Overstability and resonance
Annales de l'Institut Fourier, Volume 53 (2003) no. 1, p. 227-264

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.

On considère l’équation différentielle non linéaire singulièrement perturbée $\epsilon {u}^{\text{'}}=f\left(x\right)u+\epsilon P\left(x,u,\epsilon \right)$ qu’on suppose réelle et analytique pour $x$ proche de $\left[a,b\right]$ et $|u|$, $|\epsilon |$ asez petits. On suppose que 0 est un point tournant, c’est-à-dire $xf\left(x\right)>0$ si $x\ne 0$. On démontre que l’existence de solutions locales (en $x=0$) ou globales, analytiques réelles ou ${C}^{\infty }$ bornées quand $\epsilon \to 0$ est équivalente à l’existence d’une solution série formelle $\sum {u}_{n}\left(x\right){\epsilon }^{n}$ avec ${u}_{n}$ analytiques en $x=0$. L’outil principal de la démonstration est un nouveau “principe de prolongement analytique” pour de telles solutions dites surstables. On applique ce résultat à l’équation d’ordre deux $\epsilon {y}^{\text{'}\text{'}}+\varphi \left(x,\epsilon \right){y}^{\text{'}}+\psi \left(x,\epsilon \right)y=0$$\varphi$ et $\psi$ sont analytiques réelles pour $x$ proche de $\left[a,b\right]$ et $|\epsilon |$ assez petit. On suppose que $-x\varphi \left(x,0\right)>0$ si $x\ne 0$ et que la fonction ${\psi }_{0}:x↦\psi \left(x,0\right)$ a un zéro en $x=0$ d’ordre au moins égal à celui de ${\varphi }_{0}:x↦\varphi \left(x,0\right)$. On montre que l’existence de solutions locales ou globales, analytiques réelles ou ${C}^{\infty }$, tendant vers une solution non triviale de l’équation réduite $\varphi \left(x,0\right){y}^{\text{'}}+\psi \left(x,0\right)y=0$ est équivalente à l’existence d’une solution série formelle non triviale $\stackrel{^}{y}\left(x,\epsilon \right)=\sum {y}_{n}\left(x\right){\epsilon }^{n}$ avec ${y}_{n}$ analytiques en $x=0$. Ceci améliore et généralise un résultat de C.H. Lin concernant le phénomène de “résonance au sens d’Ackerberg-O’Malley”. Dans le dernier paragraphe, on construit des exemples d’ordre deux qui présentent une résonance et tels que la solution formelle $\stackrel{^}{y}$ ait une dérivée logarithmique prescrite $\stackrel{^}{y}\left(0,\epsilon \right)/\stackrel{^}{y}\left(0,\epsilon \right)$ en $x=0$, divergente d’ordre Gevrey 1.

DOI : https://doi.org/10.5802/aif.1943
Classification:  34E
Keywords: resonance, canard solution, overstability, singular perturbation
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author = {Fruchard, Augustin and Sch\"afke, Reinhard},
title = {Overstability and resonance},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {53},
number = {1},
year = {2003},
pages = {227-264},
doi = {10.5802/aif.1943},
zbl = {1037.34047},
language = {en},
url = {http://www.numdam.org/item/AIF_2003__53_1_227_0}
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Fruchard, Augustin; Schäfke, Reinhard. Overstability and resonance. Annales de l'Institut Fourier, Volume 53 (2003) no. 1, pp. 227-264. doi : 10.5802/aif.1943. http://www.numdam.org/item/AIF_2003__53_1_227_0/

[1] R.C. Ackerberg; R.E. O'Malley Boundary layer Problems Exhibiting Resonance, Studies in Appl. Math., Tome 49 (1970) no. 3, pp. 277-295 | MR 269940 | Zbl 0198.12901

[2] É. Benoît Asymptotic expansions of canards with poles. Application to the stationary unidimensional Schrödinger equation, Bull. Belgian Math. Soc., suppl. `Nonstandard Analysis' (1996), pp. 71-90 | MR 1409643 | Zbl 0896.34069

