Integrability and periodic orbits of a 3D jerk system with two quadratic nonlinearities
Álvarez Ramírez, Martha (Universidad Autónoma Metropolitana - Iztapalapa. Departamento de Matemáticas)
García Saldaña, Johanna Denise (Universidad Católica de la Santísima Concepción. Departamento de Matemática y Física Aplicadas)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2026
Abstract:	In mechanics jerk is the rate of change of an object's acceleration over time. Thus a jerk equation is a differential equation of the form x⃛ = f(x, ẋ, ẍ), where x, ẋ, ẍ and x⃛ represent the position, velocity, acceleration, and jerk, respectively. The jerk differential equation can be written as the jerk differential system ẋ = y, ẏ = z, ż = f(x, y, z), in R3. In this paper we study the jerk differential system with f(x, y, z) = -ax(1-x) - y + by2, previously studied by other authors showing that this system can exhibit chaos for some values of its parameters. When the parameters a = b = 0 the x-axis is filled with zero-Hopf equilibria, and all the other orbits are periodic. Here we prove analytically the existence of two families of periodic orbits for sufficiently small values of the parameters a and b. One family bifurcates from the non-isolated zero-Hopf equilibrium (1, 0, 0) of the jerk system with a = b = 0, while the other family bifurcates from a periodic orbit of the jerk system with a = b = 0.
Grants:	Agencia Estatal de Investigación PID2022-136613NB-I00 Generalitat de Catalunya 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, la comunicació pública de l'obra i la creació d'obres derivades, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Jerk system ; Integrability ; Periodic orbit ; Zero-Hopf bifurcation
Published in:	Nonlinear Analysis: Real World Applications, Vol. 88 (April 2026) , art. 104491, ISSN 1468-1218

DOI: 10.1016/j.nonrwa.2025.104491

6 p, 1.0 MB

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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