THE GLOBAL DYNAMICS OF A 3-DIMENSIONAL DIFFERENTIAL SYSTEM IN $\mathbb{R}^3$ VIA A DARBOUX INVARIANT

doi:10.1007/s10473-025-0204-9

Acta mathematica scientia,Series B ›› 2025, Vol. 45 ›› Issue (2): 338-346.doi: 10.1007/s10473-025-0204-9

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THE GLOBAL DYNAMICS OF A 3-DIMENSIONAL DIFFERENTIAL SYSTEM IN $\mathbb{R}^3$ VIA A DARBOUX INVARIANT

Jaume Llibre^1,*, Claudia Valls²

1. Departament de Matematiques, Universitat Autò-noma de Barce-lona, 08193 Bellaterra, Barcelona, Catalonia, Spain;
2. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049-001, Lisboa, Portugal

Received:2023-04-20 Online:2025-03-25 Published:2025-05-08
Contact: *Jaume Llibre,E-mail: jaume.llibre@uab.cat
About author:Claudia Valls, E-mail: cvalls@math.ist.utl.pt
Supported by:
The first author's research was partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, the H2020 European Research Council grant MSCA-RISE-2017-777911, AGAUR (Generalitat de Catalunya) grant 2021SGR00113 and the Reial Acadèmia de Ciències i Arts de Barcelona. The second author's research was partially supported by FCT/Portugal through CAMGSD, IST-ID, projects UIDB/04459/2020 and UIDP/04459/2020.

Abstract

Abstract: The differential system $\dot x= ax -y z, \ \dot y = -b y + x z, \ \dot z= -c z + x^2,$ where $a$ , $b$ and $c$ are positive real parameters, has been studied numerically due to the big variety of strange attractors that it can exhibit. This system has a Darboux invariant when $c=2b$ . Using this invariant and the Poincaré compactification technique we describe analytically its global dynamics.

Key words: invariant, Poincaré ball, Poincaré disc, $\omega$ -limit, $\alpha$ -limit, differential system in $\mathbb{R}^3$

CLC Number:

34C29

Jaume Llibre, Claudia Valls. THE GLOBAL DYNAMICS OF A 3-DIMENSIONAL DIFFERENTIAL SYSTEM IN $\mathbb{R}^3$ VIA A DARBOUX INVARIANT[J].Acta mathematica scientia,Series B, 2025, 45(2): 338-346.

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THE GLOBAL DYNAMICS OF A 3-DIMENSIONAL DIFFERENTIAL SYSTEM IN $\mathbb{R}^3$ VIA A DARBOUX INVARIANT

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THE GLOBAL DYNAMICS OF A 3-DIMENSIONAL DIFFERENTIAL SYSTEM IN R3\mathbb{R}^3 VIA A DARBOUX INVARIANT

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THE GLOBAL DYNAMICS OF A 3-DIMENSIONAL DIFFERENTIAL SYSTEM IN $\mathbb{R}^3$ VIA A DARBOUX INVARIANT