LIMIT CYCLES FOR REGULARIZED DISCONTINUOUS DYNAMICAL SYSTEMS WITH A HYPERPLANE OF DISCONTINUITY

. We consider an n dimensional dynamical system with discontinuous right-hand side (DRHS), whereby the vector ﬁeld changes discontinuously across a co-dimension 1 hyperplane S . We assume that this DRHS system has an asymptotically stable periodic orbit γ , not fully lying in S . In this paper, we prove that also a regularization of the given system has a unique, asymptotically stable, periodic orbit, converging to γ as the regularization parameter goes to 0.


1.
Introduction. Systems with discontinuous right-hand side (also called piecewise smooth, PWS, systems), have been actively investigated during the last 20-30 years, because of their relevance in many applications, such as in control theory, mechanical systems with dry frictions, biological models. See [1], [3] , [9] and [11], for important theoretical and modeling work. Among many ways in which one can study DRHS systems, the regularization method is very appealing, since it replaces the discontinuous system with a smooth system. The first authors to formally introduce this technique were Sotomayor and Teixeira in 1995, see [19], and recently this method has been exploited to study singularities, Filippov sliding vector fields and dynamical behavior near sliding regions of the PWS vector field. See [12], [13], [14] and [16].
Limit cycles of PWS vector fields also are of considerable interest. Several authors have studied the persistence of limit cycles for regularized planar vector fields; e.g., see [18] and [5]. However, these results rely heavily on the planar nature of the 3092 LUCA DIECI, CINZIA ELIA AND DINGHENG PI problem; in particular, they make use of the Poincaré Bendixson Theorem. Our own interest in this paper is to establish existence (and stability) of limit cycles of regularization of n-dimensional PWS vector fields (n ≥ 2), having a hyperplane of discontinuity. We will do so in the cases where the discontinuous system has a periodic orbit with sliding and/or crossing segments.
The work [2] is a precursor of our results. In that work, the authors considered the case of a regularized vector field for a discontinuous problem with a limit cycle having a sliding segment. No crossing case was considered in [2]. Moreover, even for the sliding case, our treatment is different from [2] . Rather than relying on the implicit function theorem applied to the Poincaré map, as the authors proposed in [2], we will first use Brouwer's fixed point theorem to establish the existence of a periodic orbit for the regularized problem, and then study its stability properties by exploiting the associated monodromy matrix. Moreover, we provide a unified treatment of limit cycles with sliding and/or crossing. Let x = (x 1 , x 2 , · · · , x n ) ∈ R n , and let h(x) ≡ x 1 . Define the switching manifold as S = h −1 (0), and let R + = {q ∈ R n : h(q) > 0}, R − = {q ∈ R n : h(q) < 0}. Consider the following system with discontinuous right-hand side: Here, F − and F + are C r functions, where r ≥ 1, which we assume to be well defined in R ∓ , on S, and in a neighborhood of S. Write F 0 = (F − , F + ). A smooth function φ : R → R is a transition function if φ(x) = −1 for x ≤ −1, φ(x) = 1 for x ≥ 1 and φ ′ (x) > 0 if x ∈ (−1, 1). To fix ideas, we consider the following C 1 function The φ-regularization of F 0 = (F − , F + ) is a 1-parameter family of vector fields F ǫ , connecting F − and F + , and giving the following regularized system for (1): When needed, we will use the notation φ ǫ (z) = φ( z ǫ ). The vector field F ǫ is an average of F − and F + inside the boundary layer {x ∈ R n | − ǫ < h(x) < ǫ}, while it is equal to either F − or F + outside the boundary layer.
In [18] and [6], the authors consider discontinuous planar systems. They show that if γ is a hyperbolic periodic orbit of (1) in R 2 , then, under suitable assumptions, the regularized vector field (3) has a hyperbolic limit cycle γ ǫ , converging to γ as ǫ → 0.
In this paper, we consider vector fields in R n , with a co-dimension 1 hyperplane of discontinuity, for any n ≥ 2. Under appropriate assumptions, we will prove that, if the original PWS system has an asymptotically stable periodic orbit, then so will the regularized system. We mention that this is not a trivial generalization. There are two main difficulties. The first is that for non-planar problems we do not have a Poincaré-Bendixson Theorem to help us in establishing existence of the limit cycle (cfr. with [4,5,6,18]); extensions of the Poincaré-Bendixson Theorem for systems in R n , see [17,21], require special type of systems (competitive or monotone systems), which do not fit our type of problem. The second difficulty is to establish the stability of the limit cycle of the regularized problem. We will do this by using the monodromy matrices of the discontinuous and regularized problems, and showing that the latter converges (as ǫ → 0) to the former.
The remainder of this paper is organized as follows. We give some definitions and state our main result in section 2. In section 3, we prove our main result. Conclusions are in section 4. In Appendix A, we give proofs of some technical results needed in section 3.

2.
Basic definitions and main result. In this section, we will give definitions and assumptions.
