Transcritical Bifurcations and Algebraic Aspects of Quadratic Multiparametric Families

TL;DR

This paper analyzes quadratic multiparametric systems, focusing on their classification, stability, bifurcations, and phase portraits, integrating algebraic and nonlinear systems theory for comprehensive understanding.

Contribution

It provides a detailed classification and stability analysis of quadratic multiparametric families, including algebraic aspects and bifurcation behavior.

Findings

Identification and classification of quadratic systems

Analysis of stability of critical points in finite and infinite planes

Graphical phase portraits of the studied families

Abstract

This article reveals an analysis of the quadratic systems that hold multiparametric families therefore, in the first instance the quadratic systems are identified and classified in order to facilitate their study and then the stability of the critical points in the finite plane, its bifurcations, stable manifold and lastly, the stability of the critical points in the infinite plane, afterwards the phase portraits resulting from the analysis of these families are graphed. To properly perform this study it was necessary to use some results of the non-linear systems theory, for this reason vital definitions and theorems were included because of their importance during the study of the multiparametric families. Algebraic aspects are also included.