IMA Journal of Applied Mathematics (2017) 82, 561–578
doi:10.1093/imamat/hxx003
Advance Access Publication on 27 February 2017

Birth of limit cycles from a 3D triangular center of a piecewise smooth vector field

Tiago Carvalho
∗

Departamento de Matemática, Faculdade de Ciências, UNESP, Av. Eng. Luiz Edmundo Carrijo Coube
14-01, CEP 17033-360, Bauru, SP, Brazil.

∗Corresponding author: tcarvalho@fc.unesp.br

Rodrigo D. Euzébio

Departamento de Matemática, Universidade Federal de Goiás, IME, CEP 74001-970,
Caixa Postal 131, Goiânia, Goiás, Brazil

email: euzebio@ime.unicamp.br

Marco Antonto Teixeira

Departamento de Matemática, IMECC–UNICAMP, CEP 13083–970, Campinas, SP, Brazil
email: teixeira@ime.unicamp.br

and

Durval José Tonon

Departamento de Matemática, Universidade Federal de Goiás, IME, CEP 74001-970,
Caixa Postal 131, Goiânia, Goiás, Brazil

email: djtonon@ufg.br

[Received on 22 December 2015; revised on 18 October 2016; accepted on 27 January 2017]

We consider a piecewise smooth vector field in R
3, where the switching set is on an algebraic variety

expressed as the zero of a Morse function. We depart from a model described by piecewise constant vector
fields with a non-usual center that is constant on the sliding region. Given a positive integer k, we produce
suitable nonlinear small perturbations of the previous model and we obtain piecewise smooth vector fields
having exactly k hyperbolic limit cycles instead of the center. Moreover, we also obtain suitable nonlinear
small perturbations of the first model and piecewise smooth vector fields having a unique limit cycle of
multiplicity k instead of the center. As consequence, the initial model has codimension infinity. Some
aspects of asymptotical stability of such system are also addressed in this article.

Keywords: periodic solutions; limit cycles; bifurcation; piecewise smooth vector fields

1. Introduction

1.1. Setting the problem

Going a step beyond the classical theory of topological dynamical systems, the theory of the so called
piecewise smooth vector fields (PSVFs, for short) has been used to model a large number of phenomena.
Indeed, both applications and theory of such a brand new area has been widely explored recently.
Applications of piecewise smooth vector fields include but not limited to physics, control systems,
electrical engineering and problems involving impact, among others. Some landmarks we can quote
of such a theory are the work of Teixeira concerning manifolds with boundary (see Teixeira, 1977)
and the book of Filippov, see Filippov (1988). One can also see the works of Broucke et al. (2001);

© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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562 T. CARVALHO ET AL.

Fig. 1. The center contained in the switching manifold.

Chillingworth (2002); di Bernardo et al. (2008); Jacquemard et al. (2012); Kozlova (1984); Luo (2006);
Teixeira (1990, 2008).

Basically, the difference between the classical theory concerning smooth dynamics and our approach
lies in the fact that the late assume the existence of a codimension one variety Σ separating the phase
portrait in two or more regions, being defined in each region a distinct vector field.

In Filippov (1988), Filippov provided a convention to induce a vector field even on Σ , which is
usually called ‘Filippov (or sliding)’ vector field. Generally, the first articles addressing PSVFs dealt
with a variety Σ generated by the preimage of a regular value of some smooth function. Meanwhile such
a theory is enhanced by a huge range of applications, even when the discontinuity set is an algebraic
variety. Models of PSVFs having Σ in such a shape can be found in books of control or in articles taking
into account relay systems (see, for instance, the book of Barbashin, 1970).

In this article, we analyse some qualitative aspects of a PSVF in R
3 whose flow becomes confined

on the ‘switching manifold’ Σ after a finite time. Indeed, while in dimension 2 the dynamics on Σ does
not present any tricky behaviour, in dimension 3 one may observe some very interesting phenomena. In
the present work, the switching manifold is formed by 12 parts (semi-planes), which give rise to eight
different vector fields, defined in eight octants of R

3.
In this scenario, we depart from a piecewise constant model with a non-usual center (see Fig. 1)

that is constant on the sliding regions. So, we perform perturbations in order to study the existence of
minimal sets as well as aspects concerning asymptotical and structural stability. Related articles that
treat about the asymptotical stability in the context of PSVF are Teixeira (1990) and Jacquemard et al.
(2013), for example.