[3] É. Benoît Enlacements de canards, Publications IHES, Tome 72 (1990), pp. 63-91 | Numdam | Zbl 0737.34018

[4] É. Benoît; J.-L. Callot; F. Diener; M. Diener Chasse au canard, Collect. Math., Tome 31 (1981) no. 1-3, pp. 37-119 | Zbl 0529.34046

[5] É. Benoît; A. Fruchard; R. Schäfke; G. Wallet Solutions surstables des équations différentielles complexes lentes-rapides à point tournant, Ann. Fac. Sci. Toulouse, Tome VII (1998) no. 4, pp. 1-32 | Numdam | MR 1693589 | Zbl 0981.34084

[6] É. Benoît; A. Fruchard; R. Schäfke; G. Wallet Overstability : toward a global study, C.R. Acad. Sci. Paris, série I, Tome 326 (1998), pp. 873-878 | MR 1648552 | Zbl 0922.34048

[7] J.-L. Callot Bifurcation du portrait de phase pour des équations différentielles linéaires du second ordre ayant pour type l'équation d'Hermite (1981) (Thèse de Doctorat d'Etat, Strasbourg)

[8] J.-L. Callot Champs lents-rapides complexes à une dimension lente, Ann. Sci. École Norm. Sup., 4e série, Tome 26 (1993), pp. 149-173 | Numdam | MR 1209706 | Zbl 0769.34005

[9] M. Canalis-Durand; J.-P. Ramis; R. Schäfke; Y. Sibuya Gevrey solutions of singularly perturbed differential equations, J. reine angew. Math., Tome 518 (2000), pp. 95-129 | Article | MR 1739408 | Zbl 0937.34075

[10] L.P. Cook; W. Eckhaus Resonance in a boundary value problem of singular perturbation type, Studies in Appl. Math., Tome 52 (1973), pp. 129-139 | MR 342799 | Zbl 0264.34070

[11] P.P.N. De Groen The nature of resonance in a singular perturbation problem of turning point type, SIAM J. Math. Anal., Tome 11 (1980), pp. 1-22 | Article | MR 556493 | Zbl 0424.34021

[12] F. Diener Méthode du plan d'observabilité (1981) (Thèse de Doctorat d'Etat, prépublication IRMA, 7, rue René Descartes, 67084 Strasbourg Cedex (France))

[13] L. Hörmander An introduction to complex analysis in several variables, Elsevier Science B.V., Amsterdam (1966, revised 1973, 1990)

[14] N. Kopell A geometric approach to boundary layer problems exhibiting resonance, SIAM. J. Appl. Math., Tome 37 (1979) no. 2, pp. 436-458 | Article | MR 543963 | Zbl 0417.34051

[15] W.D. Lakin Boundary value problems with a turning point, Studies in Appl. Math., Tome 51 (1972), pp. 261-275 | MR 355236 | Zbl 0257.34015

[16] C.H. Lin The sufficiency of Matkowsky-condition in the problem of resonance, Trans. Amer. Math. Soc., Tome 278 (1983) no. 2, pp. 647-670 | Article | MR 701516 | Zbl 0513.34055

[17] B.J. Matkowsky On boundary layer problems exhibiting resonance, SIAM Review, Tome 17 (1975), pp. 82-100 | Article | MR 358004 | Zbl 0276.34055

[18] F.W.J. Olver Sufficient conditions for Ackerberg-O'Malley resonance, SIAM J. Math. Anal., Tome 9 (1978), pp. 328-355 | Article | MR 470383 | Zbl 0375.34034

[19] Y. Sibuya A theorem concerning uniform simplification at a transition point and the problem of resonance, SIAM J. Math. Anal., Tome 12 (1981), pp. 653-668 | Article | MR 625824 | Zbl 0463.34030

[20] W. Wasow Asymptotic expansions for ordinary differential equations, Interscience, New York (1965) | MR 203188 | Zbl 0133.35301

[21] W. Wasow Linear Turning Point Theory, Springer, New York (1985) | MR 771669 | Zbl 0558.34049