We assume that F ± are C r , r ≥ 1, in R ± and in a neighborhood of S. We denote the flow of (1) as ϕ t 0 (x) and the flow of (3) as ϕ t ǫ (x). Definition 1. A subset U of S is said to be an attractive (repulsive) sliding subset if for all x ∈ U the following occurs A point x that verifies (4) is said to be an attractive (repulsive) sliding point. If a solution of (1) slides on an attractive (repulsive) subset of S, we say that attractive (repulsive) sliding occurs along the solution.
If a solution intersects S at an attractive sliding point x then it must remain on S. However the vector field F 0 is not defined on S and a sliding vector field needs to be defined. We follow Filippov (see [9]) and for each x ∈ S that verifies the first condition in (4) we define the sliding vector field as where the value of φ * (x) in (5) is such that ∇h T F S (x) = 0. In this paper we will also make use of the following definitions.
Definition 3. Assume that a solution of (3) reaches S at an attractive sliding point x. Then x is said to be a transversal entry point.
Let γ be a periodic orbit of (1). In this paper we consider one of the following forms for γ (see [10] for planar systems). See Figure 2. i) γ has a finite number of sliding arcs on S and contains no crossing points.
Then, γ is called a sliding periodic orbit. We further exclude the case in which γ is entirely contained in S. ii) γ meets S only at a finite number of crossing points. Then, γ is called a crossing periodic orbit. iii) γ has a finite number of sliding arcs and a finite number of crossing points.
Then, γ is called a crossing and sliding periodic orbit.  As for smooth dynamical systems, stability properties of a periodic orbit γ can be studied via the eigenvalues of the monodromy matrix, the Floquet multipliers. As we will see in Section 3, to take into account the jumps in the derivatives of the solution, the monodromy matrix along γ is defined with the aid of suitable saltation matrices. These matrices are full rank in the case of crossing, but they are singular (of rank (n − 1)) in case γ has one or more sliding arcs on S. Definition 4. Let γ be a periodic orbit of (1). Let µ 1 , µ 2 , · · · , µ n be the corresponding Floquet multipliers. We say that γ is asymptotically stable if one of the multipliers is 1, say µ 1 = 1, and all other µ i 's are less than 1 in modulus.
When γ is a sliding or sliding and crossing periodic orbit, the associated monodromy matrix has one multiplier equal to 0. This witnesses that there is sliding on a co-dimension 1 region of R n . Notice that when (1) is planar, a sliding or sliding and crossing periodic orbit has a multiplier at 1 and one at 0. In this case, γ is said to be stable in finite time.
2.1. Basic assumptions. Before we state our main result, we make the following basic assumptions. H 1 Sliding subsets of S are attractive (see Definition 1 ). H 2 Entry points on S are transversal (see Definition 3). H 3 Exit points from S are first order tangential exit points (see Definition 2). H 4 For each solution of (1), only a finite number of crossings/exits/entries can occur. Our main result is the following Theorem 5. Assume that hypotheses H 1 − H 4 hold. Let γ be an asymptotically stable periodic orbit of (1) not entirely contained in S. Then, for ǫ > 0 sufficiently small, system (3) has a unique asymptotically stable limit cycle γ ǫ , and γ ǫ → γ when ǫ → 0.
Of course, Theorem 5 above holds true also when (1) is planar and γ is stable in finite time.
3. Proof of main result. In this section, we prove Theorem 5. We will treat the crossing and sliding cases separately.
3.1. Crossing periodic orbit. We consider the case in which γ meets S in just two crossing points, denote them withx 1 andx 2 . In Remark 15 we discuss the case of a finite number of crossings.
Theorem 6. Assume that (1) has an asymptotically stable periodic orbit γ with two transversal crossing points with S. Then, for ǫ sufficiently small, there exists one and only one periodic orbit γ ǫ of (3) in a neighborhood of γ. Moreover γ ǫ is asymptotically stable and lim ǫ→0 γ ǫ = γ.
We will prove Theorem 6 according to the following steps.
(1) Prove that (3) has at least one limit cycle. To do this, we will define a Poincaré map P ǫ , and use Brouwer's fixed point Theorem to show that it has a fixed point. This will give at least one limit cycle γ ǫ of (3), and we will show that γ ǫ → γ 0 when ǫ → 0. (2) Then, we will show that γ ǫ is asymptotically stable, so γ ǫ is the unique limit cycle of (3), for ǫ sufficiently small. Let us define a Poincaré map associated to γ. Without loss of generality, we will assume that the periodic solution associated to γ crosses S atx 1 coming from R + and entering in R − , and then again (at a later time) crosses S atx 2 coming from R − and entering in R + , as in Figure 3.
