Open AccessSymbolic Methods for Analysing Bifurcations and Chaos of Two Five-Parameter Families of Planar Quadratic Maps
Author(s) -
Sarbast H. Mikaeel,
Bewar H. Othman
Publication year - 2020
Publication title -
science journal of university of zakho
Language(s) - English
Resource type - Journals
eISSN - 2663-6298
pISSN - 2663-628X
DOI - 10.25271/sjuoz.2020.8.2.723
Subject(s) - symbolic computation , mathematics , bifurcation , chaotic , quadratic equation , planar , algebraic number , bifurcation theory , stability (learning theory) , equilibrium point , chaos (operating system) , computation , pure mathematics , mathematical analysis , algorithm , computer science , nonlinear system , geometry , differential equation , artificial intelligence , physics , computer graphics (images) , computer security , quantum mechanics , machine learning
In this work, we analyze the dynamical behaviors of two five-parameter families of planar quadratic maps by utilizing strategies of symbolic computation. We are going to use computer algebra methods to clarify how to detect the stability of equilibrium points to analyze chaos and also the bifurcation of planar maps. Based on strategies for solving the systems in types of semi-algebraic and by utilizing an algorithmic approach, we obtain respectively for the two maps, sufficient conditions on the parameters to have a prescribed number of (stable) equilibrium points; necessary conditions on the parameters to undergo a certain type of bifurcation or to have chaotic behavior induced by snapback repeller.