Name: Pitchfork bifurcation and heteroclinic connections in the Kuramoto–Sivashinsky PDE
Keywords: Galerkin projection, FOS: Mathematics, Pitchfork bifurcation, dissipative PDEs, computer–assisted proof, rigorous numerics, heteroclinic connection, Dynamical Systems (math.DS), Mathematics - Dynamical Systems

descriptionPublicationkeyboard_double_arrow_right Article , Preprint 01 Jan 2025Embargo end date: 01 Jan 2023Publisher:American Institute of Mathematical Sciences (AIMS)Journal:Journal of Computational Dynamics, volume 12, pages 325-392 (issn: 2158-2491, eissn: 2158-2505,

Authors: Kubica, Jacek; Zgliczyński, Piotr; Kalita, Piotr;

doi: 10.3934/jcd.2025001 , 10.48550/arxiv.2312.08125

arXiv: http://arxiv.org/abs/2312.08125

Pitchfork bifurcation and heteroclinic connections in the Kuramoto–Sivashinsky PDE

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Abstract

We present a method for the complete analysis of the dynamics of dissipative Partial Differential Equations (PDEs) undergoing a pitchfork bifurcation. We apply our technique to the Kuramoto--Sivashinsky PDE on the line to obtain a computer-assisted proof of the creation of two symmetric branches of non-symmetric fixed points and heteroclinic connections between the symmetric fixed point and the new ones. The range of parameters is given explicitly and is large enough to allow for the rigorous continuation of the fixed points and heteroclinic connections created during the bifurcation.

Related Organizations

Jagiellonian University
Poland

Keywords

Galerkin projection, FOS: Mathematics, Pitchfork bifurcation, dissipative PDEs, computer–assisted proof, rigorous numerics, heteroclinic connection, Dynamical Systems (math.DS), Mathematics - Dynamical Systems

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