Fold-Regular Polycycle in Filippov Systems

• Fold-Regular Polycycle is a self-connection at a fold-regular singularity on the switching manifold in a three-dimensional Filippov system.
• It emerges from codimension-two unfolding of cusp-fold singularities, where anti-collinearity induces a cubic return-map degeneracy.
• This structure delineates the boundary between pure crossing cycles and mixed sliding-crossing dynamics, with applications in models like controlled boost converters.

A fold-regular polycycle is, in the terminology of the three-dimensional Filippov analysis of cusp-fold singularities, a special case of polycycle associated with a fold-regular singularity on the switching manifold $\Sigma=f^{-1}(0)$. In that setting, a polycycle is a closed curve $\Gamma$ made of finitely many $\Sigma$-singularities or equilibria and regular oriented orbit arcs, carrying a nonconstant first return map on a two-dimensional section; the fold-regular case is the configuration composed of a single fold-regular singularity $p_1\in\Sigma$ and a unique regular orbit $\gamma_1$ returning to it. Geometrically, it is a self-connection at a fold-regular singularity. In the codimension-two anti-collinear cusp-fold unfolding studied in three dimensions, it appears as the terminal object of a branch of crossing limit cycles: a crossing limit cycle degenerates into a fold-regular polycycle and then disappears (Cespedes et al., 14 Jul 2025).

1. Definition and geometric framework

In the relevant Filippov setting, one considers a piecewise smooth vector field $Z=(X,Y)$ on an open subset $M\subset \mathbb R^3$, with switching manifold

$\Sigma=f^{-1}(0),$

where $0$ is a regular value of $f$. The manifold splits $\Gamma$0 into $\Gamma$1, and the crossing region is

$\Gamma$2

while the tangency sets are $\Gamma$3 and $\Gamma$4 (Cespedes et al., 14 Jul 2025).

A point $\Gamma$5 is a fold of $\Gamma$6 if

$\Gamma$7

and it is a cusp of $\Gamma$8 if

$\Gamma$9

together with

$\Sigma$0

A fold-regular singularity of $\Sigma$1 is a point $\Sigma$2 that is a fold of one side vector field and a regular point of the other. More precisely, $\Sigma$3 is fold-regular if it is a fold of $\Sigma$4 and a regular point of $\Sigma$5; dually, one may have a regular-fold point if it is a fold of $\Sigma$6 and regular for $\Sigma$7 (Cespedes et al., 14 Jul 2025).

Within this vocabulary, a fold-regular polycycle is a $\Sigma$8-polycycle composed by a fold-regular singularity $\Sigma$9 and a unique regular orbit $p_1\in\Sigma$0. The geometric content is that the loop closes only because one end of the orbit lands exactly on a fold point of one field while the other field is transverse there. The object is therefore not a smooth periodic orbit, and it is not a generic crossing cycle: it is a singular self-connection at a fold-regular point (Cespedes et al., 14 Jul 2025).

2. Emergence from cusp-fold singularities

The principal setting in which the term is introduced is the unfolding of an invisible cusp-fold singularity. A cusp-fold singularity of $p_1\in\Sigma$1 is a point $p_1\in\Sigma$2 such that $p_1\in\Sigma$3 is a cusp of $p_1\in\Sigma$4, a fold of $p_1\in\Sigma$5, and $p_1\in\Sigma$6 at $p_1\in\Sigma$7. The analysis focuses on the case in which the fold of $p_1\in\Sigma$8 is invisible and the cusp belongs to $p_1\in\Sigma$9 (Cespedes et al., 14 Jul 2025).

Two distinct degeneracy classes are separated. In the codimension-one case, the invisible cusp-fold singularity is of degree $\gamma_1$0 when

$\gamma_1$1

and $\gamma_1$2 at $\gamma_1$3. In that generic situation, nearby systems have no crossing limit cycles in a neighborhood of $\gamma_1$4. The geometric interpretation given there is that $\gamma_1$5 is equivalent to $\gamma_1$6 and $\gamma_1$7 not being anti-collinear in $\gamma_1$8, and that non-anti-collinearity blocks the local crossing-limit-cycle mechanism (Cespedes et al., 14 Jul 2025).