We should stress that while we present an abstract model in order to study some aspects of PSVFs
having a tricky switching manifold, the model we present is inspired in a formalism that is known in
the literature as ‘complementarity problem’ (CP) (see Stewart, 2011 and references therein), which has
a strongly connection with PSVFs and its importance in applications has been largely explored. More
precisely, our approach in this work has some resemblance with the setting presented in Bernard and el
Kharroubi (1991) and briefly discussed in Stewart (2011) (see pages 161–166). Observe that comple-
mentarity problems as DVIs (differential variational inequalities) or DCIs (differential complementarity
problem) seems to be very important in describing physical situations presenting mechanical impacts or
Coulomb friction. The connection between both, piecewise smooth’s formalism and complementarity

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 563

has been reported in the books of Acary and Brogliato Acary & Brogliato (2008) (Chapters 1 and 2), di
Bernardo et al. (2008) (Section 2.3.1) and Stewart (2011) (Chapter 1).

1.2. Statements of the main results

Consider the PSVF Z0 defined in R
3 by

Z0(x, y, z) = Xi(x, y, z), if (x, y, z) ∈ Ri, i = 1, 2, . . . , 8, (1.1)

where each region Ri, i = 1, 2, . . . , 8 is determined according to the following table:

R1 R2 R3 R4 R5 R6 R7 R8

sign of x ≥ 0 ≤ 0 ≤ 0 ≥ 0 ≥ 0 ≤ 0 ≤ 0 ≥ 0
sign of y ≥ 0 ≥ 0 ≤ 0 ≤ 0 ≥ 0 ≥ 0 ≤ 0 ≤ 0
sign of z ≥ 0 ≥ 0 ≥ 0 ≥ 0 ≤ 0 ≤ 0 ≤ 0 ≤ 0

By considering a coherent arrangement of the regions listed previously, we get the switching region
Σ = ⋃

{j,k}∈I Σj,k , where Σj,k is the boundary separating two adjacent regions Rj and Rk , for j, k ∈ I =
{1, 2, . . . , 8}. We call Ω r the space of vector fields Z : R

3 → R
3 having the form of (1.1) and observe

that on Σ we have defined two or more vector fields, in this case the classical Filippov’s convention
does not apply. Two approaches taking into account piecewise smooth systems having more than two
vector fields defined on can be found in Dieci & Lopez (2011) and Llibre (2015).

In our model, we consider the following choice of constant (linear) vector fields Xi, i = 1, 2, . . . , 8:

X1(x, y, z) = (−1, −1, −1),

X2(x, y, z) = (1, −1, 3),

X3(x, y, z) = (1, −1, 1),

X4(x, y, z) = (3, 1, −1),

X5(x, y, z) = (−1, 3, 1),

X6(x, y, z) = (−1, 1, 1),

X7(x, y, z) = (1, 1, 1),

X8(x, y, z) = (1, 1, −1).

(1.2)

In what follows, we write

S = Σ1,2 ∪ Σ1,4 ∪ Σ1,5.

Latter on, we show that S is the sliding region of Σ and it is a global attractor of System (1.1)
(see Lemmas A and B, respectively).

Now we state the main results of the article.

Theorem A. Let Z0 ∈ Ω r be given by (1.1). For each integer k ≥ 0, there exists an one-parameter family
Zε ∈ Ω r satisfying:

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564 T. CARVALHO ET AL.

(a) Zε → Z0 when ε → 0;

(b) Restricted to S, Zε has exactly k hyperbolic limit cycles in a neighbourhood of the origin. The same
holds for k = ∞ and,

(c) Z0 possesses the invariant subset S ⊂ Σ as a global attractor.

In order to announce the next result, let us roughly define the classical notion of codimension of vector
fields.

Definition 1 Consider Θ(W) the set of all small perturbations of a vector field W , defined on a compact
set K . We say that W has codimension k if it appears exactly k distinct topological types of vector fields
in Θ(W).

An immediate consequence of Theorem A is:

Corollary B. The PSVF Z0 ∈ Ω r , given by (1.1), has infinite codimension.

Moreover, the next theorem brightens the scenario around Z0.

