Resurgent functions and splitting problems
David Sauzin (IMCCE)
TL;DR
This paper introduces Ecalle's theory of resurgent functions and alien calculus, focusing on their application to exponentially small separatrix splitting and nonlinear difference equations in complex dynamics.
Contribution
It provides an accessible introduction to resurgent analysis and alien calculus with examples from nonlinear difference equations and separatrix splitting problems.
Findings
Resurgent functions effectively analyze exponentially small phenomena.
Alien calculus offers tools for understanding complex dynamical systems.
Examples demonstrate the application of theory to nonlinear difference equations.
Abstract
The present text is an introduction to \'Ecalle's theory of resurgent functions and alien calculus, in connection with problems of exponentially small separatrix splitting. An outline of the resurgent treatment of Abel's equation for resonant dynamics in one complex variable is included. The emphasis is on examples of nonlinear difference equations, as a simple and natural way of introducing the concepts.
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Taxonomy
TopicsQuantum chaos and dynamical systems · Advanced Differential Equations and Dynamical Systems · Nonlinear Dynamics and Pattern Formation