Small-amplitude periodic solutions in the polynomial jerk equation of arbitrary degree
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Sun, Xianbo (Hangzhou Normal University)

Date:	2025
Abstract:	A zero-Hopf singularity for a 3-dimensional differential system is a singularity for which the Jacobian matrix of the differential system evaluated at it has eigenvalues zero and ± ω i with ω ≠ 0. In this paper we investigate the periodic orbits that bifurcate from a zero-Hopf singularity of the nth-degree polynomial jerk equation x⃛- ϕ(x,ẋ,ẍ) = 0, where ϕ(∗,∗,∗) is an arbitrary nth-degree polynomial in three variables. We obtain sharp upper bounds on the maximum number of limit cycles that can emerge from such a zero-Hopf singularity using the averaging theory up to the second order. The result improves upon previous findings reported in the literature on zero-Hopf singularities and averaging theory. As an application we characterize small-amplitude periodic traveling waves in a class of generalized non-integrable Kawahara equations. This is accomplished by transforming the partial differential models into a five-dimensional dynamical system and subsequently analyzing a jerk system on a normally hyperbolic critical manifold, leveraging the averaging method and singular perturbation theory.
Grants:	Agencia Estatal de Investigación PID2022-136613NB-I00 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Note:	Altres ajuts: Reial Acadèmia de Ciències i Arts de Barcelona
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Published in:	Physica D: Nonlinear Phenomena, Vol. 476 (June 2025) , art. 134628, ISSN 0167-2789

DOI: 10.1016/j.physd.2025.134628

Available from: 2027-06-30 Postprint

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles