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| Consider a planar family of planar differential systems with a center at $p$. The period function assigns to each periodic orbit in the period annulus its period. The problem of bifurcation of critical periods have been studied and there are three different situations to consider: bifurcations from the center, bifurcations from the interior of the period annulus and bifurcation from the outer boundary of the period annulus. The bifurcation of critical periods from the inner boundary is completely understood by using the so-called period constants. We study the bifurcation of critical periods from the outer boundary, which has an additional difficulty since the period function can not be analytically extended. We give some results to ensure that the period function extends to the outer boundary and identify the parameter regions where there are no bifurcations associated in the case of families of potential vector fields.
Particularly, we found the bifurcation curves associated to the outer boundary of the bi-parametric family of potential system $X_{p,q}=-y\partial_x+V_{p,q}'(x)\partial_y$ with $V_{p,q}'(x)=(x+1)^p-(x+1)^q$. |
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