With ϕ t ± we denote the flows of F ± , and with ϕ t 0 (x 0 ) the solution of (1) at time t, with initial condition x 0 at time t = 0. Recalling that in the present crossing case if x 0 ∈ S, then ∇h T F + (x 0 ) and ∇h T F − (x 0 ) have the same sign, we note that ϕ t 0 (x 0 ), for t ≥ 0 sufficiently small, is the solution oḟ Let B δ (x 1 ) = {x ∈ R n : x −x 1 < δ} be the open ball centered atx 1 and of radius δ; here, and later on, the norm is always the Euclidean norm. Denote with B δ (x 1 , S) its intersection with S and with B δ (x 1 , S) its closure. Then for δ sufficiently small the Poincaré map P − (x) = ϕ t−(x) − (x), where t − (x) is the first return time to S, is well defined and smooth in x and it takes a point x in B δ (x 1 , S) into a neighborhood ofx 2 . Similarly, we can define a Poincaré map P + (x) that, due to the asymptotic stability of γ, takes a point x in a neighborhood ofx 2 into B δ (x 1 , S). Let P = P + • P − : B δ (x 1 , S) → B δ (x 1 , S) be the Poincaré map of system (1). Then P is well defined and smooth with its inverse in B δ (x 1 , S) and since γ is asymptotically stable, P satisfies P (B δ (x 1 , S)) ⊂ B δ (x 1 , S) for δ sufficiently small. Let ψ δ be the boundary of B δ (x 1 , S), then ψ δ is the intersection of the (n−1)-sphere of centerx 1 and radius δ with S. The set P (ψ δ ) is a diffeomorphic image of ψ δ . Let V be the union of all trajectories of (1) with initial point on ψ δ and endpoint on P (ψ δ ) together withB δ = B δ (x 1 , S)\P (B δ (x 1 , S)) and letV be the compact subset of R n whose boundary is V . Then all solution trajectories of (1) that intersectB δ will do so transversally, will enterV and will remain inside it. The periodic orbit γ attracts all trajectories insideV .
Stability of γ can be studied via the monodromy matrix X(T ) at γ. Let T be the period of γ and assume that at time t =t, ϕt 0 (x 1 ) =x 2 . Then, X(T ) can be written as the composition of the following matrices (e.g., see [8,15]): are so-called saltation matrices, while the fundamental matrix solutions X(t, 0) and X(t,t) satisfyẊ (t, 0) = DF − (ϕ t 0 (x 1 ))X(t, 0), X(0, 0) = I; X(t,t) = DF + (ϕ t 0 (x 1 ))X(t,t), X(0,t) = I. The two saltation matrices in (6) are nonsingular and hence X(T ) has an eigenvalue at 1 and all the other eigenvalues are less than 1 in modulus because of asymptotic stability of γ.
To prove the existence of a periodic orbit of (3) in a neighborhood of γ we employ the Poincaré map of (3). In a neighborhood ofx 1 andx 2 solutions of (3) intersect S transversally and hence we will show that the following Poincaré map, P ǫ : B δ (x 1 , S) → B δ (x 1 , S) that associates to a point in B δ (x 1 , S) its first return to B δ (x 1 , S), is well defined. The following proposition establishes the existence of at least one periodic orbit of (3).

Proposition 7.
For ǫ sufficiently small, the map P ǫ has at least one fixed point in In order the prove this proposition we will need the following Lemma.
uniformly for t in a compact interval.
Proof. Denote with ϕ t 0 and ϕ t ǫ the flows of (1) and (3) respectively. Together with S, consider also the hyperplanes In what follows, for x 0 ∈ B δ (x 1 ), we want to estimate the distance between ϕ t 0 (x 0 ) and ϕ t ǫ (x 0 ) at their intersection points with S, S ǫ and S −ǫ . Without loss of generality assume that x 0 ∈ B δ (x 1 , S). Then for δ and ǫ sufficiently small ǫ (x 0 ) and show that it goes to zero when ǫ → 0. To fix ideas, assume t ǫ , then the following inequality holds where the last inequality follows from Gronwall's Lemma. Moreover using In particular lim ǫ→0 t 1 = 0 and, in a similar way, lim ǫ→0 t ǫ 1 = 0. This, together with the bound for ϕ t . But it will eventually change direction and it will meet S −ǫ again in a neighborhood ofx 2 . Let t 2 be such that ) be the local Lipschitz constant for F − , then the following bound holds . In a similar way we can show that ϕ t 0 (x 0 ) − ϕ tǫ ǫ (x 0 ) → 0 up to returning in a neighborhood ofx 1 . This proves the Lemma.
As a consequence of Lemma 8, this Corollary holds.

LUCA DIECI, CINZIA ELIA AND DINGHENG PI
Proof. Proposition 7 ensures existence of a fixed point of P ǫ in B δ (x 1 , S). We need to exclude the possibility that the fixed point is an equilibrium point for F ǫ . Assume by contradiction that there existsx contradiction. An invariant region V ǫ for F ǫ can be built by considering the union of all trajectories with initial points on ψ δ and endpoints on P ǫ (ψ δ ) together with LetV ǫ be the compact subset of R n with boundary V ǫ . Then all trajectories of (1) that crossB ǫ δ will do so transversally, will enterV ǫ and will remain inside it. See Figure 4. We will show that (3) has a unique periodic orbit γ ǫ inV ǫ and γ ǫ attracts all the solutions insideV ǫ .
Next, for each ǫ, select a fixed point of P ǫ and denote it with , t ∈ R}, be the corresponding periodic orbit. In order to prove Theorem 6 we will show Main steps ii) for ǫ sufficiently small, γ ǫ is asymptotically stable and this allows us to exclude that (3) has a family of periodic orbits that converges to γ.
