Homoclinic Loop and Multiple Limit Cycle Bifurcation Surfaces

Abstract

This paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces $\mathcal {H}$ and multiplicity-two, limit cycle bifurcation surfaces $\mathcal {C}$ for planar systems depending on two or more parameters; it determines the side of $\mathcal {H}$ or $\mathcal {C}$ on which limit cycles occur; and it shows that if $\mathcal {H}$ and $\mathcal {C}$ intersect, then typically they do so at a flat contact.