Slow divergence integral in regularized piecewise smooth systems

In this paper we define the notion of slow divergence integral along sliding segments in regularized planar piecewise smooth systems. The boundary of such segments may contain diverse tangency points. We show that the slow divergence integral is invariant under smooth equivalences. This is a natural generalization of the notion of slow divergence integral along normally hyperbolic portions of curve of singularities in smooth planar slow-fast systems. We give an interesting application of the integral in a model with visible-invisible two-fold of type $VI_3$. It is related to a connection between so-called Minkowski dimension of bounded and monotone"entry-exit"sequences and the number of sliding limit cycles produced by so-called canard cycles.


Introduction
The notion of slow divergence integral [6,Chapter 5] has proved to be an important tool to study lower and upper bounds of limit cycles in smooth planar slow-fast systems (see e.g.[5-7, 10, 11] and references therein).In this paper by "smooth", we mean differentiable of class C ∞ .One of the main goals of this paper is to define the slow divergence integral in regularized planar piecewise smooth (PWS) systems with sliding and to prove its invariance under smooth equivalences (by smooth equivalence we mean smooth coordinate change and a multiplication by a smooth positive function).This is a natural generalization of [25] where the slow divergence integral is defined only for a PWS two-fold bifurcation of type visible-invisible called V I 3 and the switching boundary is a straight line (for more details about two-fold singularity V I 3 see [28] and Sections 2 and 3).In this paper we define the slow divergence integral along sliding segments with a regular sliding vector field [16] (Section 2.1), and extend it to tangency points when only one vector field is tangent to switching boundary (Section 2.3), two-fold singularities of sliding type V V 1 , II 1 , V I 2 and V I 3 [28], and to a visible-invisible two-fold singularity when the sliding vector field vanishes at the two-fold point (Section 2.2).
Consider a smooth planar slow-fast system defined on open set V ⊂ R 2 , where 0 < ϵ ≪ 1 is the singular perturbation parameter, λ ∼ λ 0 ∈ R l , f λ is a smooth family of functions and Y λ and Q λ are smooth families of vector fields.We suppose that X 0,λ has a curve of singularities C λ for all λ ∼ λ 0 (Fig. 1).We further assume that ∇f λ (p) ̸ = (0, 0) for all p ∈ {(x, y) ∈ V | f λ (x, y) = 0} and that Y λ has no singularities for each λ ∼ λ 0 .Then we have C λ = {f λ = 0} and C λ is a smooth family of 1-dimensional manifolds.
The orbits of the flow of X 0,λ are located inside the leaves of a smooth 1-dimensional foliation F λ on V tangent to Y λ (F λ is called the fast foliation of X 0,λ ).If p ∈ C λ , then DX 0,λ (p) has one eigenvalue equal to zero, with eigenspace T p C λ , and the other one equal to div X 0,λ (p) (i.e., the trace of DX 0,λ (p)) with eigenspace T p l λ,p (l λ,p is the leaf through p).We say that p ∈ C λ is normally hyperbolic if div X 0,λ (p) ̸ = 0 (attracting when div X 0,λ (p) < 0 and repelling when div X 0,λ (p) > 0).We can define the notion of slow vector field on normally hyperbolic segments of C λ .Let p ∈ C λ be a normally hyperbolic singularity and let Qλ (p) ∈ T p C λ be the projection of Q λ (p) on T p C λ in the direction of T p l λ,p .Qλ is called the slow vector field and its flow the slow dynamics.The time variable of the slow dynamics is the slow time t = ϵt where t is the time variable of the flow of X ϵ,λ .We point out that the classical definition of the slow vector field using center manifolds is equivalent to this definition.For more details see [6,Chapter 3].We define now the notion of slow divergence integral (see [6,Chapter 5]).If m λ ⊂ C λ is a normally hyperbolic segment not containing singularities of Qλ , then the slow divergence integral along m λ is defined by where ) and z λ ( t1 ) and z λ ( t2 ) are the end points of m λ (we parameterize m λ by t).This definition is independent of the choice of parameterization z λ of m λ and the slow divergence integral is invariant under smooth equivalences (see [6,Section 5.3]).
