Effective chaos for the Kirchhoff equation on tori
Pietro Baldi
Università di Napoli Federico II, ItalyFilippo Giuliani
Politecnico di Milano, ItalyMarcel Guardia
Universitat de Barcelona, Spain; Centre de Recerca Matemàtica, Bellaterra, SpainEmanuele Haus
Università Roma Tre, Italy
Abstract
We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillate in a chaotic way on certain long timescales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, which we name effective chaos (it occurs over a long, but finite, timescale), is a consequence of the existence of symbolic dynamics for an effective system. Since the first-order dynamics has been proved to be essentially stable, we need to perform a second-order to find an effective model displaying chaotic dynamics. More precisely, after some nontrivial reductions, this model behaves as two weakly coupled pendulums.
Cite this article
Pietro Baldi, Filippo Giuliani, Marcel Guardia, Emanuele Haus, Effective chaos for the Kirchhoff equation on tori. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2024), published online first
DOI 10.4171/AIHPC/110