these manifolds on the parameters of the system.

Tangency bifurcation of invariant manifolds
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Guckenheimer and Meerkamp[@guckenheimer2012siam] drew bifurcation diagrams of the system in a two-dimensional slice of the parameter space defined by ${ \varepsilon}= 0.01$, $b = -1$, and $c = 1$. Codimension-one tangencies of $S^r_{{ \varepsilon}}$ and $W^u$ are represented in Figure 5.1 of their paper by smooth curves (labeled T) in $(\nu,a)$ space. For fixed $a$ and increasing $\nu$, this tangency occurs after $p_{eq}$ undergoes a supercritical Hopf bifurcation. A parametric family of stable limit cycles emerges from this bifurcation. Henceforth we refer to ‘the’ small-amplitude stable periodic orbit $\Gamma$ to refer to the corresponding member of this family at a particular parameter set. The two-dimensional stable manifolds of $\Gamma$ interact with the other invariant manifolds of the system. Guckenheimer and Meerkamp identify a branch of period-doubling bifurcations as $\nu$ continues to increase after the first slow-manifold tangency. We show that the basin of attraction of the periodic orbit has a significant influence on the global returns of the system.

Fixing $a = -0.03$, the tangency occurs within the range $\nu \in \left[ 0.00647, 0.00648\right]$. The location of the tangency may be approximated by studying the asymptotics 