In the next result, consider Ω r(K) the space of the PSVF, with domain restricted to a compact subset
K ⊂ R

3 around the origin.

Theorem C. Let Z0 ∈ Ω r(K) be given by (1.1). For each neighbourhood V of Z0 and for each integer
k > 0 there exists Zk

ε ∈ V having codimension k.

Next result also provides informations about the asymptotical stability of the origin.

Theorem D. There exist Z0
ε ∈ Ω r , an one-parameter family of perturbations of the PSVF Z0, for which

the origin is either asymptotically stable or unstable, depending on the sign of ε.

Theorems A and C and Corollary B are proved in Section 5. Theorem D is proved in Section 3.
The article is organized as follows. In Section 2, we introduce the terminology, some definitions and

the basic theory about PSVFs. In Section 3, we study the main properties of System Z0. In Section 4,
suitable perturbations of this system are considered. Moreover, the birth of limit cycles are explicitly
exhibited. In Section 5, we prove the main results of the article. Finally, in Section 6, we present a brief
conclusion.

2. Preliminaries

Inspired in the models presented in Bernard and el Kharroubi (1991) (see also Stewart, 2011) we first
consider the particular PSVF Z0 given by (1.1). At the sequel, we formalize some theory about this kind
of vector field.

Denote πi : R
3 −→ R the projection onto the i−th coordinate, i = 1, 2, 3 and observe that

Σ =
⋃

i=1,2,3

π−1
i (0).

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 565

We remark that once Σ is not regular, the coordinate planes Ai = π−1
i (0) itself, i = 1, 2, 3, do not play

any special role in our approach. However, they allows us to define the function h : R
3 \ F1 −→ R

given by

h = χA1π1 + χA2π2 + χA3π3,

where χAi is the characteristic function associated to the set Ai, i = 1, 2, 3, and F1 is given by

F1 =
⋃

i,j∈{1,2,3}

[
π−1

i (0) ∩ π−1
j (0)

]
.

Note that the set of three coordinate axes F1 is not a codimension one variety of R
3. So, the Filippov

convention does not apply on F1 ⊂ Σ directly.
Moreover, let Xk : R

3 −→ R
3 be Cr-vector fields, k = 1, . . . , 8 and designate by Xr the space

of Cr-vector fields on R
3 endowed with the Cr-topology with r = ∞ or r ≥ 1 large enough for our

purposes.
In order to classify the points on Σ , for each vector field Xi, i = 1, . . . , 8, we denote Xih(p) =

〈Xi(p), ∇h(p)〉 and (Xn
i h(p) = 〈Xi(p), ∇Xn−1

i h(p)〉, the Lie’s derivatives, where 〈·, ·〉 denote the canonical
inner product. We may consider Ω r endowed with the product topology and denote any element in Ω r by
Z = (X1, . . . , X8), which we will accept to be multivalued in points of Σ . The basic results of differential
equations, in this context, were stated by Filippov in Filippov (1988). Related theories can be found in
di Bernardo et al. (2008); Orlov (2009); Teixeira (2008) and references therein.

On Σij \ F1 we generically distinguish three regions according to the Filippov’s convention (see
Filippov, 1988):

• Sewing Region: Σ c
ij = {p ∈ Σij | Xih(p)Xjh(p) > 0}.

• Sliding Region: Σ s
ij = {p ∈ Σij | Xih(p) < 0 and Xjh(p) > 0}.

• Escaping Region: Σ e
ij = {p ∈ Σij | Xih(p) > 0 and Xjh(p) < 0}.

We stress that for those points q ∈ Σ s
ij ∪Σ e

ij, it was established by Filippov (see Filippov, 1988) a manner
to induce a vector field Zs on Σij which is tangent to Σ s

ij ∪ Σ e
ij and writes

Zs
ij = 1

〈(Xj − Xi), ∇h〉
[〈Xj, ∇h〉Xi − 〈Xi, ∇h〉Xj

] = XjhXi − XihXj

Xjh − Xih
. (2.1)

In this work, the vector field Zs
ij will be called ‘sliding vector field’ or ‘Filippov vector field’.

By a time rescaling, the sliding vector field Zs
ij is orbitally equivalent on Σ s

ij to the ‘normalized sliding
vector field’

Z̃ s
ij = [〈Xj, ∇h〉Xi − 〈Xi, ∇h〉Xj

] = XjhXi − XihXj.