Let x ∈ B δ (x 1 , S) and let ϕ t 0 (x) be the solution of (1) with initial condition x. Denote with x 2 its intersection with S in a neighborhood ofx 2 and let The following Lemma holds.
Lemma 11. Denote with X(t, 0, x) and X ǫ (t, 0, x) respectively the fundamental matrix solution of (1) along the solution ϕ t 0 (x) and the fundamental matrix solution of (3) In what follows we will omit the explicit dependence of X(t, 0, x) and X ǫ (t, 0, where I is the identity matrix and DF ǫ (x) is the Jacobian of F ǫ and it is given by Similarly to what we have done in the proof of Lemma 8, we consider the intersections of ϕ t ǫ (x) with S, S ǫ and S −ǫ and use these intersection points to rewrite X ǫ (T ǫ (x), 0) as product of transition matrices. It will be handy to take the initial condition on S ǫ instead of S. So, let We evaluate the monodromy matrix X ǫ (T ǫ (x), 0) along the shifted solution ϕ t ǫ (x ǫ 0 ). Notice that in a whole neighborhood of x the following inequality is satisfied ∇h T F ǫ (x) < 0 so that ϕ t (x ǫ 0 ) intersects S and then S −ǫ at two isolated points: . Then the trajectory enters the set {x ∈ R n |h(x) < −ǫ} and remains 3100 LUCA DIECI, CINZIA ELIA AND DINGHENG PI in this set until, at time t ǫ 3 , it meets S −ǫ again in x ǫ 3 . In a neighborhood of x ǫ 3 the following inequality is satisfied With these notations, we can rewrite X ǫ as follows We want to show that all the factors making up X ǫ in (10) converge to the corresponding factors in X(t, 0, x) (see (6) as well).
We first look at the factor X ǫ (t ǫ 2 , 0), rewritten as , as ǫ → 0, will exist if the limits of the other two factors do. We have The first integral goes to 0 as ǫ → 0, since t ǫ 1 → 0 and the integrand is bounded. The second integral is dealt with by noticing that φ ′ ( h ǫ ) is positive and thus we can consider the change of variable t → x 1 , along with 1 2ǫ 0 ǫ φ ′ ( x1 ǫ )dx 1 = − 1 2 , and the mean value Theorem for integrals to obtain where the entries of the matrix on the right-hand side are evaluated at some valuē t, 0 ≤t < t ǫ 1 , possibly different for each matrix entry. Since all objects on the righthand side are smooth in [0, t ǫ 1 ), t ǫ 1 → 0, F ǫ at t = 0 is F + , and X ǫ (t ǫ 1 , 0) = I + O(t ǫ 1 ), by expanding around t = 0 (or x 1 = ǫ), we conclude that the limit as ǫ → 0 of X ǫ (t ǫ 1 , 0) exists and it is: For X ǫ (t ǫ 2 , t ǫ 1 ), we have Again, the first integral goes to 0 as ǫ → 0, since t ǫ 2 , and t ǫ 1 → 0 and the integrand is bounded. For the second integral, since φ ′ ( h ǫ ) is positive, we can consider the change of variable t → x 1 , along with 1 2ǫ −ǫ 0 φ ′ ( x1 ǫ )dx 1 = − 1 2 , and the mean value Theorem for integrals to obtain 1 2ǫ where the entries of the matrix on the right-hand side are evaluated at some valuē t, t ǫ 1 <t ≤ t ǫ 2 , possibly different for each matrix entry. Since all objects on the righthand side are smooth in (t ǫ 1 , t ǫ 2 ], t ǫ 1,2 → 0, F ǫ at t = t ǫ 2 is F − , then by expanding around t = t ǫ 2 (or x 1 = −ǫ), we conclude that the limit as ǫ → 0 of X ǫ (t ǫ 2 , t ǫ 1 ) exists, call it M , and it is defined by the relation: Note that the matrix defining M is invertible, and we have Finally, using (11) and (13), we can then conclude that lim ǫ→0 X ǫ (t ǫ 2 , 0) exists, and it is given by Doing the algebra, this simplifies to lim ǫ→0 X ǫ (t ǫ 2 , 0) = I + Similarly to (15), we can prove lim ǫ→0 X ǫ (t ǫ 5 , t ǫ This, together with the following limits: x ǫ 3 , x ǫ 4 , x ǫ 5 → x 2 , and t ǫ 3 , t ǫ 4 , t ǫ 5 → t 2 , and T ǫ (x) → T (x), imply: lim ǫ→0 X ǫ (t ǫ 3 , t ǫ 2 ) = X(t 2 , 0), and lim ǫ→0 X ǫ (T ǫ (x), 0) = X(T (x), 0).
Lemma 11 implies in particular that X ǫ (T ǫ (x), 0, x) is bounded for all ǫ. At first, this might come as a surprise, since the derivative with respect to x 1 of F ǫ is multiplied by the factor 1 ǫ in the boundary layer, (see also the first column of DF ǫ in (9)). However, the boundary layer and the time spent inside the boundary layer go to zero linearly with ǫ, so that the derivative of the vector field in the boundary layer remains finite in the limit. Moreover, boundedness of X ǫ (T ǫ (x), 0, x) implies Lipschitzianity of the solution for each ǫ > 0, as the following corollary states.