If both eigenvalues of the linear part of X 0,λ at p ∈ C λ are zero, then we say that p is a (nilpotent) contact point between C λ and F λ .The slow divergence integral can also be defined along parts of C λ that contain contact points, using its invariance under smooth equivalences and normal forms near contact points (see [6,Section 5.5]).
We come now to a natural question: can we define the notion of slow divergence integral if we replace the slow-fast system X ϵ,λ with a regularized planar PWS system?In Section 2 we give a positive answer to the question.Instead of X 0,λ we consider a λ-family of planar PWS systems (2) defined in Section 2. The switching boundary Σ λ defined after (2) plays the role of the curve of singularities C λ , and the Filippov sliding vector field Z sl λ on sliding subsets of Σ λ (see (3)) plays the role of the slow vector field Qλ on normally hyperbolic portions of C λ (see Proposition 2.1).The function that will be integrated (Definition 1 in Section 2.1) is the divergence of a smooth slow-fast vector field visible in the scaling chart of a cylindrical blow-up applied to regularized PWS system (5) (for more details see [25] and the proof of Proposition 2.1).The notion of slow divergence integral in the PWS setting is well-defined when the sliding vector field Z sl λ has no singularities (see Definition 1).We show that the slow divergence integral from Definition 1 is invariant under smooth equivalences (see Theorem 2.2 in Section 2.1).
Figure 2: Limit periodic sets in planar PWS systems through two-fold points with sliding (the V I 3 case and the V V 1 case).They can be located in a region with invisible fold point (green) or in a region with visible fold point (red).
In Sections 2.2 and 2.3 we define the slow divergence integral near tangency points, as already mentioned above (tangency points in Σ λ play the role of contact points between C λ and F λ ).We use the invariance of the slow divergence integral under smooth equivalence.The extension of the slow divergence integral to tangency points has proved to be crucial when we study the number of sliding limit cycles (i.e., isolated closed orbits with sliding segments) of a regularized planar PWS system produced by so-called canard limit periodic sets or canard cycles (for more details see [25]).In [25] only the V I 3 case has been studied, with canard cycles located in the region with invisible fold point (see the green closed curve in Fig. 2).Canard cycles contain both stable and unstable sliding portions of the discontinuity manifold (often called switching boundary).A canard cycle can also be located in a region with visible fold point (for example, red closed curves in Fig. 2), and again the slow divergence integral associated to the segment of the switching boundary contained in the canard cycle plays an important role when studying sliding limit cycles (see [24]).
Besides sliding cycles, crossing limit cycles can exist for PWS systems and for example J. Llibre and co-workers have obtained upper bounds for a number of classes [13,29,32].See also [2,4,17,18,20,21,30,31] and references therein.Section 3 is devoted to applications of the slow divergence integral from Section 2. In Section 3 we focus on the model used in [25] (visible-invisible two-fold V I 3 ) and read upper bounds of the number of sliding limit cycles and type of bifurcations near so-called generalized canard cycles (Fig. 7) from fractal properties of a bounded and monotone sequence in R defined using the slow divergence integral and the notion of slow relation function (see Section 3.1).The main advantage of this fractal approach is that, instead of computing the multiplicity of zeros of the slow divergence integral (like in [25]), it suffices to find the Minkowski dimension [14] of the sequence.There is a bijective correspondence between the multiplicity and the Minkowski dimension (see Section 4).A similar fractal approach has been used in [8,22,23,26] where one deals with smooth slow-fast systems.See also [12,37] and the references therein.We point out that there exist simple formulas for numerical computation of the Minkowski dimension of the sequence (see e.g.[23]).In Section 3.2 we state the main fractal results (Theorems 3.1-3.3),and in Section 4 we prove them.