Moreover, since the denominator in (2.1) could be negative on Σ e
ij, we get that Z̃ s

ij is orbitally equivalent
to −Zs

ij on Σ e
ij.

Note that Z̃ s
ij can be Cr extended beyond the boundary of Σ s

ij ∪ Σ e
ij.

In this article, we deal with the following notion of flow.

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566 T. CARVALHO ET AL.

Definition 2 The flow of Z = (X1, . . . , X8) ∈ Ω r is obtained by the concatenation of flows of X1, . . . , X8

or Xs
ij, i, j ∈ {1, . . . , 8}.

Let p be a sewing point contained in Σj1,k1 . In our approach, we allow the trajectory of Xi by p ∈ Σj1,k1

to hit Σj2,k2 with {j1, k1} �= {j2, k2} after a finite time. Let t1(p) > 0 be the Xi-‘flight time’ employed in
such journey. So we define the ‘return map associated to Xi’ by ϕZi(p) = φZi(t1(p), p) = p1 ∈ Σj2,k2 .
When it is possible, i.e. p1 is still a sewing point, we repeat this procedure for it. Assuming that the
trajectory which started in p0 hits only sewing point, the ‘complete return map’ associated to Z0, given
by (1.1), is defined by the composition of these return maps, i.e.

ϕZ0(p) = ϕZ8 ◦ . . . ◦ ϕZ1(p).

The starting vector field X1 can be changed by any other Xj, j = 2, . . . , 8 since the orientation is
preserved.

Alternatively, inspired in the previous ideas, if the starting point p belongs to a sliding or escaping
region, one can still define a complete return map. Indeed, in the next section, we construct such a map
for the region S, since the results throughout this article address this region. The construction of such
maps provide an important object in order to study the behaviour of Z0 around the origin. The proofs of
the main results require a detailed analysis of the complete return map.

In this article, we study smooth nonlinear perturbations of the model (1.1) and a complete picture
of its dynamics is exhibited. It is worth to say that some constructions and ideas of Buzzi et al. (2014)
are very useful in our approach.

3. Auxiliary results

In this section, we describe some important features about the PSVF expressed by Equation (1.1).
Indeed, in Subsection 3.1 we describe its behaviour in Σ and the sliding vector field associated to (1.1)
(Lemmas A and B). Then, we study the sliding region S of Σ and the asymptotical stability at origin
of an one-parameter family of perturbation of Z0. Later on we analyse the way in which the trajectories
converge to a limit set. This last analysis permits us to detect a global attractor for the trajectories
of (1.1).

3.1. Properties of System Z0 given by (1.1)

In this subsection, we analyse the region of discontinuity Σ . Indeed, the behaviour of Z0 in each
component of Σ is presented by the following two lemmas.

Lemma A. The set Σ \ S is contained in Σ c.

Proof. It is not difficult to verify that Σ \ S coincides with the sewing region. For instance, if p =
(x, 0, z) ∈ Σ2,3 then h(p) = π2(p) and consequently

X2π2(p) · X3π2(p) = 〈(1, −1, 3), (0, 1, 0)〉 · 〈(1, −1, 1), (0, 1, 0)〉 = 1 > 0.

The other cases are analogous. �

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 567

Lemma B. The set S = Σ1,2 ∪ Σ1,4 ∪ Σ1,5 is the sliding region Σ s and the respective sliding vectors
fields Zs

12, Zs
14 and Zs

15 are given by

Zs
12(x, y, z) = (0, −1, 1),

Zs
14(x, y, z) = (1, 0, −1),

Zs
15(x, y, z) = (−1, 1, 0).

Proof. The proof of Lemma B is straightforward following Lemma A and using an algebraic
manipulation of the formula (2.1). �

Now we define a complete return map associated to the sliding vector fields of Z0 given by (1.1).
Indeed, consider the regions

S1 = Σ1,2 ∩ Σ1,5,

S2 = Σ1,2 ∩ Σ1,4,

S3 = Σ1,4 ∩ Σ1,5,

which are transversal sections of the flows Zs
12, Zs

14, Zs
15.

Given a point s1 ∈ S1, consider the transition map P1 : S1 → S2 such that P1(s1) = s2 is the point
of S2 where the Zs

12-trajectory passing through s1 intersects S2. Analogously, define the transition maps
P2 : S2 → S3 and P3 : S3 → S1 using the flows of Zs

14 and Zs
15 respectively.

