# Overstability and resonance

• [1] Université de La Rochelle, Laboratoire de Mathématiques Calcul Asymptotique, Pôle Sciences et Technologie, Avenue Michel Crépeau, 17042 La Rochelle Cedex (France)
• [2] Université Louis Pasteur, Département de Mathématiques, 7 rue René Descartes, 67084 Strasbourg Cedex (France)
• Volume: 53, Issue: 1, page 227-264
• ISSN: 0373-0956

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## Abstract

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We consider a singularity perturbed nonlinear differential equation $\epsilon {u}^{\text{'}}=f\left(x\right)u++\epsilon P\left(x,u,\epsilon \right)$ which we suppose real analytic for $x$ near some interval $\left[a,b\right]$ and small $|u|$, $|\epsilon |$. We furthermore suppose that 0 is a turning point, namely that $xf\left(x\right)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $ϵ\to 0$) local (at $x=0$) or global, real analytic or ${C}^{\infty }$ solutions is equivalent to the existence of a formal series solution $\sum {u}_{n}\left(x\right){\epsilon }^{n}$ with ${u}_{n}$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions. We apply this result to the second order linear differential equation $\epsilon {y}^{\text{'}\text{'}}+\varphi \left(x,\epsilon \right){y}^{\text{'}}+\psi \left(x,\epsilon \right)y=0$ with $\varphi$ and $\psi$ real analytic for $x$ near some interval $\left[a,b\right]$ and small $|\epsilon |$. We assume that $-x\varphi \left(x,0\right)$ is positive if $x\ne 0$ and that the function ${\psi }_{0}:x↦\psi \left(x,0\right)$ has a zero at $x=0$ of at least the same order as ${\varphi }_{0}↦\varphi \left(x,0\right)$. For this equation, we prove that the existence of local or global, real analytic or ${C}^{\infty }$ solutions tending to a nontrivial solution of the reduced equation $\varphi \left(x,0\right){y}^{\text{'}}+\psi \left(x,0\right)y=0$ is equivalent to the existence of a non trivial formal series solution $\stackrel{^}{y}\left(x,\epsilon \right)=\sum {y}_{n}\left(x\right){\epsilon }^{n}$ with ${y}_{n}$ analytic at $x=0$. This improves and generalizes a result of C.H. Lin on this so-called " Ackerberg-O’Malley resonance" phenomenon. In the proof, the problem is reduced to the preceding problem for the corresponding Riccati equation In the final section, we construct examples of such second order equations exhibiting resonance such that the formal solution $\stackrel{^}{y}$ has a prescribed logarithmic derivative ${\stackrel{^}{y}}^{\text{'}}\left(0,\epsilon \right)/\stackrel{^}{y}\left(0,\epsilon \right)$ at $x=0$ which is divergent of Gevrey order 1.

## How to cite

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Fruchard, Augustin, and Schäfke, Reinhard. "Overstability and resonance." Annales de l’institut Fourier 53.1 (2003): 227-264. <http://eudml.org/doc/116035>.