For the sake of readability, in this paper we work in R 2 using the Euclidean metric.We believe that the notion of slow divergence integral in regularized PWS systems on a smooth surface could also be defined.We point out that the slow divergence integral [6] is defined for slow-fast systems on a smooth surface.
2 The slow divergence integral in PWS systems with sliding First we recall the basic definitions in PWS theory [9,19] (switching boundary, sliding set, crossing set, sliding vector field, tangency point, two-fold singularity, etc.).Then we define the notion of slow divergence integral of a regularized PWS system along a sliding segment (not containing singularities of the sliding vector field) and prove that the integral is invariant under smooth equivalences (see Section 2.1).In Section 2.2 we extend the definition of the slow divergence integral to segments consisting of a stable sliding region, an unstable sliding region and a two-fold singularity between them.If the two-fold singularity is visible-invisible, then we assume that the sliding vector field is regular or has a hyperbolic singularity in the two-fold point.In Section 2.3 we define the slow divergence integral near a tangency point where the tangency (quadratic or more degenerate) appears only in one vector field.Consider a λ-family of PWS systems in the plane where z = (x, y), λ ∼ λ 0 ∈ R l and the switching boundary is a smooth λ-family of 1-dimensional manifolds Σ λ given by with an open set V and a smooth family of functions h λ such that ∇h λ (z We assume that the λ-family of vector fields Z The subset Σ sl λ ⊂ Σ λ consisting of all points z ∈ Σ λ such that is the set of all stable (resp.unstable) sliding points.In Σ s λ (resp.Σ u λ ) the vector fields Z ± λ point toward (resp.away from) the switching boundary.We call the set Σ cr λ ⊂ Σ λ of all points z ∈ Σ λ such that At each point z ∈ Σ cr λ the orbit of (2) crosses the switching boundary Σ λ (it is the concatenation of the orbit of Z + λ and the orbit of Z − λ through z).Along the sliding set Σ sl λ , the flow is given by the Filippov sliding vector field [16] The sliding vector field We λ and invisible in Z − λ or visible in Z − λ and invisible in Z + λ .Following [28], there exist 7 (generic) cases for two-fold singularities taking into account the direction of the flow of Z ± λ and Z sl λ : 2 visible-visible cases (denoted by V V 1 and V V 2 in [28]), 2 invisible-invisible cases (II 1 and II 2 ) and 3 visibleinvisible cases (V I 1 , V I 2 and V I 3 ).For more details we refer to [1,19,27,28].In Section 2.2 we define the notion of slow divergence integral near two-fold singularities of sliding type (V V 1 , II 1 , V I 2 and V I 3 ).The four sliding cases are illustrated in Fig. 4. We also treat a visible-invisible two-fold singularity where the sliding vector field points toward (or away from) the two-fold singularity on both sides (Fig. 5).
We consider a regularized PWS system [25] where 0 < ϵ ≪ 1 and ϕ : R → R is a smooth regularization function.We assume that ϕ is strictly monotone, i.e., and ϕ has the following asymptotics for u → ±∞: Moreover, we assume that ϕ is smooth at ±∞ in the following sense: The functions Using the asymptotics of ϕ given in (7), the system (5) becomes the PWS system (2) in the limit ϵ → 0. Combining this with the fact that ϕ is smooth at ±∞, we have that, for z kept in any fixed compact set in V \ Σ λ , the right hand side of ( 5) is an o(1)-perturbation of the right hand side of (2) where o(1) is a smooth function in (z, ϵ, λ) and tends to 0 as ϵ → 0, uniformly in (z, λ).Thus, the PWS system (2) describes the dynamics of ( 5), for ϵ > 0 small, as long as z is kept uniformly away from Σ λ .

