![Figure 1. The topological structure of a saddled Swiss-roll attractor [Scu1].](attachment:84392de3-cd30-4a41-97ae-ea433823ff2f:image.png)
C. Hinsley
3 May 2025
I gave a poster presentation about this topic at the SIAM DS25 conference in Denver, Colorado. The poster PDF can be found below.
Swiss-roll attractors generally cannot be studied through the lens of geometric singular perturbation theory (GSPT) despite seeming the perfect use for the theory’s tools. However, if a Swiss-roll attractor happens to be the $\omega$-limit set of a saddle equilibrium with two stable directions and one unstable direction, then GSPT typically becomes viable. This is because, loosely speaking, the attractor is guaranteed to be compact under relatively straightforward requirements on the two forward orbits of that saddle and the attractor flattens out nicely in the appropriate singular limit. I’ll refer to such attractors as saddled Swiss-roll attractors.
To make this notion clearer, the Rössler attractor is an unsaddled Swiss-roll attractor, so it does not exhibit certain nice topological properties as does a saddled Swiss-roll attractor like in the 5D modified Plant neuronal burster model from my lab's 2025 paper in Chaos (see the figure below).
Figure 1. The topological structure of a saddled Swiss-roll attractor [Scu1].
The first important part of this picture is that the unstable manifold $W_{\rm SF}^u$ of the saddle-focus, SF, can be decomposed into two parts: $M_{\rm Q}$, the quiescent manifold, and $M_{\rm PO}$, the spiking manifold. Since there’s a heteroclinic connection (in black) from SF to the saddle equilibrium, SD, and since the attractor can usually (under most parameter values of interest) be assumed to be the $\omega$-limit set of SD, this means that the attractor itself can be decomposed into (subsets of) $M_{\rm Q}$ and $M_{\rm PO}$.
We’ll speak of the attractor and $W_{\rm SF}^u$ as if they are equivalent, and so we can say that $W_{\rm SF}^u = M_{\rm Q} \cup M_{\rm PO}$. There’s a unique orbit $\gamma_0$ heteroclinic from SF to SD which forms part of the upper boundary of $M_{\rm Q}$ and is the only such heteroclinic orbit which does not wind around $M_{\rm PO}$. This orbit can be seen in black in Figure 1, originating at SF and intersecting the left upper boundary of $M_{\rm Q}$ between the 0 and 1-spike bands before proceeding up to SD.
Suppose that at least one branch $\Gamma_{\rm SD}^\pm$ of $W_{\rm SD}^u$ spikes at least twice on its first burst (i.e., after its first transition from $M_{\rm Q}$ to $M_{\rm PO}$). In particular, suppose each of $\Gamma_{\rm SD}^\pm$ traverses the interior of some spike-count band (but not on the blue-colored center point) as it transitions from $M_{\rm Q}$ to $M_{\rm PO}$ and denote by $|k^\pm|$ the associated spike counts of those first transitions, where the sign of $k^\pm \in \mathbb{Z}$ is negative if the subsequent return from $M_{\rm PO}$ to $M_{\rm Q}$ is on the rear side of $M_{\rm PO}$, and positive if the return is on the front side. Then $k^\pm$ is negative if $\Gamma_{\rm SD}^\pm$ crosses the $k$-spike-count band nearer to SF than the blue-colored center point, and is positive if $\Gamma_{\rm SD}^\pm$ crosses the band farther from SF than the blue-colored center point. Unless otherwise stated, we will assume that $\Gamma_{\rm SD}^\pm$ traverse the same spike-count band at the same location, so that $k^- = k^+$. In the singular limit for systems with a 2D slow expanding subsystem and 1D fast contracting subsystem, this will always be exactly the case anyway.
For any positive integer $i$ less than $\max |k^\pm|$, there’s a unique orbit $\gamma_i$ heteroclinic from SF to SD. These heteroclinic orbits can be seen (also in black) in Figure 1, and $\gamma_i$ first transitions from $M_{\rm Q}$ to $M_{\rm PO}$ between the $i$-spike-count and $(i+1)$-spike-count bands. After one revolution around $M_{\rm PO}$ (i.e., one spike), $\gamma_i$ becomes close to $\gamma_{i-1}$ and, in the singular limit, exactly unites with it. Hence as the timescale separation between fast and slow subsystems grows towards the singular limit, the stable manifold $W_{\rm SD}^s$ of the saddle SD collapses locally into a single direction, that direction along which $\gamma_0$ enters (though this direction may change as the timescale separation is varied). When not in the singular limiting regime, $W_{\rm SD}^s$ looks something like this locally, to give you a rough idea:
Figure 2. Rough sketch of $W_{\rm SD}^s$. This is from a photo of my idle scribbles during an unrelated lecture, so please forgive me if it’s very ugly. $\Gamma_{\rm SD}^\pm$ are the branches of $W_{\rm SD}^u$, which is transverse to $W_{\rm SD}^s$ so they “come out” of the disc $W_{\rm SD}^s$.
We may take the orbits $\Gamma_{\rm SD}^\pm$ to form the periphery of the attractor, so that whichever is the furthest outward (i.e., initially transitions from $M_{\rm Q}$ to $M_{\rm PO}$ farthest from SF) forms one half of the boundary of the attractor. The inner boundary of the attractor is a little more difficult to characterize.
Figure 3. A sketch of the structure of the meeting of the zero-spike and one-spike bands as they return from $M_{\rm PO}$ to $M_{\rm Q}$. The dotted curve $\beta$ is the reinsertion loop on the 0-spike and 1-spike portions of the attractor.
In the paper [Scu1], a major accomplishment was to identify by what exactly this inner boundary is formed. However, only an informal geometric characterization of it was given, where a reinsertion loop ($\beta$ in Figure 3) was identified as the interface for the return from $M_{\rm PO}$ to $M_{\rm Q}$ and the “crease point” $T$ was taken to be the point on the reinsertion loop nearest the equilibrium SF. But neither an exact definition of the reinsertion loop nor of the notion of a nearest point to SF was given.
We can go further now to properly define $\beta$ and $T$.
Let $\beta_0$ be a smooth curve in $M_{\rm Q} \subset W_{\rm SD}^u$ with endpoints SF, SD and transverse to the flow everywhere on its interior. In the singular limit, a return map $\varphi_0: \beta_0 \to \beta_0$ can be defined on $\beta_0$.