@article{Fruchard2003,
abstract = {We consider a singularity perturbed nonlinear differential equation $\varepsilon u^\{\prime \}=f(x)u + +\varepsilon P(x,u,\varepsilon )$ which we suppose real analytic for $x$ near some interval $[a,b]$ and small $\vert u\vert$, $\vert \varepsilon \vert$. We furthermore suppose that 0 is a turning point, namely that $xf(x)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $\epsilon \rightarrow 0$) local (at $x=0$) or global, real analytic or $C^\infty$ solutions is equivalent to the existence of a formal series solution $\sum u_n(x)\varepsilon ^n$ with $u_n$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions. We apply this result to the second order linear differential equation $\varepsilon y^\{\prime \prime \} +\varphi (x,\varepsilon )y^\{\prime \}+\psi (x,\varepsilon )y=0$ with $\varphi$ and $\psi$ real analytic for $x$ near some interval $[a,b]$ and small $\vert \varepsilon \vert$. We assume that $-x\varphi (x,0)$ is positive if $x\ne 0$ and that the function $\psi _0:x\mapsto \psi (x,0)$ has a zero at $x=0$ of at least the same order as $\varphi _0\mapsto \varphi (x,0)$. For this equation, we prove that the existence of local or global, real analytic or $C^\infty$ solutions tending to a nontrivial solution of the reduced equation $\varphi (x,0)y^\{\prime \}+\psi (x,0)y=0$ is equivalent to the existence of a non trivial formal series solution $\hat\{y\}(x,\varepsilon )=\sum y_n(x)\varepsilon ^n$ with $y_n$ analytic at $x=0$. This improves and generalizes a result of C.H. Lin on this so-called " Ackerberg-O’Malley resonance" phenomenon. In the proof, the problem is reduced to the preceding problem for the corresponding Riccati equation In the final section, we construct examples of such second order equations exhibiting resonance such that the formal solution $\hat\{y\}$ has a prescribed logarithmic derivative $\hat\{y\}^\{\prime \}(0,\varepsilon ) /\hat\{y\} (0,\varepsilon )$ at $x=0$ which is divergent of Gevrey order 1.},
affiliation = {Université de La Rochelle, Laboratoire de Mathématiques Calcul Asymptotique, Pôle Sciences et Technologie, Avenue Michel Crépeau, 17042 La Rochelle Cedex (France); Université Louis Pasteur, Département de Mathématiques, 7 rue René Descartes, 67084 Strasbourg Cedex (France)},
author = {Fruchard, Augustin, Schäfke, Reinhard},
journal = {Annales de l’institut Fourier},
keywords = {resonance; canard solution; overstability; singular perturbation},
language = {eng},
number = {1},
pages = {227-264},
publisher = {Association des Annales de l'Institut Fourier},
title = {Overstability and resonance},
url = {http://eudml.org/doc/116035},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Fruchard, Augustin
AU - Schäfke, Reinhard
TI - Overstability and resonance
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 1
SP - 227
EP - 264
AB - We consider a singularity perturbed nonlinear differential equation $\varepsilon u^{\prime }=f(x)u + +\varepsilon P(x,u,\varepsilon )$ which we suppose real analytic for $x$ near some interval $[a,b]$ and small $\vert u\vert$, $\vert \varepsilon \vert$. We furthermore suppose that 0 is a turning point, namely that $xf(x)$ is positive if $x\ne 0$. We prove that the existence of nicely behaved (as $\epsilon \rightarrow 0$) local (at $x=0$) or global, real analytic or $C^\infty$ solutions is equivalent to the existence of a formal series solution $\sum u_n(x)\varepsilon ^n$ with $u_n$ analytic at $x=0$. The main tool of a proof is a new “principle of analytic continuation” for such “overstable” solutions. We apply this result to the second order linear differential equation $\varepsilon y^{\prime \prime } +\varphi (x,\varepsilon )y^{\prime }+\psi (x,\varepsilon )y=0$ with $\varphi$ and $\psi$ real analytic for $x$ near some interval $[a,b]$ and small $\vert \varepsilon \vert$. We assume that $-x\varphi (x,0)$ is positive if $x\ne 0$ and that the function $\psi _0:x\mapsto \psi (x,0)$ has a zero at $x=0$ of at least the same order as $\varphi _0\mapsto \varphi (x,0)$. For this equation, we prove that the existence of local or global, real analytic or $C^\infty$ solutions tending to a nontrivial solution of the reduced equation $\varphi (x,0)y^{\prime }+\psi (x,0)y=0$ is equivalent to the existence of a non trivial formal series solution $\hat{y}(x,\varepsilon )=\sum y_n(x)\varepsilon ^n$ with $y_n$ analytic at $x=0$. This improves and generalizes a result of C.H. Lin on this so-called " Ackerberg-O’Malley resonance" phenomenon. In the proof, the problem is reduced to the preceding problem for the corresponding Riccati equation In the final section, we construct examples of such second order equations exhibiting resonance such that the formal solution $\hat{y}$ has a prescribed logarithmic derivative $\hat{y}^{\prime }(0,\varepsilon ) /\hat{y} (0,\varepsilon )$ at $x=0$ which is divergent of Gevrey order 1.
LA - eng
KW - resonance; canard solution; overstability; singular perturbation
UR - http://eudml.org/doc/116035
ER -

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