Formal expansion of van der pol equation canard solutions are gevrey

Canalis-Durand, Mireille

doi:10.1007/bfb0085022

Cited by 15 publications

(12 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The modified Nagumo norm (see [15,22]), was first used in the context of singularly perturbed equations by R. Schäfke in [19] and by Canalis-Durand in [5,7,6].…”

mentioning

confidence: 99%

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Mégret¹,

Demongeot²

2020

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

“…The modified Nagumo norm (see [15,22]), was first used in the context of singularly perturbed equations by R. Schäfke in [19] and by Canalis-Durand in [5,7,6].…”

mentioning

confidence: 99%

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Mégret¹,

Demongeot²

2020

Discrete &Amp; Continuous Dynamical Systems - S

View full text Add to dashboard Cite

“…Ce résultat constitue une généralisation de [2] qui traite l'exemple de l'équation de Van der Pol et de [3] qui traite du cas unidimensionnel. Pour cette généralisation, nous nous sommes placés dans les hypothèses où G. Wallet démontre l'existence de solutions surstables [10].…”

Section: Nous Faisons Les Hypothèses Supplémentairesunclassified

“…Pour cette généralisation, nous nous sommes placés dans les hypothèses où G. Wallet démontre l'existence de solutions surstables [10]. Remarquons que le caractère Gevrey 1 des solutions formelles permet également, par des procédés de sommation approchée inspirés des théorèmes de sommation [4], [5], [8], d'obtenir l'existence de telles solutions modulo une décroissance exponentielle ainsi que les véritables solutions comme on le montre dans [2] sur un exemple .…”

Section: Nous Faisons Les Hypothèses Supplémentairesunclassified

Solution formelle Gevrey d'une équation singulièrement perturbée : le cas multidimensionel

Canalis-Durand¹

1993

Annales De L’institut Fourier

Self Cite

View full text Add to dashboard Cite

“…Even if we start with an analytic problem-indeed polynomial!-several objects of interest are only Gevrey. The phenomenon that analytic problems have only Gevrey solutions has appeared in other contexts in dynamics, notably in the study of singular perturbations [CDRSS00], the regularity of attractors and fast-slow systems [Bae95,CD91,FT89]. Closer to us, in dependence on parameters of solutions of nonlinear problems, [Lin92,Sau92], dependence of KAM tori in the frequency [Pop00], or in the theory of parabolic manifolds [BFM17,BH08].…”

Section: Introductionmentioning

confidence: 99%

Gevrey estimates for asymptotic expansions of Tori in weakly dissipative systems*

Bustamante

Llave

2022

Nonlinearity

View full text Add to dashboard Cite

We consider a singular perturbation for a family of analytic symplectic maps of the annulus possessing a KAM torus. The perturbation introduces dissipation and contains an adjustable parameter. By choosing the adjustable parameter, one can ensure that the torus persists under perturbation. Such models are common in celestial mechanics. In field theory, the adjustable parameter is called the counterterm and in celestial mechanics, the drift. It is known that there are formal expansions in powers of the perturbation both for the quasi-periodic solution and the counterterm. We prove that the asymptotic expansions for the quasiperiodic solutions and the counterterm satisfy Gevrey estimates. That is, the nth term of the expansion is bounded by a power of n!. The Gevrey class (the power of n!) depends only on the Diophantine condition of the frequency and the order of the friction coefficient in powers of the perturbative parameter. The method of proof we introduce may be of interest beyond the problem considered here. We consider a modified Newton method in a space of power expansions. As is custumary in KAM theory, each step of the method is estimated in a smaller domain. In contrast with the KAM results, the domains where we control the Newton method shrink very fast and the Newton method does not prove that the solutions are analytic. On the other hand, by examining carefully the process, we can obtain estimates on the coefficients of the expansions and conclude the series are Gevrey.

show abstract

Formal expansion of van der pol equation canard solutions are gevrey

Cited by 15 publications

References 0 publications

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Gevrey solutions of singularly perturbed differential equations, an extension to the non-autonomous case

Solution formelle Gevrey d'une équation singulièrement perturbée : le cas multidimensionel

Gevrey estimates for asymptotic expansions of Tori in weakly dissipative systems*

Contact Info

Product

Resources

About