Invisible Cusp-Fold Singularity

• Invisible cusp-fold singularity is a configuration in 3D Filippov systems where one vector field shows cubic (cusp) and the other quadratic (fold) contact with the switching manifold, leading to structural instability.
• Normal form analysis and canonical representations reveal distinct sign conditions on Lie derivatives that determine sliding, escaping, and crossing dynamics near the singularity.
• Codimension-one and codimension-two unfoldings govern bifurcation behavior, with applications such as boost-converter models demonstrating limit cycle creation and transitions into polycycles.

Searching arXiv for the specified papers to ground the article in current sources. An invisible cusp-fold singularity is a local singular configuration of a $3$-dimensional Filippov system $Z=(X,Y)$ on a switching manifold $\Sigma=\{f=0\}$ in which one vector field has cusp-type contact with $\Sigma$ and the other has fold-type contact at the same point. In the formulation used for recent bifurcation analysis, a point $p\in\Sigma$ is a cusp-fold singularity when $Xf(p)=0$, $X^2f(p)=0$, $X^3f(p)\neq0$, $Yf(p)=0$, $Y^2f(p)\neq0$, and the tangency curves $Z=(X,Y)$0 and $Z=(X,Y)$1 intersect transversally at $Z=(X,Y)$2. Around such points the local dynamics is structurally unstable under small perturbations, and the unfolding organizes crossing dynamics, sliding geometry, and, in specific degenerate cases, the birth of crossing limit cycles and their degeneration into fold-regular polycycles (Cespedes et al., 14 Jul 2025).

1. Definition and sign structure

Let $Z=(X,Y)$3 be a Filippov vector field on an open $Z=(X,Y)$4 with smooth switching function $Z=(X,Y)$5, and let $Z=(X,Y)$6. A cusp-fold singularity is defined by cubic contact of $Z=(X,Y)$7 with $Z=(X,Y)$8 and quadratic contact of $Z=(X,Y)$9 with $\Sigma=\{f=0\}$0: $\Sigma=\{f=0\}$1

$\Sigma=\{f=0\}$2

The intersection of the tangency sets must be transversal at $\Sigma=\{f=0\}$3. The contact is called invisible for $\Sigma=\{f=0\}$4 if $\Sigma=\{f=0\}$5 (Cespedes et al., 14 Jul 2025).

In the local coordinates used in the stand-alone summary of the bifurcation analysis, $\Sigma=\{f=0\}$6, and on $\Sigma=\{f=0\}$7 one has

$\Sigma=\{f=0\}$8

$\Sigma=\{f=0\}$9

At the singular value $\Sigma$0, the same summary states that invisibility of the $\Sigma$1-fold means $\Sigma$2 (Cespedes et al., 14 Jul 2025). This sign-based description is the operational criterion used there to distinguish the invisible configuration in the chosen normal-form expansion.

The central dynamical significance of the invisible cusp-fold is that it lies at the interface of crossing and tangential dynamics. Small perturbations separate the cusp and fold contacts, and the resulting local return mechanism determines whether a crossing periodic orbit can or cannot close.

2. Local normal forms and canonical representatives

Near an invisible cusp-fold, there exist smooth coordinates $\Sigma$3 centered at $\Sigma$4 and a time rescaling such that $\Sigma$5, and, up to third order,

$\Sigma$6

while for $\Sigma$7,

$\Sigma$8

Here $\Sigma$9 unfolds the cusp-fold coincidence, with $p\in\Sigma$0 corresponding to the singularity, and $p\in\Sigma$1 is the coefficient singled out in the expansion (Cespedes et al., 14 Jul 2025).

A complementary canonical description is given by the classification of $p\in\Sigma$2D cusp-fold singularities into $p\in\Sigma$3 topologically distinct normal forms. For the invisible cusp-fold one convenient representative is obtained by choosing

$p\in\Sigma$4

so that $p\in\Sigma$5 and $p\in\Sigma$6. The vector fields are

$p\in\Sigma$7

$p\in\Sigma$8

At the origin,

$p\in\Sigma$9

which gives an invisible cusp on the $Xf(p)=0$0-side, while

$Xf(p)=0$1

which gives an invisible fold on the $Xf(p)=0$2-side. In the notation of the classification, the type

$Xf(p)=0$3

is the only one among the $Xf(p)=0$4 canonical forms for which both the cusp and the fold are invisible (Carvalho et al., 2022).

