DOI : 10.1016/j.anihpc.2018.04.001

Classification : 37J20, 37J25
Mots-clés : Floquet multipliers, Symplectic matrices, Krein type, Krein–Lyubarskii theorem, Bifurcation

@article{AIHPC_2019__36_1_75_0,
     author = {Chang, Yinshan and Long, Yiming and Wang, Jian},
     title = {On bifurcation of eigenvalues along convex symplectic paths},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {75--102},
     publisher = {Elsevier},
     volume = {36},
     number = {1},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.04.001},
     mrnumber = {3906866},
     zbl = {1409.37058},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.04.001/}
}

TY  - JOUR
AU  - Chang, Yinshan
AU  - Long, Yiming
AU  - Wang, Jian
TI  - On bifurcation of eigenvalues along convex symplectic paths
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 75
EP  - 102
VL  - 36
IS  - 1
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2018.04.001/
DO  - 10.1016/j.anihpc.2018.04.001
LA  - en
ID  - AIHPC_2019__36_1_75_0
ER  -

%0 Journal Article
%A Chang, Yinshan
%A Long, Yiming
%A Wang, Jian
%T On bifurcation of eigenvalues along convex symplectic paths
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 75-102
%V 36
%N 1
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2018.04.001/
%R 10.1016/j.anihpc.2018.04.001
%G en
%F AIHPC_2019__36_1_75_0

Chang, Yinshan; Long, Yiming; Wang, Jian. On bifurcation of eigenvalues along convex symplectic paths. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 1, pp. 75-102. doi : 10.1016/j.anihpc.2018.04.001. http://www.numdam.org/articles/10.1016/j.anihpc.2018.04.001/

Bibliographie
Cité par

[1] Ekeland, I. Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Results in Mathematics and Related Areas (3), vol. 19, Springer-Verlag, Berlin, 1990 | DOI | MR | Zbl

[2] Gel'fand, I.M.; Lidskiĭ, V.B. On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Am. Math. Soc. Transl., Volume 2 (1958) no. 8, pp. 143–181 | MR | Zbl

[3] Kato, T. Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995 (reprint of the 1980 edition) | MR | Zbl

[4] Kreĭn, M.G. A generalization of some investigations of A. M. Lyapunov on linear differential equations with periodic coefficients, Dokl. Akad. Nauk SSSR (N.S.), Volume 73 (1950), pp. 445–448 | MR | Zbl

[5] Kreĭn, M.G. On certain problems on the maximum and minimum of characteristic values and on the Lyapunov zones of stability, Akad. Nauk SSSR. Prikl. Mat. Meh., Volume 15 (1951), pp. 323–348 | MR

[6] Kreĭn, M.G.; Ljubarskiĭ, G.J. Analytic properties of the multipliers of periodic canonical differential systems of positive type, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 26 (1962), pp. 549–572 | MR

[7] Kuwamura, M.; Yanagida, E. Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems, J. Differ. Equ., Volume 230 (2006) no. 2, pp. 446–464 | DOI | MR | Zbl

[8] Long, Y. Bott formula of the Maslov-type index theory, Pac. J. Math., Volume 187 (1999) no. 1, pp. 113–149 | DOI | MR | Zbl

[9] Long, Y. Index Theory for Symplectic Paths with Applications, Progress in Mathematics, vol. 207, Birkhäuser Verlag, Basel, 2002 | DOI | MR | Zbl

[10] Moser, J. New aspects in the theory of stability of Hamiltonian systems, Commun. Pure Appl. Math., Volume 11 (1958), pp. 81–114 | DOI | MR | Zbl

[11] Winitzki, S. Linear Algebra via Exterior Products, 2010 lulu.com

[12] Yakubovich, V.A.; Starzhinskii, V.M. Linear Differential Equations with Periodic Coefficients. 1, 2, Halsted Press [John Wiley & Sons], New York–Toronto, Ont., 1975 (Israel Program for Scientific Translations, Jerusalem–London, translated from Russian by D. Louvish) | MR | Zbl

Cité par Sources :