Definition and invariance of the slow divergence integral
Proposition 2.1.Suppose that the PWS system (2) has a stable (resp.unstable) sliding point p ∈ Σ sl λ0 .Then, for each 0 < ϵ ≪ 1 and λ ∼ λ 0 , (5) has a locally invariant manifold near p with foliation by stable (resp.unstable) fibers, and the reduced dynamics on this manifold (when ϵ → 0) is given by sliding vector field Z sl λ defined in (3).
We get the same expression (10) if we use the definition of the slow vector field introduced in Section 1.This completes the proof.
Following [6,Chapter 5] or Section 1 in the smooth slow-fast system (8) one can define the notion of slow divergence integral along normally hyperbolic curve of singularities ỹ = ϕ −1 (τ λ (x, f λ (x))) when the sliding vector field in (10) has no singularities: it is the integral of the divergence in (9) where the variable of integration is the time variable of the flow of the sliding vector field.This is our motivation for the definition of the notion of slow divergence integral of regularized PWS system (5) (see also [25]).
Definition 1 (Slow divergence integral).Let m λ ⊂ Σ sl λ be a bounded segment (Fig. 3) not containing pseudo-equilibria of the PWS system (2).Let where z λ (t 1 ) and z λ (t 2 ) are the end points of m λ (z λ is a parameterization of m λ ).Then we define the slow divergence integral of regularized PWS system (5) associated to m λ as where Note that the definition of the slow divergence integral given by (11) is independent of the choice of z λ .Indeed, if ẑλ is another solution to z ′ (t) = Z sl λ (z(t)) and p ∈ m λ , then there exist t ∈ [t 1 , t 2 ] and t such that z λ ( t) = ẑλ ( t) = p.Then we have z λ (t) = ẑλ (t + t − t) due to uniqueness of solutions.Now, we get where we use the change of variable s = t + t − t.
The slow divergence integral from Definition 1 is invariant under smooth equivalences (Theorem 2.2.1 and Theorem 2.
2. Let g be a smooth strictly positive function defined in a neighborhood of m λ .Then the slow divergence integral of Z ϵ,λ along m λ is equal to the slow divergence integral of the equivalent vector field g.Z ϵ,λ along m λ .