The fold-regular polycycle arises only in the codimension-two anti-collinear case. There, the singularity is of degree $\gamma_1$9 when:

1. $Z=(X,Y)$0 at $Z=(X,Y)$1;
2. $Z=(X,Y)$2;
3. an explicit cubic coefficient combination defines an index $Z=(X,Y)$3;
4. $Z=(X,Y)$4.

In this case, anti-collinearity creates the local return-map degeneracy required for the birth of a crossing limit cycle and for its later degeneration into a fold-regular polycycle (Cespedes et al., 14 Jul 2025).

A local normal form is available after a smooth coordinate change and, if necessary, time reversal. Near the singularity,

$Z=(X,Y)$5

and

$Z=(X,Y)$6

In this form, the parameter $Z=(X,Y)$7 separates cusp and fold, while $Z=(X,Y)$8 measures anti-collinearity breaking (Cespedes et al., 14 Jul 2025).

3. Return maps, displacement equations, and the bifurcation curve

The local dynamics is encoded by two half-return maps: $Z=(X,Y)$9, induced by the flow of the cusp side $M\subset \mathbb R^3$0, and $M\subset \mathbb R^3$1, induced by the flow of the fold side $M\subset \mathbb R^3$2. Closed crossing orbits are characterized by the equation

$M\subset \mathbb R^3$3

which is rephrased through a displacement map $M\subset \mathbb R^3$4. In the notation of the cited analysis,

$M\subset \mathbb R^3$5

An orbit through $M\subset \mathbb R^3$6 is closed and intersects $M\subset \mathbb R^3$7 only at two points iff

$M\subset \mathbb R^3$8

(Cespedes et al., 14 Jul 2025).

In the codimension-two case, lower-order terms cancel and the reduced displacement acquires a cubic leading term with coefficient $M\subset \mathbb R^3$9. This produces a square-root splitting of return-map fixed points and leads to an explicit bifurcation curve

$\Sigma=f^{-1}(0),$0

The cited theorem states that, under the corresponding sign and smallness conditions, there is a unique one-loop crossing limit cycle in the interior parameter region, whereas on the curve

$\Sigma=f^{-1}(0),$1

there is a unique one-loop critical crossing cycle, namely the fold-regular polycycle (Cespedes et al., 14 Jul 2025).

The bifurcation sequence is accordingly organized as follows. Before the birth curve there is no crossing limit cycle. On the birth curve

$\Sigma=f^{-1}(0),$2

there is a critical crossing cycle collapsed at the double tangency. Between that curve and the polycycle curve there is a unique one-loop crossing limit cycle. On

$\Sigma=f^{-1}(0),$3

that crossing cycle becomes a fold-regular polycycle. Beyond it, one branch of fixed points leaves the crossing region and the crossing limit cycle disappears (Cespedes et al., 14 Jul 2025).

This clarifies a common ambiguity. The fold-regular polycycle is not itself a crossing limit cycle; it is the singular boundary configuration at which a crossing branch terminates. In the same analysis it is also described as the boundary object between pure crossing periodic behavior and mixed sliding-crossing behavior (Cespedes et al., 14 Jul 2025).

4. Relation to broader nonsmooth polycycle theory

The phrase “fold-regular polycycle” is specific, but it sits naturally within a wider theory of nonsmooth polycycles. In planar non-smooth vector fields, a graphic is formed by singularities $\Sigma=f^{-1}(0),$4 and regular orbits $\Sigma=f^{-1}(0),$5 such that $\Sigma=f^{-1}(0),$6 is a stable characteristic orbit of $\Sigma=f^{-1}(0),$7 and an unstable characteristic orbit of $\Sigma=f^{-1}(0),$8; a polycycle is a graphic with a return map. A semi-elementary polycycle allows hyperbolic saddles, semi-hyperbolic singularities, and tangential singularities, while an elementary polycycle allows hyperbolic saddles and tangential singularities (Santana, 2021).

In that framework, tangential singularities are assigned stable and unstable contact orders $\Sigma=f^{-1}(0),$9 and $0$0, and a local hyperbolicity ratio

$0$1

when the singularity is tangential. The local transition maps satisfy expansions of the form

$0$2

$0$3

A tangential singularity is classified as stable when $0$4 and unstable when $0$5 (Santana, 2021).

