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Homoclinic Bifurcation of a Quadratic Family of Real Functions with Two Parameters

DOI: 10.4236/oalib.1107300, PP. 1-11

Subject Areas: Dynamical System

Keywords: Local Unstable Set, Unstable Set, Homoclinic Point, Homoclinic Orbit, Non-Degenerate, Homoclinic Tangency, Homoclinic Bifurcation

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Abstract

In this work the homoclinic bifurcation of the family H={h(a,b)(x)=ax2 b:a∈R/{0},b∈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 Wu(P1) does not intersect the backward orbit of P1 for b>-2/a, but has intersection for b<-2/a with a>0. So H has this type of the bifurcation.

Cite this paper

Farris, S. M. and Abdul-Kareem, K. N. (2021). Homoclinic Bifurcation of a Quadratic Family of Real Functions with Two Parameters. Open Access Library Journal, 8, e7300. doi: http://dx.doi.org/10.4236/oalib.1107300.

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