Let p
Then we have where pλ is a smooth λ-family of nowhere zero functions satisfying Proof.Statement 1.The pullback of the vector field Z ϵ,λ | Vz can be written as where . It is not difficult to see that the Liederivative of h λ • T with respect to the vector field W ± λ is given by Using ( 12) and Definition 1 we find that the slow divergence integral of T * (Z ϵ,λ | Vz ) along T −1 (m λ ) is given by where ), the result follows.
Statement 3. The proof of Statement 3 is similar to the proof of Statement 2.
Remark 2. It follows directly from Definition 1 that the slow divergence integral of −Z ϵ,λ along m λ is equal to the slow divergence integral of Z ϵ,λ along m λ multiplied by −1.
We will use the invariance of the slow divergence integral under smooth equivalences in Section 2.2 and Section 2.3.
If m λ ⊂ Σ sl λ contains pseudo-equilibria, then the slow divergence integral associated to m λ is not well-defined.

The slow divergence integral near two-fold singularities
In this section we suppose that the sliding vector field Z sl λ , given by (3), is defined around a two-fold singularity.Our goal is to define the notion of slow divergence integral near such a two-fold singularity.Since the slow divergence integral ( 11) is invariant under smooth equivalences (Theorem 2.2), we use a normal form of (2), locally near the two-fold singularity, in which h λ (x, y) = y and the two-fold point corresponds to the origin p = (0, 0).Notice that such normal form coordinates exist because ∇h λ (z) ̸ = (0, 0), ∀z ∈ Σ λ , in (2).
Using h λ (x, y) = y the two-fold p satisfies and the sliding vector field Z sl λ near p can be written as where det ). Remark 3. The notation det Z λ comes from [1].In [25] a similar notation has been used for − ).Since we assumed that the sliding vector field X sl λ is defined around the two-fold p, we find that X + λ (0)X − λ (0) > 0 if the folds have the same visibility (visible-visible or invisible-invisible) and X + λ (0)X − λ (0) < 0 if the folds have opposite visibility.We have p ∈ ∂Σ s λ ∩ ∂Σ u λ .These properties follow directly from (14) and the definition of visible and invisible folds (see [1, Lemma 2.8]), and imply that ∂ is clear that the sliding vector field in (15) has a removable singularity in x = 0 and From [1, Lemma 2.9] and [1, Corollary 2.10] it follows that ν ̸ = 0 and sgn(ν) = sgn(X + λ (0)) if the folds have the same visibility (V V 1 and II 1 in Fig. 4), and that ν ̸ = 0 and sgn(ν) = − sgn X + λ (0)(det Z λ ) ′ (0) if the folds have opposite visibility and (det Z λ ) ′ (0) ̸ = 0 (V I 2 and V I 3 in Fig. 4).If the folds have opposite visibility, we assume that ν ̸ = 0 in (16) or x = 0 is a hyperbolic singularity of the sliding vector field X sl λ (or, equivalently, x = 0 is a zero of multiplicity 1 or 2 of the function det Z λ ).We refer to Fig. 5 (the multiplicity of the zero x = 0 of det Z λ is 2). From 0) < 0 it follows that the function τ λ defined in (4) has the following property when x → 0: Let us now compute the slow divergence integral along [x 0 , x 1 ], with 0 < x 0 < x 1 .We assume that x 1 is small enough such that [x 0 , x 1 ] does not contain any singularities of the sliding vector field X sl λ .We use Theorem 2.2.3.We take This implies (18) Finally, we define the slow divergence integral along [0, x 1 ] (the left end point of [0, x 1 ] is the two-fold point).where I([x 0 , x 1 ]) is given in (18).
Remark 4. Notice that the function x → ϕ ′ ϕ −1 (τ λ (x, 0)) in (18) can be defined at x = 0 such that this function is smooth and positive on the segment m λ (see ( 6), ( 7) and ( 17)).If the folds have the same visibility, then I(m λ ) is well-defined (finite) because ν ̸ = 0 in (16).Since we assume that the multiplicity of the zero x = 0 of det Z λ does not exceed 2 when the folds have opposite visibility, I(m λ ) is finite.

The slow divergence integral near one-sided tangency points
In this section we define the slow divergence integral near a tangency point p ∈ ∂Σ s λ ∪ ∂Σ u λ where both vectors Z ± λ (p) are nonzero and precisely one of them is tangent to Σ λ at p (see e.g.Fig. 6).Like in Section 2.2, the switching boundary Σ λ is locally given by h λ (x, y) = y and p = (0, 0).Since we suppose that p ∈ ∂Σ s λ ∪∂Σ u λ , there is a side of p (without loss of generality we take x > 0) where the sliding vector field is defined, and given by (15).If the (nonzero) vector Since X + λ (0) ̸ = 0 (resp.X − λ (0) ̸ = 0), the sliding vector field X sl λ in ( 19) is regular near x = 0. Thus, the segment [x 0 , x 1 ], with 0 < x 0 < x 1 , does not contain any singularities of X sl λ if x 1 is small enough and we can define the slow divergence integral along [x 0 , x 1 ] exactly in the same way as in Section 2.2.The slow divergence integral is given by ( 18) and we use the same notation I([x 0 , x 1 ]).
We can now define the slow divergence integral near the tangency point p.
Then the slow divergence integral along m λ is defined as Remark 6.The slow divergence integral I(m λ ) from Definition 3 is welldefined.Indeed, lim u→±∞ ϕ ′ (u) = 0 (due to the smoothness of ϕ at ±∞ given after (7)).Moreover, we have (a) −∞ (see (7)) and finally (c) It follows from (a), (b) and (c) that the integrand in (18) can be defined at x = 0 (0 for x = 0) and that the integrand is continuous on the segment m λ .This implies that I(m λ ) is well-defined.
3 Limit cycles and fractal analysis through visibleinvisible two-fold V I 3

