Local Bifurcation and Center Problem for a More Generalized Lorenz System

• Jingping Lu ^[3] ; Chunyong Wang ^[1] ; Wentao Huang ^[2] ; Qinlong Wang ^[3]
  1. [1] Hezhou University

     Hezhou University

     China

     
  2. [2] Guangxi Normal University

     Guangxi Normal University

     China

     
  3. [3] Guilin University of Electronic Technology & Guilin University of Technology

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 4, 2022
• Idioma: inglés
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• Resumen

  • In this paper, Hopf bifurcation and center problem are investigated for a class of more generalized Lorenz systems, which are Z2 symmetric and quadratic threedimensional systems. Firstly, the singular point quantities of one equilibrium are calculated carefully, and the two symmetric fourth-order weak foci are found. Secondly, the corresponding invariant algebraic surfaces are figured out, and the center conditions on a center manifold are determined. In this way it is proved that there exist at most eight small limit cycles from the two symmetric equilibria via a Hopf bifurcation, which is a new result for general Lorenz models. At the same time, when the center conditions are satisfied, the complete classification of Darboux invariants is established for this system.

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