This canonical representative is topological rather than asymptotic in purpose: it captures the side of $Xf(p)=0$5 on which each tangency sits, the relative placement of sliding and escaping sectors, and the qualitative way in which nearby orbits encounter the switching manifold.

3. Geometry on the switching manifold

For the canonical invisible model, the switching surface is $Xf(p)=0$6, and the Lie derivatives reduce to

$Xf(p)=0$7

This divides $Xf(p)=0$8 into sliding, escaping, and crossing regions (Carvalho et al., 2022).

Region	Defining inequalities	Location on $Xf(p)=0$9
Sliding $X^2f(p)=0$0	$X^2f(p)=0$1 and $X^2f(p)=0$2	$X^2f(p)=0$3
Escaping $X^2f(p)=0$4	$X^2f(p)=0$5 and $X^2f(p)=0$6	$X^2f(p)=0$7
Crossing $X^2f(p)=0$8	$X^2f(p)=0$9	Complement of the two sectors

Since $X^3f(p)\neq0$0, the crossing region splits into subregions $X^3f(p)\neq0$1 and $X^3f(p)\neq0$2 according to the signs of the two inner products. The Filippov sliding vector field on $X^3f(p)\neq0$3 is

$X^3f(p)\neq0$4

and, for this normal form,

$X^3f(p)\neq0$5

On $X^3f(p)\neq0$6, this defines a well-posed $X^3f(p)\neq0$7-dimensional flow (Carvalho et al., 2022).

The topological placement of the tangency curves is correspondingly rigid. On the $X^3f(p)\neq0$8-side, the curve $X^3f(p)\neq0$9 is the $Yf(p)=0$0-axis and carries the cusp at the origin. On the $Yf(p)=0$1-side, the curve $Yf(p)=0$2 is the $Yf(p)=0$3-axis and carries the invisible fold. The sliding region lies in the quadrant $Yf(p)=0$4 (Carvalho et al., 2022).

4. Generic codimension-one unfolding

In the recent bifurcation analysis, the generic codimension-one case is the non-anti-collinear situation: $Yf(p)=0$5 is fixed and $Yf(p)=0$6 and $Yf(p)=0$7 are not anti-collinear. Equivalently, the linear form $Yf(p)=0$8 has no double zero at $Yf(p)=0$9. The scalar parameter $Y^2f(p)\neq0$0 then moves the cusp out of the $Y^2f(p)\neq0$1-fold (Cespedes et al., 14 Jul 2025).

The local closure problem is expressed through two half-return maps on $Y^2f(p)\neq0$2: $Y^2f(p)\neq0$3 where $Y^2f(p)\neq0$4 is the first positive solution of $Y^2f(p)\neq0$5 and $Y^2f(p)\neq0$6 is the small nonzero root of $Y^2f(p)\neq0$7. A one-loop crossing cycle solves

$Y^2f(p)\neq0$8

Defining the displacement

$Y^2f(p)\neq0$9

the analysis shows, by the Implicit Function Theorem, that for sufficiently small $Z=(X,Y)$00 there is exactly one solution $Z=(X,Y)$01, corresponding to the double tangency. No new small crossing cycles bifurcate in this codimension-one scenario (Cespedes et al., 14 Jul 2025).

A second codimension-one description appears in the canonical-form treatment. There one perturbs only the third component of the lower-side vector field: $Z=(X,Y)$02 The fold line shifts from $Z=(X,Y)$03 to

$Z=(X,Y)$04

For $Z=(X,Y)$05, the origin is no longer a cusp-fold; instead there is a folded-folded pair at $Z=(X,Y)$06. The bifurcation occurs at $Z=(X,Y)$07 and is codimension $Z=(X,Y)$08. For $Z=(X,Y)$09, the entire $Z=(X,Y)$10 branch of $Z=(X,Y)$11 has moved into $Z=(X,Y)$12, producing one visible-invisible fold-fold and one invisible-invisible fold-fold; for $Z=(X,Y)$13, the picture is symmetric; for $Z=(X,Y)$14, one recovers the genuine cusp-fold (Carvalho et al., 2022).

Taken together, these two descriptions show that codimension-one unfoldings separate the cusp and fold tangencies but, in the generic non-anti-collinear case, do not generate a small crossing limit cycle.

