# Overstability and resonance

Augustin Fruchard^{[1]}; Reinhard Schäfke^{[2]}

- [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)

Annales de l’institut Fourier (2003)

- Volume: 53, Issue: 1, page 227-264
- ISSN: 0373-0956

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topFruchard, 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 -

## References

top- R.C. Ackerberg, R.E. O'Malley, Boundary layer Problems Exhibiting Resonance, Studies in Appl. Math. 49 (1970), 277-295 Zbl0198.12901MR269940
- É. BenoÎt, Asymptotic expansions of canards with poles. Application to the stationary unidimensional Schrödinger equation, Bull. Belgian Math. Soc., suppl. `Nonstandard Analysis' (1996), 71-90 Zbl0896.34069MR1409643
- É. BenoÎt, Enlacements de canards, 72 (1990), 63-91, Publications IHES Zbl0737.34018
- É. BenoÎt, J.-L. Callot, F. Diener, M. Diener, Chasse au canard, Collect. Math. 31 (1981), 37-119 Zbl0529.34046
- É. BenoÎt, A. Fruchard, R. Schäfke, G. Wallet, Solutions surstables des équations différentielles complexes lentes-rapides à point tournant, Ann. Fac. Sci. Toulouse VII (1998), 1-32 Zbl0981.34084MR1693589
- É. BenoÎt, A. Fruchard, R. Schäfke, G. Wallet, Overstability : toward a global study, C.R. Acad. Sci. Paris, série I 326 (1998), 873-878 Zbl0922.34048MR1648552
- J.-L. Callot, Bifurcation du portrait de phase pour des équations différentielles linéaires du second ordre ayant pour type l'équation d'Hermite, (1981)
- J.-L. Callot, Champs lents-rapides complexes à une dimension lente, Ann. Sci. École Norm. Sup., 4e série 26 (1993), 149-173 Zbl0769.34005MR1209706
- M. Canalis-Durand, J.-P. Ramis, R. Schäfke, Y. Sibuya, Gevrey solutions of singularly perturbed differential equations, J. reine angew. Math. 518 (2000), 95-129 Zbl0937.34075MR1739408
- L.P. Cook, W. Eckhaus, Resonance in a boundary value problem of singular perturbation type, Studies in Appl. Math. 52 (1973), 129-139 Zbl0264.34070MR342799
- P.P.N. de Groen, The nature of resonance in a singular perturbation problem of turning point type, SIAM J. Math. Anal. 11 (1980), 1-22 Zbl0424.34021MR556493
- F. Diener, Méthode du plan d'observabilité, (1981)
- L. Hörmander, An introduction to complex analysis in several variables, (1966, revised 1973, 1990), Elsevier Science B.V., Amsterdam Zbl0138.06203
- N. Kopell, A geometric approach to boundary layer problems exhibiting resonance, SIAM. J. Appl. Math. 37 (1979), 436-458 Zbl0417.34051MR543963
- W.D. Lakin, Boundary value problems with a turning point, Studies in Appl. Math. 51 (1972), 261-275 Zbl0257.34015MR355236
- C.H. Lin, The sufficiency of Matkowsky-condition in the problem of resonance, Trans. Amer. Math. Soc. 278 (1983), 647-670 Zbl0513.34055MR701516
- B.J. Matkowsky, On boundary layer problems exhibiting resonance, SIAM Review 17 (1975), 82-100 Zbl0276.34055MR358004
- F.W.J. Olver, Sufficient conditions for Ackerberg-O'Malley resonance, SIAM J. Math. Anal. 9 (1978), 328-355 Zbl0375.34034MR470383
- Y. Sibuya, A theorem concerning uniform simplification at a transition point and the problem of resonance, SIAM J. Math. Anal. 12 (1981), 653-668 Zbl0463.34030MR625824
- W. Wasow, Asymptotic expansions for ordinary differential equations, (1965), Interscience, New York Zbl0133.35301MR203188
- W. Wasow, Linear Turning Point Theory, (1985), Springer, New York Zbl0558.34049MR771669

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