This suggests that fold-regular polycycles can be viewed as simple-order specializations of the tangential-singularity formalism. The cited nonsmooth theory does not define “fold-regular” explicitly, but it states that a fold corresponds to the lowest-order tangential case, and it analyzes global stability of an elementary polycycle through the product

$0$6

If $0$7, the polycycle attracts nearby trajectories on the relevant side; if $0$8, it repels them. For polycycles whose singularities are all stable or all unstable in the sense above, the cyclicity is one, and any bifurcating limit cycle is unique and hyperbolic (Santana, 2021).

The fold-regular polycycle studied in three dimensions is therefore more specific than the general planar tangential theory, but the shared language of local power-law transition maps, return maps, and stability ratios places it in the same conceptual lineage.

5. Regularization and local fold-based building blocks

Fold-regular polycycles are also connected to the regularization theory of planar Filippov singularities. A generic codimension-one fold-fold singularity is studied for systems

$0$9

with switching manifold $f$0, fold conditions

$f$1

and sliding field

$f$2

(Bonet-Revés et al., 2016).

Its Sotomayor–Teixeira regularization is

$f$3

with a monotone transition function $f$4. After the rescaling $f$5, one obtains a slow-fast system whose critical manifold is defined by

$f$6

Over sliding or escaping regions this manifold is

$f$7

and the reduced flow on it is

$f$8

(Bonet-Revés et al., 2016).

A plausible implication is that regularized fold-regular polycycles inherit their singular local passages from this fold-fold machinery. The regularization theory shows that sliding or escaping pieces are replaced by slow invariant manifolds, and that visible-invisible configurations can generate Hopf bifurcation, maximal canards, and saddle-node bifurcations of cycles. In the visible-invisible case with linear regularization, the regularized system is equivalent in the strip to a general slow-fast normal form of Krupa–Szmolyan type, and the maximal canard occurs on a curve

$f$9

(Bonet-Revés et al., 2016).

This does not by itself define a fold-regular polycycle, but it provides local and semi-local blocks for its regularized analysis. In particular, it demonstrates that fold-based singular geometries can organize periodic orbits and canard-type returns even when the nonsmooth limiting system has no ordinary smooth cycle.

6. Illustrative models and dynamical significance

The explicit toy model in the three-dimensional cusp-fold analysis makes the fold-regular polycycle visible in closed form. There,

$\Gamma$00

with $\Gamma$01, so that

$\Gamma$02

and hence

$\Gamma$03

When $\Gamma$04 and $\Gamma$05, the double tangency is an invisible cusp-fold singularity of degree $\Gamma$06 with index $\Gamma$07, and anti-collinearity is simply

$\Gamma$08

In this model the crossing limit cycle exists in a parameter strip bounded on one side by the T-singularity birth line $\Gamma$09 and on the other by

$\Gamma$10

where it becomes the fold-regular polycycle. After that, the closed object becomes a limit cycle with a sliding segment (Cespedes et al., 14 Jul 2025).

The engineering application is a controlled boost converter modeled by

$\Gamma$11

with explicitly given matrices $\Gamma$12 and parameter vector $\Gamma$13. In that system the polycycle bifurcation curve is obtained numerically by combining the closing equations with a tangency condition on $\Gamma$14. The completed $\Gamma$15-diagram has a blue branch corresponding to the T-singularity or crossing-limit-cycle birth and a red branch corresponding to the fold-regular polycycle. The region between those branches is exactly the region where the converter has a crossing limit cycle; outside it no crossing limit cycle exists (Cespedes et al., 14 Jul 2025).

These examples fix the dynamical role of the notion. A fold-regular polycycle is not merely a degenerate orbit. It is the codimension-one boundary configuration at which a branch of crossing periodic motion reaches a fold-regular singularity, after which the crossing character is lost. In that sense it is an organizing object for the termination of crossing cycles and for transitions between crossing dynamics and sliding-crossing dynamics in nonsmooth systems (Cespedes et al., 14 Jul 2025).