Model and assumptions
We consider a PWS system (2) where we assume that λ ∼ λ 0 ∈ R, and h λ (x, y) = y (the switching boundary is the line y = 0).
Assumption A implies that Y ± λ0 (0) = 0 and the origin z = 0 is therefore a tangency point (see Section 2).We assume that z = 0 for λ = λ 0 is a twofold singularity.Moreover, we suppose that the two-fold singularity is visible from "above" and invisible from "below", i.e., the orbit of Z + λ0 through z = 0 is contained within y > 0 near z = 0, and the orbit of Z − λ0 through z = 0 is not contained within y < 0 (Section 2).Assumption B. We assume that the origin z = 0 in the PWS system (2) is a visible-invisible two-fold for λ = λ 0 : Y ± λ0 (0) = 0 and Additionally, we assume that (det Z λ0 ) ′ (0) < 0 where det Z λ is defined in (15).
Assumption E. We suppose that Γ 0 is balanced and that s = 0 is an isolated zero of I(s), meaning that s = 0 has a small neighborhood ] − s, s[ (s > 0) that does not contain any other zero of I(s).
Assumption E implies that I is either negative or positive for s > 0 (I is continuous).Using the above mentioned property of the integrand in (23) it can be easily seen that 0 < G(s) < s for s > 0 when I is negative and G(s) > s for s > 0 when I is positive.Let s 0 > 0 be small and fixed.Thus, if I is negative (resp.positive), then the orbit of s 0 defined by s n+1 = G(s n ) (resp.s n+1 = G −1 (s n )), n ≥ 0, tends monotonically to the fixed point s = 0 of G.We want to study the Minkowski dimension of U 0 .Let us first define the notion of Minkowski (or box) dimension (see [14,34] and references therein).Let U ⊂ R N be a bounded set.We define the δneighborhood of U : and denote by |U δ | the Lebesgue measure of U δ .The lower u-dimensional Minkowski content of U , for u ≥ 0, is defined by and analogously the upper u-dimensional Minkowski content M * u (U ) (we replace lim inf δ→0 with lim sup δ→0 ).We define lower and upper Minkowski dimensions of U : We have dim B U ≤ dim B U and, if dim B U = dim B U , we call it the Minkowski dimension of U , and denote it by dim B U .The upper Minkowski dimension is finitely stable.More precisely, < ∞ for some d, then we say that U is Minkowski nondegenerate.In this case we have necessarily that d = dim B U .Recall also that the notion of being Minkowski nondegenerate is invariant under bi-Lipschitz maps.Namely, if Φ is a bi-Lipschitz map and U is Minkowski nondegenerate, then Φ(U ) is also Minkowski nondegenerate (see [35,Theorem 4.1]).

Statement of results
In this section we consider the family (22) that satisfies Assumptions A through E and assume that λ = λ 0 + ϵ λ with λ ∼ 0. Theorem 3.1.Let s 0 > 0 be small and fixed and let U 0 be the orbit of s 0 defined in (25).If dim B U 0 = 0, then the following statements hold.
Theorem 3.1 will be proved in Section 4.1.
Theorem 3.3.Let U 0 be the orbit of s 0 defined in (25), for a small s 0 > 0, and dim B U 0 = 0.The following statements are true.
1.For λ = λ(ϵ) given in Theorem 3.1.1and for each small ϵ > 0, the Minkowski of any spiral trajectory accumulating (in forward or backward time) on the unique limit cycle of (22) near Γ 0 is equal to 1.