5. Anti-collinearity and codimension-two cycle creation

The codimension-two regime arises when the vector fields are anti-collinear at the cusp-fold singularity. In the normal form, this is imposed by

$Z=(X,Y)$15

The unfolding then requires two parameters $Z=(X,Y)$16, where $Z=(X,Y)$17. The displacement function admits the expansion

$Z=(X,Y)$18

with nonzero index

$Z=(X,Y)$19

After setting $Z=(X,Y)$20, the Malgrange Preparation Theorem yields that the vanishing of $Z=(X,Y)$21 occurs along two smooth branches for

$Z=(X,Y)$22

(Cespedes et al., 14 Jul 2025).

The local bifurcation picture is then organized by the signs of $Z=(X,Y)$23 and $Z=(X,Y)$24:

• For $Z=(X,Y)$25: no small cycles.
• At $Z=(X,Y)$26: a double (Teixeira) singularity births two symmetric roots at the origin.
• For $Z=(X,Y)$27: there is one crossing limit cycle.
• At $Z=(X,Y)$28: the cycle grows until it hits the fold line $Z=(X,Y)$29 and degenerates into a fold-regular polycycle (Cespedes et al., 14 Jul 2025).

The same summary gives the polycycle-bifurcation curve in $Z=(X,Y)$30-space as

$Z=(X,Y)$31

Within the scope of that analysis, this codimension-two configuration is precisely the one that produces a bifurcating crossing limit cycle. This distinguishes it sharply from the generic codimension-one case, where no local crossing limit cycle bifurcates from the invisible cusp-fold.

6. Boost-converter realization

The codimension-two scenario is realized in a boost-converter model studied previously in the literature. In that application, parameters $Z=(X,Y)$32 are identified so that an invisible cusp-fold appears at $Z=(X,Y)$33, and the resulting two-parameter unfolding exactly matches the codimension-two mechanism described above (Cespedes et al., 14 Jul 2025).

By numerically solving the closing equations, two curves are traced in the $Z=(X,Y)$34-plane:

• the Teixeira-cycle bifurcation curve $Z=(X,Y)$35, corresponding to the birth of a crossing limit cycle;
• the polycycle curve $Z=(X,Y)$36, corresponding to degeneration into a fold-regular polycycle.

Between these curves, the converter exhibits a unique crossing limit cycle. Outside them, no small crossing cycles occur. The completed bifurcation set is given as Figure 1 of the paper (Cespedes et al., 14 Jul 2025).

This application is significant because it places the local singularity theory in a concrete nonsmooth dynamical system where the cusp-fold is not merely a local degeneracy but an organizing center for observable phase-portrait transitions. A plausible implication is that the invisible cusp-fold provides a compact bifurcation descriptor for converter regimes in which orbit closure, collision with tangency sets, and disappearance through polycycles are all present within a small parameter wedge.

7. Terminological distinctions and related uses of “invisibility”

The expression “invisible cusp-fold singularity” belongs primarily to the theory of $Z=(X,Y)$37D Filippov systems, but closely related terminology also appears in singularity theory and differential geometry. In the study of folded cuspidal edges, the “invisible” cuspidal fold or cuspidal $Z=(X,Y)$38-singularity is obtained by folding a generic cuspidal edge along the plane $Z=(X,Y)$39 via

$Z=(X,Y)$40

so that the resulting map $Z=(X,Y)$41 is a cuspidal cross-cap (Sinha et al., 2017).

In that geometric setting, invisibility has a different meaning. All of $Z=(X,Y)$42 lies in the half-space $Z=(X,Y)$43, and $Z=(X,Y)$44 forces the branch that would be present in a genuine Whitney umbrella on $Z=(X,Y)$45 to be absent. The paper further states that $Z=(X,Y)$46, $Z=(X,Y)$47, and $Z=(X,Y)$48 guarantee that the osculating plane of the edge is transverse to the tangent cone, so in projection the forbidden branch never appears (Sinha et al., 2017).

This distinction prevents a common conflation. In Filippov dynamics, invisibility is a sign condition on Lie derivatives at tangency and determines how trajectories meet the switching manifold. In the geometry of folded cuspidal edges, invisibility refers to the absence of a branch after folding and to one-sidedness relative to the tangent cone. The shared vocabulary reflects a common theme of hidden or non-accessible local structure, but the objects, mechanisms, and invariants are different.