Corollary 12. There exists β > 0 such that for each t ≥ 0 and for each ǫ > 0 The proof follows from the mean value Theorem for integrals applied to ϕ t ǫ (x) − ϕ t ǫ (y) = 1 0 X ǫ (t, 0, sx + (1 − s)y)(x − y)ds . Corollary 13. There exists α > 0 such that for each ǫ > 0 Proof. The result follows from the equality where x ∈ B δ (x 1 , S), E = (e 2 , . . . , e n ), e j is the j-th canonical vector and T ǫ (x) is the first return time of ϕ t ǫ (x) to B δ (x 1 , S). Then DP ǫ (x) ≤ X ǫ (T ǫ (x), 0, x) and the norm on the right is bounded as ǫ → 0 because of Lemma 11.
The following Lemma is the first part of Main step i) in the proof of Theorem 6.
Proof. Let us denote with x k ǫ andx k 1 the k-th component of x ǫ andx 1 , k ≥ 2. (The first components are 0.) Let and let x ǫi and x ǫs be two sequences such that lim ǫi→0 x 2 ǫi = x 2 and lim ǫs→0 x 2 ǫs = x 2 . From x ǫi and x ǫs we can extract convergent subsequences which, with abuse of notation, we sill denote as x ǫi and x ǫs . Let Notice that lim ǫi→0 P ǫi (x ǫi ) = lim ǫi→0 x ǫi = x, and lim ǫs→0 P ǫs (x ǫs ) = lim ǫs→0 x ǫs = x. Then The three terms to the right of the inequality go to zero for ǫ i → 0. Indeed we have P ǫi (x) → P (x), P ǫi (x ǫi ) → x and P ǫi (x) − P ǫi (x ǫi ) ≤ α x − x ǫi because of Corollary 13. It then follows that x = P (x), so that x =x 1 . Similarly, x =x 1 . As a consequence of this reasoning, lim ǫ→0 x 2 ǫ =x 2 1 . To show convergence of x k ǫ , k = 3, . . . , n, we use reasonings analogous to the ones used for (16) together with the reasonings in this proof. Lemma 14 and Corollary 12 imply the following inequality In particular, (17) and Lemma 8 insure γ ǫ → γ and this proves Main step i) above. Lemma 11 and Lemma 14 together with continuity of X ǫ with respect to x ∈ B δ (x 1 , S), imply that for all µ > 0 there exists ǫ µ sufficiently small so that for ǫ < ǫ µ the following inequality is satisfied As a consequence, for ǫ sufficiently small, X ǫ (T ǫ , 0, x ǫ ) has all eigenvalues less than Remark 15. The proofs of the results in this section do not change in case of a periodic orbit γ with a finite number of transversal crossings. The Poincaré map P can be defined in the neighborhood of one of the transversal crossing points and viewed as composition of two or more smooth maps and it retains its smoothness. The attractivity of γ insures contractivity of P . Pointwise convergence of P ǫ to P can be proved as in Corollary 9 and all related results can as well, in particular obtaining contractivity of P ǫ for ǫ sufficiently small. The monodromy matrix is given by the composition of saltation matrices of full rank with fundamental matrix solutions and the proof of convergence for the monodromy matrix remains essentially the same as we have given.
Theorem 6 together with Remark 15 suffice to prove Theorem 5 for a crossing periodic orbit γ with a finite number of transversal crossings.

3.2.
Sliding periodic orbit. Let γ be the periodic orbit of (1). We will first assume that γ is the union of two arcs, one in R − , or R + , and one in S. In Remark 26 we will discuss how to generalize the proof to the case of an orbit with a finite number of sliding arcs and a finite number of crossing transversal points.
Theorem 16. Assume that system (1) has an asymptotically stable sliding periodic orbit γ given by two arcs: one arc in the region R + or R − and one arc on S. Assume moreover that the entry point in S is a transversal entry point and that the exit point from S is a first order tangential exit point. Then, for ǫ sufficiently small, there exists one and only one periodic orbit γ ǫ of problem (3) in a neighborhood of γ. Moreover γ ǫ is asymptotically stable and lim ǫ→0 γ ǫ = γ.