2.
For each small ϵ > 0, the Minkowski dimension of any spiral trajectory accumulating (in forward or backward time) on the limit cycle of multiplicity 2 of (22), born in a saddle-node bifurcation of limit cycles Hausdorff close to Γ 0 , is equal to 3 2 and moreover, the spiral is Minkowski nondegenerate.Theorem 3.3 will be proved in Section 4.2.A small (Hausdorff) neighborhood of Γ 0 in which we consider spiral trajectories in Theorem 3.   Let s > 0 be small and fixed.Suppose that F is a smooth function on [0, s[, F (0) = 0 and 0 < F (s) < s for all s ∈]0, s[.We define H(s) := s − F (s) and the orbit of s 0 ∈]0, s[ by H: U := {s n = H n (s 0 ) | n = 0, 1, . . .} where H n denotes n-fold composition of H.It is clear that s n tends monotonically to zero.We say that the multiplicity of the fixed point s = 0 of H is equal to m if s = 0 is a zero of multiplicity m of F (F (0) = • • • = F (m−1) (0) = 0 and F (m) (0) ̸ = 0).If F (n) (0) = 0 for each n = 0, 1, . . ., then we say that the multiplicity of s = 0 of H is ∞.Theorem 4.1 ( [12]).Let F be a smooth function on [0, s[, F (0) = 0 and 0 < F (s) < s for each s ∈]0, s[.Let H = id − F and let U be the orbit of s 0 ∈]0, s[ by H. Then dim B U is independent of the initial point s 0 and, for 1 ≤ m ≤ ∞, the following bijective correspondence holds: where m is the multiplicity of s = 0 of H (if m = ∞, then dim B U = 1).
The first part of Theorem 3.3.2can be proved in the same way as Theorem 3.3.1.Since the non-hyperbolic limit cycle is generated by a saddle-node bifurcation of limit cycles we have dim B Γ = dim B Γ = dim B Γ = 3  2 (see [37,Theorem 10] and [36,Theorem 1]).To prove the claim about Minkowski nondegeneracy; first observe that M 3/2 ( Γ) = 0 since dim B ( Γ) = 1 < 3/2 so that this part does not affect the upper and lower Minkowski content of Γ = Γ ∪ Γ; hence, it is enough to show that Γ is Minkowski nondegenerate.To see this, we observe that Γ can be partitioned into finitely many pieces Γi ; i = 1, . . ., k such that each Γi is bi-Lipschitz equivalent to [0, 1[×U by the Lipschitz flow-box Theorem [3].Note also that dim B U = 1/2 and it is Minkowski nondegenerate which implies that [0, 1[×U is also Minkowski nondegenerate; see the proof of [36,Theorem 4(b)].Finally, the finite stability of Minkowski dimension and of Minkowski nondegeneracy now complete the proof exactly as in the proof of [36,Theorem 4(b)].

Figure 1 :
Figure 1: The dynamics of X 0,λ with the curve of singularities C λ (blue) and the fast foliation F λ .A normally hyperbolic segment m λ ⊂ C λ (red) along which the slow divergence integral can be defined.

Theorem 2 . 2 (
Invariance of the slow divergence integral).Let us denote the family of vector field in (5) by Z ϵ,λ and let m λ ⊂ Σ sl λ be as in Definition 1.The following statements are true.

Figure 4 :
Figure 4: The different types of two-fold singularities with sliding: the folds in V V 1 and II 1 have the same visibility, while the folds in V I 2 and V I 3 have opposite visibility.

Figure 5 :
Figure 5: Non-generic visible-invisible two-fold singularities.The (extended) sliding vector field has a hyperbolic singularity at the two-fold points.(a) The sliding vector field points toward the two-fold singularity.(b) The sliding vector field points away from the two-fold singularity.

Figure 6 :
Figure 6: (a) The sliding vector field is defined on one side of the tangency point.(b) The sliding vector field is defined on both sides of the tangency point.(c) A crossing region around the tangency point (in this case the slow divergence integral near the tangency point is not defined).

Figure 7 :
Figure 7: A fractal sequence (s n ) n∈N near the canard cycle Γ 0 .