Qualitative analysis of polynomial differential systems with the line at infinity of maximal multiplicity: exploring linear, quadratic, cubic, quartic, and quintic cases

Abstract

This article investigates the phase portraits of polynomial differential systems with maximal multiplicity at the line at infinity. The study explores theoretical foundations, including algebraic multiplicity definitions, to establish the groundwork for qualitative analyses of dynamical systems. Spanning polynomial degrees from linear to quintic, the article systematically presents transformations and conditions to achieve maximal multiplicity of the invariant lines at infinity. Noteworthy inclusions of systematic transformations, such as Poincaré transformations, simplify analysis and enhance the accessibility of phase portraits.