Without loss of generality, we assume that γ has an arc in R − , and the other on S. Further, we let x in and x out be respectively the entry point in S, from R − , and the exit point from S, onto R − , of γ. The point x out is a first order tangential exit point, then in particular ∇h(x out ) T F − (x out ) = 0. Assume that the point x in is a transversal entry point. See Figure 5; although the time arrow is for convenience only, we will work with this figure in mind when defining the Poincaré map. S X in X out π out Σ X 1 Figure 5. Sliding periodic orbit We first define a Poincaré map for the nonsmooth system in a neighborhood of γ. To do so, letx 1 ∈ γ ∩ R − and take a cross section Σ to γ atx 1 . Let B δ (x 1 , Σ) = B δ (x 1 ) ∩ Σ. Then, for δ sufficiently small, all solutions with initial condition in B δ (x 1 , Σ) reach S in a neighborhood of x in and they will start sliding on S since x in is a transversal entry point. Let g(x) = ∇h T F − (x) and consider the set Π out = {x ∈ B η (x out )| g(x) = 0}, with η > 0 and small; note that this is a small neighborhood of x out . Then ∇g T F − (x out ) < 0 (see Definition 2) implies that all solutions that slide on S in a neighborhood of γ will cross Π out transversally and enter R − . Then they will reach Σ again. We define the map P : 0 (x), with t(x) first return time to Σ. P is given by the composition of three smooth maps: where with ϕ t S we denote the flow of the sliding Filippov vector field (5) and t S (x) is the time at which ϕ t S (x) meets Π out ; Remark 17. The Poincaré map P defined in this way is smooth. The cross section in R − allows us to consider only solutions of (1) with orbits in R − ∪ S. [We remark that taking any section along the sliding arc (for example Π out ), forces us to consider also orbits with an arc in R + and this has two drawbacks: i) the corresponding Poincaré map is continuous but not smooth (it is defined in a different way for points in R − or in R + ); ii) the orbits of (1) might cross the section more than once (if we consider the section Π out for example, this happens if ∇g T F − (x) < 0 while ∇g T F + (x) > 0) and a definition of the Poincaré map via the first return time to the section is not possible. Also, notice that considering the cross section S ∩ B δ (x in ) would remove the concerns i) and ii) above for the non-smooth problem. However, a section that lies on S cannot be used for the regularized problem because solutions of (3) might never reach it.] P is a contraction so that P (B δ (x 1 , Σ)) is a proper subset of B δ (x 1 , Σ). We study the stability of γ using the monodromy matrix of (1) along γ. Let t in be such that ϕ tin 0 (x 1 ) = x in , t out be such that ϕ tout 0 (x 1 ) = x out , and T be such that ϕ T 0 (x 1 ) =x 1 . Then the monodromy matrix X(T, 0) along γ is given by the following expression (e.g., see [8]) , and the three fundamental matrices in (18) solve the following Cauchy problems: The matrix S −S (x in ) has rank (n − 1) and hence X(T, 0) has an eigenvalue at 1, one at 0, and, due to the asymptotic stability of γ, all the other eigenvalues are less than 1 in modulus. The eigenvalue at 0 is a direct consequence of the fact that solution trajectories reach S in finite time and then slide on S for some time.
To define the Poincaré map, the next result is needed.
uniformly for t in a compact interval.
Proof. The proof of this Lemma is in Appendix A, and it relies on a singular perturbation analysis based on [20, pp. 249-260].
We now define the Poincaré map P ǫ for (3) as the map P ǫ : B δ (x 1 , Σ) → Σ so that for every point x ∈ B δ (x 1 , Σ), P ǫ (x) is the first return point of ϕ tǫ(x) ǫ (x) to Σ. Lemma 18 insures that P ǫ is well defined and it implies the following Proposition.
Propositions 20 and 21 below will allow us to view the Poincaré map as the composition of three maps, similarly to what we did for P . This is desirable since it will lead to a decomposition of the fundamental matrix solution of (3) into four different factors, as for the discontinuous case (see (18)).

Proposition 20.
For ǫ sufficiently small, orbits of (3) corresponding to solutions with initial conditions in a neighborhood of γ must intersect Π out transversally.
and S + −ǫ , such that for all x ∈ S ± −ǫ , ∇h T F ǫ (x) = ∇h T F − (x) ≷ 0. Lemma 18 implies that an orbit Γ ǫ of (3) in a neighborhood of γ must intersect S − −ǫ in a point x ǫ out in a neighborhood of x out . Moreover, since ∇h T F ǫ (x ǫ out ) = ∇h T F − (x ǫ out ) < 0, x ǫ out must be isolated. Uniform convergence of solutions of (3) to solutions of (1) implies that there are points of Γ ǫ in a neighborhood of x out that satisfy ∇h T F − (x) > 0, while ∇h T F − (x ǫ out ) < 0. Continuity of solutions with respect to x imply that there must be a pointx ǫ of Γ ǫ so that g(x ǫ ) = ∇h T F − (x ǫ ) = 0 and ∇g T F ǫ (x ǫ ) < 0. The statement of the proposition follows.
Proposition 21. Solution trajectories of (3) corresponding to solutions with initial conditions in B δ (x 1 , Σ) must intersect Π out only in one point, before returning to Σ.
Proof. The proof uses some of the tools needed for the proof of Lemma 18 and hence is included in Appendix A.
Remark 22. The condition that x ∈ B δ (x 1 , Σ) is essential in Proposition 21. It guarantees that the corresponding solutions of (3) can not cross the boundary layer to enter {x ∈ R n |h(x) > ǫ}. Without this property, the statement of the proposition might be false. Consider for example the case ∇g T F + (x out ) > 0 and take an initial condition x ∈ Π out ∩ S ǫ .
