We consider a singularity perturbed nonlinear differential equation which we suppose real analytic for near some interval and small , . We furthermore suppose that 0 is a turning point, namely that is positive if . We prove that the existence of nicely behaved (as ) local (at ) or global, real analytic or solutions is equivalent to the existence of a formal series solution with analytic at . 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 with and real analytic for near some interval and small . We assume that is positive if and that the function has a zero at of at least the same order as . For this equation, we prove that the existence of local or global, real analytic or solutions tending to a nontrivial solution of the reduced equation is equivalent to the existence of a non trivial formal series solution with analytic at . 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 has a prescribed logarithmic derivative at which is divergent of Gevrey order 1.
On considère l’équation différentielle non linéaire singulièrement perturbée qu’on suppose réelle et analytique pour proche de et , asez petits. On suppose que 0 est un point tournant, c’est-à-dire si . On démontre que l’existence de solutions locales (en ) ou globales, analytiques réelles ou bornées quand est équivalente à l’existence d’une solution série formelle avec analytiques en . 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 où et sont analytiques réelles pour proche de et assez petit. On suppose que si et que la fonction a un zéro en d’ordre au moins égal à celui de . On montre que l’existence de solutions locales ou globales, analytiques réelles ou , tendant vers une solution non triviale de l’équation réduite est équivalente à l’existence d’une solution série formelle non triviale avec analytiques en . 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 ait une dérivée logarithmique prescrite en , divergente d’ordre Gevrey 1.
Keywords: resonance, canard solution, overstability, singular perturbation
Mot clés : résonance, solution du canard, surstabilité, perturbation singulière
Fruchard, Augustin 1; Schäfke, Reinhard 2
@article{AIF_2003__53_1_227_0, author = {Fruchard, Augustin and Sch\"afke, Reinhard}, title = {Overstability and resonance}, journal = {Annales de l'Institut Fourier}, pages = {227--264}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {1}, year = {2003}, doi = {10.5802/aif.1943}, zbl = {1037.34047}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1943/} }
TY - JOUR AU - Fruchard, Augustin AU - Schäfke, Reinhard TI - Overstability and resonance JO - Annales de l'Institut Fourier PY - 2003 SP - 227 EP - 264 VL - 53 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1943/ DO - 10.5802/aif.1943 LA - en ID - AIF_2003__53_1_227_0 ER -
%0 Journal Article %A Fruchard, Augustin %A Schäfke, Reinhard %T Overstability and resonance %J Annales de l'Institut Fourier %D 2003 %P 227-264 %V 53 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1943/ %R 10.5802/aif.1943 %G en %F AIF_2003__53_1_227_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.1943/
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