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Journal of Dynamical and Control Systems
Published in: Journal of Dynamical and Control Systems 2/2016

29-07-2015

Confluence of Singularities of Nonlinear Differential Equations via Borel–Laplace Transformations

Author: Martin Klimeš

Published in: Journal of Dynamical and Control Systems | Issue 2/2016

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Abstract

Borel summable divergent series usually appear when studying solutions of analytic ODE near a multiple singular point. Their sum, uniquely defined in certain sectors of the complex plane, is obtained via the Borel–Laplace transformation. This article shows how to generalize the Borel–Laplace transformation in order to investigate bounded solutions of parameter dependent nonlinear differential systems with two simple (regular) singular points unfolding a double (irregular) singularity. We construct parametric solutions on domains attached to both singularities, that converge locally uniformly to the sectoral Borel sums. Our approach provides a unified treatment for all values of the complex parameter.
previous article Conformal Equivalence of 3D Contact Structures on Lie Groups
next article Boundary Partial Stabilization of a Coupled Wave Equations on an Exterior Bounded Obstacle

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Footnotes
1
If instead f(x, 𝜖,0) was only O(|x|+|𝜖|), and \(u_{\pm \sqrt \epsilon }\in \mathbb {C}^{m}\) were the unique solutions of \(0=M u_{\pm \sqrt \epsilon }+f(\pm \sqrt \epsilon ,u_{\pm \sqrt \epsilon },\epsilon )\), with u ±0 = 0, then the change of variable \(y\,\mapsto \, y-\tfrac {1}{2\sqrt \epsilon }\left (u_{+\sqrt \epsilon }(x\,+\,\sqrt \epsilon ) -\!\right .\) \(\left . \! u_{-\sqrt \epsilon }(x\,-\,\sqrt \epsilon )\right )\), analytic in (x, 𝜖), would bring the system (10) to a one with f(x, 𝜖,0)=O(x 2 −𝜖).
 
2
For m = 1, it’s been shown in [22, Proposition 3.1], cf. also [10, Lemma 1], that the family (13) is in fact locally orbitally analytically equivalent to a family (10).
 
3
These α will later correspond to the direction of the unfolded Laplace integrals (8), and \(\mathbf {T}_{\alpha }^{\pm }(\varLambda , \sqrt \epsilon )\) to their strips of convergence.
 
4
More precisely to a covering space of the x-plane ramified at \(\{\sqrt \epsilon ,-\!\sqrt \epsilon \}\), the Riemann surface of t(x, 𝜖)(9).
 
5
Zhang also unfolds the Laplace integral (3), unlike us he chooses to unfold the kernel \(e^{-\frac {\xi }{x}}d\xi \) by \(\left (\frac {x-\sqrt \epsilon \xi }{x+\sqrt \epsilon \xi }\right )^{\frac {1}{2 \sqrt \epsilon }}d\xi =e^{-t(x,\xi ^{2}\epsilon )\cdot \xi }d\xi \), in our notation.
 
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Metadata
Title
Confluence of Singularities of Nonlinear Differential Equations via Borel–Laplace Transformations
Author
Martin Klimeš
Publication date
29-07-2015
Publisher
Springer US
Published in
Journal of Dynamical and Control Systems / Issue 2/2016
Print ISSN: 1079-2724
Electronic ISSN: 1573-8698
DOI
https://doi.org/10.1007/s10883-015-9290-7

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