The following is a consequence of Brouwer fixed point Theorem and of Lemma 19.
Proposition 23. For ǫ sufficiently small, P ǫ has at least a fixed point in B δ (x 1 , Σ).
Proof. See the proof of Proposition 7. Moreover, (3) cannot have equilibria in a neighborhood of γ. Indeed outside the boundary layer F ǫ = F ± , while if there is an equilibriumx inside the boundary layer, then F ǫ (x) = 0 and φ(x) is such that (5) and hence F S (x) = F ǫ (x) = 0, a contradiction. Then, to each fixed point of P ǫ , there corresponds a periodic orbit of (3).
For each ǫ, we select a fixed point of P ǫ and we denote it as x ǫ . Let γ ǫ be the corresponding periodic orbit. What follows mimics the reasonings employed for the case of a crossing periodic orbit. We will show Main steps i) as ǫ → 0, x ǫ →x 1 , which in turn will imply γ ǫ → γ; ii) for ǫ sufficiently small, γ ǫ is asymptotically stable so that it is the unique periodic orbit of (3) in a neighborhood of γ.
Let x ∈ B δ (x 1 , Σ) and let ϕ t 0 (x) be the corresponding solution of (1). Let x in = ϕt in (x) be its intersection point with S and let ϕt out (x) =x out ∈ Π out . At time T (x) the solution reaches Σ at a point different from x, unless x =x 1 .
The following Lemma holds.
Lemma 24. Let X(t, 0, x) be the fundamental matrix solution of (1) along ϕ t 0 (x) and X ǫ (t, 0, x) be the fundamental matrix solution of (3) along ϕ t ǫ (x). Then lim where T ǫ (x) denotes the first return time of ϕ t ǫ (x) to Σ.
Proof. Below, we omit the dependence of X and X ǫ upon x. We will adopt the notation used in Appendix A. Together with S, consider the hyperplane S −ǫ = {x ∈ R n |h(x) = −ǫ}. For ǫ sufficiently small, uniform convergence of ϕ t ǫ (x) to ϕ t 0 (x) implies that ϕ t ǫ (x) meets S −ǫ in two isolated points: x ǫ in in a neighborhood ofx in and x ǫ out in a neighborhood ofx out . Let t ǫ in be such that ϕ t ǫ in ǫ (x) = x ǫ in . Then lim ǫ→0 x ǫ in =x in and lim ǫ→0 t ǫ in = t in .
For the second piece, we write In what follows we reason similarly to the proof of Lemma 11. The fundamental matrix solution X ǫ must satisfy ). When we take the limit as ǫ → 0, then ϕ and this discontinuity is reflected also in the limit of the fundamental matrix solution. Indeed, if L = lim ǫ→0 X ǫ (t ǫ in + t µ , t ǫ in ) exists, then it must satisfy Now, the first integral in (20) goes to zero as ǫ → 0 since t µ = ǫτ µ → 0 and the integrand is bounded.
For the second integral, inside the interval (t ǫ in , t ǫ in + t µ ), we can write where R ǫ is bounded, and R ǫ → 0 as t → t ǫ in . Using this in the integral above, since φ ′ ǫ > 0, we get 1 2ǫ dt, andĒ is the error matrix whose components are each computed at possibly different values oft ∈ (t ǫ in , t ǫ in + t µ ). Now, for ǫ → 0,Ē → 0 and x ǫ in →x 1 , and t ǫ in →t in from the right. Thus, for ǫ → 0, the second integral gives and so would have and since (21) holds, we must have that ∇h T F−(xin) . Thus, finally we get lim For the third transition matrix in (19), we have = (1 − ϕ * (y(t))DF − (ϕ * (y(t)) + ϕ * (y(t))DF + (ϕ * (y(t)) = DF S (t).
It follows that lim ǫ→0 X(t out ǫ , t ǫ in + t µ ) = X(t out ,t in ). Finally, the convergence of the last transition matrix in (19), X ǫ (T ǫ (x), t ǫ out ), to the corresponding term of X(T (x), 0) (see also (18)), follows from t ǫ out →t out and x ǫ out →x out . As in Section 3.1, Lemma 24 implies that ϕ t ǫ and P ǫ are Lipschitz for all ǫ. Then the following result follows.
Proof. Let us denote with x k ǫ the k-th component of x ǫ . Let lim inf ǫ→0 and let x ǫi and x ǫs be two sequences such that lim ǫi→0 x 1 ǫi = x 1 and lim ǫs→0 x 1 ǫs = x 1 . From x ǫi and x ǫs we can extract convergent subsequences which we sill denote as x ǫi and x ǫs . Let lim As in Lemma 14, using the fact that P ǫ is Lipschitz for all ǫ, we obtain that P (x) = x and P (x) = x, so that x = x =x 1 . This proves convergence of the first component of x ǫ to the first component ofx 1 . The proof for the other components is done in a similar way.
Lemma 24 and Lemma 25 together with continuity of X ǫ with respect to x implies that X ǫ (T ǫ , 0, x ǫ ) has all eigenvalues less than 1 and one equal to 1 (since γ ǫ is periodic). This implies that γ ǫ is asymptotically stable and hence isolated, and Theorem 16 follows.
