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 Mosc. Math. J., 2012, Volume 12, Number 1, Pages 77–138 (Mi mmj449)

Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1

Caroline Lambert, Christiane Rousseau

Département de Mathématiques et de Statistique, Université de Montréal, Montréal (Qc), Canada

Abstract: In this, paper, we give a complete system of analytic invariants for the unfoldings of nonresonant linear differential systems with an irregular singularity of Poincaré rank 1 at the origin over a fixed neighborhood $\mathbb D_r$. The unfolding parameter $\epsilon$ is taken in a sector $S$ pointed at the origin of opening larger than $2\pi$ in the complex plane, thus covering a whole neighborhood of the origin. For each parameter value $\epsilon\in S$, we cover $\mathbb D_r$ with two sectors and, over each sector, we construct a well chosen basis of solutions of the unfolded linear differential systems. This basis is used to find the analytic invariants linked to the monodromy of the chosen basis around the singular points. The analytic invariants give a complete geometric interpretation to the well-known Stokes matrices at $\epsilon=0$: this includes the link (existing at least for the generic cases) between the divergence of the solutions at $\epsilon=0$ and the presence of logarithmic terms in the solutions for resonance values of the unfolding parameter. Finally, we give a realization theorem for a given complete system of analytic invariants satisfying a necessary and sufficient condition, thus identifying the set of modules.

Key words and phrases: Stokes phenomenon, irregular singularity, unfolding, confluence, divergent series, monodromy, Riccati matrix differential equation, analytic classification, summability, realization.

DOI: https://doi.org/10.17323/1609-4514-2012-12-1-77-138

Full text: http://www.ams.org/.../abst12-1-2012.html
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MSC: Primary 34M35, 34M40, 34M50, 34M03; Secondary 37G10, 34E10, 37G05
Received: August 24, 2010; in revised form May 5, 2011
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Citation: Caroline Lambert, Christiane Rousseau, “Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1”, Mosc. Math. J., 12:1 (2012), 77–138

Citation in format AMSBIB
\Bibitem{LamRou12} \by Caroline Lambert, Christiane Rousseau \paper Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincar\'e rank~1 \jour Mosc. Math.~J. \yr 2012 \vol 12 \issue 1 \pages 77--138 \mathnet{http://mi.mathnet.ru/mmj449} \crossref{https://doi.org/10.17323/1609-4514-2012-12-1-77-138} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2952427} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309364900006} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Caroline Lambert, Christiane Rousseau, “Moduli space of unfolded differential linear systems with an irregular singularity of Poincaré rank 1”, Mosc. Math. J., 13:3 (2013), 529–550
2. Jacques Hurtubise, Caroline Lambert, Christiane Rousseau, “Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank $k$”, Mosc. Math. J., 14:2 (2014), 309–338
3. Rousseau C., “Analytic Moduli For Unfoldings of Germs of Generic Analytic Diffeomorphisms With a Codimension K Parabolic Point”, Ergod. Theory Dyn. Syst., 35:1 (2015), 274–292
4. M. Klime, “Confluence of singularities of nonlinear differential equations via Borel–Laplace transformations”, J. Dyn. Control Syst., 22:2 (2016), 285–324
5. J. Hurtubise, Ch. Rousseau, “Moduli space for generic unfolded differential linear systems”, Adv. Math., 307 (2017), 1268–1323
6. Cotti G. Guzzetti D., “Results on the Extension of Isomonodromy Deformations to the Case of a Resonant Irregular Singularity”, Random Matrices-Theor. Appl., 7:4, SI (2018), 1840003
7. Klimes M., “Stokes Phenomenon and Confluence in Non-Autonomous Hamiltonian Systems”, Qual. Theor. Dyn. Syst., 17:3 (2018), 665–708
8. Stoyanova Ts., “Zero Level Perturbation of a Certain Third-Order Linear Solvable Ode With An Irregular Singularity At the Origin of Poincar, Rank 1”, J. Dyn. Control Syst., 24:4 (2018), 511–539
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