New Insights on Non-integrability and Dynamics in a Simple Quadratic Differential System

Abstract

This study focuses on the integrability and qualitative behaviors of a quadratic differential system x˙=a+yz,y˙=-y+x2,z˙=b-4x.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\dot{x}=a+yz,\\quad\\dot{y}=-y + x^{2},\\quad\\dot{z}=b-4x.$$\\end{document}We provide some new perspectives on the system and reveal its diverse properties, including non-integrability in the sense of absence of first integrals, bifurcations of co-dimension one or two, Jacobi instability and dynamics at infinity.