Remark 26. In case there are multiple sliding segments, and possibly also crossing, combining together the results of Theorems 16 and 6, then Theorem 5 holds for these scenarios. It should be appreciated that, see [8,Theorem 2.8], for an asymptotically stable periodic orbit of (1) with multiple sliding segments on the plane S, the multipliers are still 1,0, all other multipliers being less than 1 in absolute value and different from zero. 4. Conclusions. In this paper we have considered n dimensional DRHS systems with a discontinuous co-dimension one plane S separating two regions R − and R + . We have further assumed that this discontinuous system has an asymptotically stable periodic orbit γ. Our main result shows that if γ consists of arcs on S and/or on R − and R + , but does not lie entirely on S, then a regularization of the discontinuous system also has a unique asymptotically stable limit cycle, converging to γ as the regularization parameter goes to 0. The case of γ lying entirely on S remains to be considered.
Finally, we stress that, our switching manifold is a codimension one plane. The case of higher co-dimension switching manifold (say, the intersection of two planes) is considerably more complex (e.g., see [7]), and remains to be considered as well.
Appendix A.
Proof of Lemma 18. Let x ∈ B δ (x 1 ) and denote with ϕ t 0 (x) the solution of (1) and with ϕ t ǫ (x) the solution of (3). Since in R − , Then both ϕ t ǫ (x) and ϕ t 0 (x) enter the boundary layerS = {x ∈ R n s.t. − ǫ < h(x) < ǫ}. In particular, there existx in ∈ S andt in > 0 such that ϕt in 0 (x) =x in and lim ǫ→0 x ǫ in =x in . For t >t in , ϕ t 0 (x) starts sliding on S until it meets Π out at timê t out : ϕt out 0 (x) =x out ∈ S ∩ Π out . In order to show uniform convergence of ϕ t ǫ to ϕ t 0 , we need to study the limiting behavior of ϕ t ǫ insideS and we do this via singular perturbation theory. Let e 1 = (1, 0, 0, . . .) T , then h(x) = e T 1 x = x 1 . Let y = E T x with E = (e 2 , . . . , e n ) ∈ R n×(n−1) , where e i is the i-th canonical vector of R n . When x is inside the boundary layer the function φ = φ( x1 ǫ ) is strictly monotone, so that x 1 in [−ǫ, ǫ] can be expressed as a function of φ in [−1, 1]. Then we can consider the new variables (φ, y) and split (3) into fast and slow motion as follows with z = x1 ǫ and with initial conditions φ(0) = φ 0 and y(0) = y 0 . The transformation is done strictly inside the boundary layer, hence the initial condition for the fast variable φ satisfies −1 < φ 0 < 1, while the initial condition for the slow variable y 0 is arbitrarily close to E T x ǫ in and it converges to E Tx in for ǫ → 0. Since φ(z) is monotone increasing in (−1, 1), then z can be rewritten in function of φ, so φ ′ (z) is a function of φ as φ ′ (z) = g(φ). Denote the solution of (23) as (φ ǫ (t), y ǫ (t)). If we set ǫ = 0 in (23), using the fact that x 1 → 0 (while −1 < z < 1), we obtain the reduced problem with initial conditions φ(0) = φ 0 , y(0) = E Tx in , and with g(φ) = φ ′ (z). The algebraic equation in (24) has a unique solution in (−1, 1) for each y and this is given by the following smooth function of y: φ * (y) = e T 1 (F−+F+) e T (φ * (y(t)), y(t)) (while lim ǫ→0 x 1 (t) = 0) uniformly in time in [t in +η,t out ] wheret out is such that φ * (y(t out )) = φ * (E Tx out ) = −1 and η > 0. The uniform convergence is not immediate to verify since we cannot use continuity of solutions with respect to the parameter ǫ. We consider the fast time τ = t ǫ and the derivative with respect to τ . We obtain the following fast system dφ dτ with initial condition φ(0) = φ 0 , y(0) = y 0 . Notice that if (φ ǫ (t), y ǫ (t)) solves (23), then (φ ǫ (ǫτ ), y ǫ (ǫτ )) solves (25). If we set ǫ = 0 we have the reduced fast system with initial condition φ(0) = φ 0 and y(0) = E Tx in and with y as parameter. We denote the solution of the reduced fast system as (φ(τ, y(0)), y(0)). Notice that for φ = 1, −1, g(φ) = φ ′ (z) = 0.
Proof. Let y be fixed and rewrite the first of (26) as so that dφ −aφ+b = φ ′ (z(τ ))dτ . The statement follows from the mean value Theorem for integrals upon noticing that φ ′ (z(τ )) is strictly positive in z ∈ (−1, 1) and it is zero only for z = 1, −1.
The proof of Lemma 27 applies also to obtain the following Corollary.
Then F ǫ (x 1 ) is arbitrarily close to F S (0, E T x 2 ), where with F S we have denoted the sliding vector field on S. But at the exit point (0, E T x 2 ) it must be ∇g T F S (E T x 2 ) < 0 and this is in contradiction with (30).