%0 Journal Article
%T Homoclinic Bifurcation of a Quadratic Family of Real Functions with Two Parameters
%A Salma M. Farris
%A Karam N. Abdul-Kareem
%J Open Access Library Journal
%V 8
%N 5
%P 1-11
%@ 2333-9721
%D 2021
%I Open Access Library
%R 10.4236/oalib.1107300
%X In this work the homoclinic bifurcation of the family H={h<sub>(a,b)</sub>(x)=ax<sup>2</sup> b:a&isin;R/{0},b&isin;R}
 is studied. We proved that this family has a homoclinic tangency associated to x=0 of P1 for b=-2/a. Also we proved that <i>W</i><sup>u</sup>(<i>P</i><sub>1</sub>) does not intersect the backward orbit of P1 for b>-2/a, but has intersection for b<-2/a with a>0. So <i>H</i> has this type of the bifurcation.
%K Local Unstable Set
%K Unstable Set
%K Homoclinic Point
%K Homoclinic Orbit
%K Non-Degenerate
%K Homoclinic Tangency
%K Homoclinic Bifurcation
%U http://www.oalib.com/paper/6527328