the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3cpng.png "fig:"){width="45.00000%"} (d) ![\[fig:retmap2\] (a) Subinterval of the return map $R: \Sigma_+ \to \Sigma)+$ of Eqs. and (b) a refinement of the subinterval. Dashed black line is the line of fixed points $R(z) = z$. (c) Periodic orbit corresponding to fixed point of $R$ at $z \approx 0.05939079$. (d) Time series of the periodic orbit. The orbit is decomposed into red, gray, green, blue, magenta, and black segments (defined as in Sec. \[sec:maps\]). Parameter set: $\nu \approx 0.00870134$, $a = -0.3$, $b = -1$, $c = 1$.](3dpng.png "fig:"){width="45.00000%"}

Approximating points on $\Sigma$ by their $z$-coordinates, the return map $R: \Sigma \to \Sigma$ is well-approximated by a one-dimensional map on an interval, also denoted $R$. In the presence of the small-amplitude stable periodic orbit $\Gamma$, we now compare our one-dimensional approximation to return maps in the case of folded nodes[@wechselberger2005] and folded 