Normalization in Banach scale Lie algebras via mould calculus and applications
Thierry Paul David Sauzin
Discrete & Continuous Dynamical Systems - A 2017, 37(8): 4461-4487 doi: 10.3934/dcds.2017191

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of Banach scale Lie algebras (this notion is defined in the article). This situation covers the case of classical and quantum normal forms in a unified way which allows a direct comparison. In particular we prove a precise estimate for the difference between quantum and classical normal forms, proven to be of order of the square of the Planck constant. Our method uses mould calculus (recalled in the article) and properties of the solution of a universal mould equation studied in a preceding paper.

keywords: Mould calculus normal forms dynamical systems quantum mechanics semiclassical approximation
Resurgence of inner solutions for perturbations of the McMillan map
Pau Martín David Sauzin Tere M. Seara
Discrete & Continuous Dynamical Systems - A 2011, 31(1): 165-207 doi: 10.3934/dcds.2011.31.165
A sequence of "inner equations" attached to certain perturbations of the McMillan map was considered in [5], their solutions were used in that article to measure an exponentially small separatrix splitting. We prove here all the results relative to these equations which are necessary to complete the proof of the main result of [5]. The present work relies on ideas from resurgence theory: we describe the formal solutions, study the analyticity of their Borel transforms and use Écalle's alien derivations to measure the discrepancy between different Borel-Laplace sums.
keywords: exponentially small phenomena splitting of separatrices. Resurgence
Exponentially small splitting of separatrices in the perturbed McMillan map
Pau Martín David Sauzin Tere M. Seara
Discrete & Continuous Dynamical Systems - A 2011, 31(2): 301-372 doi: 10.3934/dcds.2011.31.301
The McMillan map is a one-parameter family of integrable symplectic maps of the plane, for which the origin is a hyperbolic fixed point with a homoclinic loop, with small Lyapunov exponent when the parameter is small. We consider a perturbation of the McMillan map for which we show that the loop breaks in two invariant curves which are exponentially close one to the other and which intersect transversely along two primary homoclinic orbits. We compute the asymptotic expansion of several quantities related to the splitting, namely the Lazutkin invariant and the area of the lobe between two consecutive primary homoclinic points. Complex matching techniques are in the core of this work. The coefficients involved in the expansion have a resurgent origin, as shown in [14].
keywords: exponentially small phenomena splitting of separatrices asymptotic formula. McMillan map

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