Consequently the complete return map for the sliding region writes

ϕZ0 = P3 ◦ P2 ◦ P1 : S1 → S1.

Remark 1 In our particular case, the forward trajectory (which is the interesting situation here) has a
unique choice, i.e. the forward trajectory of a point in F1 is well defined and it does not depart from the
set S.

The following Propositions A and B describe the behaviour on S.

Proposition A. The set S is fulfilled with periodic trajectories of (1.1).

Proof. We make use of the first return map in order to guarantee that any trajectory through a point in
S = Σ1,2 ∪Σ1,4 ∪Σ1,5 is closed. Indeed, consider S1 the region of return of the trajectories and an initial
condition (0, y0, 0) ∈ S1.

By Lemma B, the flows ϕ
1,2
t , ϕ

1,4
t and ϕ

1,5
t starting in p0

1,2 = (0, y0, z0), p0
1,4 = (x0, 0, z0) and

p0
1,5 = (x0, y0, 0) associated to the sliding vector fields Zs

12, Zs
14 and Zs

15, respectively, are

ϕ
1,2
t (p0

12) = (0, −t + y0, t + z0),

ϕ
1,4
t (p0

14) = (t + x0, 0, −t + z0),

ϕ
1,5
t (p0

15) = (−t + x0, t + y0, 0).

(3.1)

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568 T. CARVALHO ET AL.

Fig. 2. S is fulfilled by periodic trajectories.

Now, once S1 = Σ1,2 ∩ Σ1,5 and every future trajectory reaches S1 transversally for positive time,
the trajectory enters in Σ1,2 and consequently is governed by the flow ϕ

1,2
t . Nevertheless, considering

p0 = (0, y0, 0) ∈ S1 we obtain ϕ
1,2
t (p0) = (0, −t + y0, t) and consequently P1(p0) = ϕ

1,2
t=y0

(p0) =
(0, 0, y0) ∈ S2.

From the point (0, 0, y0) the trajectory enters in Σ1,4 through the flow ϕ
1,4
t and we get ϕ

1,4
t (0, 0, y0) =

(t, 0, −t + y0). Again, taking t = y0, we have P2(0, 0, y0) = ϕ
1,4
t=y0

(0, 0, y0) = (y0, 0, 0) ∈ S3. Finally,
the flow ϕ

1,5
t starting at the point (y0, 0, 0) is ϕ

1,5
t (y0, 0, 0) = (−t + y0, t, 0), then P3(y0, 0, 0) =

ϕ
1,5
t=y0

(y0, 0, 0) = (0, y0, 0) = p0 ∈ S1, i.e. the first return map ϕZ0 : S1 → S1 is given by

ϕZ0(y0) = ϕ1,5
t=y0

◦ ϕ1,4
t=y0

◦ ϕ1,2
t=y0

(y0) = y0.

Therefore, through every point p0 = (0, y0, 0) ∈ S1 passes a periodic trajectory, which means that
S = Σ1,2 ∪ Σ1,4 ∪ Σ1,5 is fulfilled with periodic orbits. �

Figures 2 and 3 illustrate the situation described in Proposition A. As we can see, the origin of the
system of coordinates looks like a center once every neighbourhood of it in S is fulfilled with periodic
trajectories. Moreover, Proposition B tells that this continuum of periodic trajectories is a global attractor
of System (1.1).

The next proposition tell us that every trajectory of Z0 in R
3 converges to S.

Proposition B. The set S is a global attractor for Z0.

Proof. We start considering the trivial case, i.e. taking a point q ∈ S = Σ1,2 ∪ Σ1,4 ∪ Σ1,5. Then,
by Proposition A, the Z0-trajectory through q is periodic and we get the result in this case. Now,
consider a point p1 = (x1, y1, z1) ∈ R1 and ϕ1

t (p1) the flow associated to the vector field Z1. Clearly
ϕ1

t (p1) = (−t + x1, −t + y1, −t + z1). Thus, taking t1 = min{x1, y1, z1} it holds ϕ1
t1
(p1) = q ∈ S which

is periodic, so the result is true.
Now we consider a point p in

⋃
i=2,...,8 Ri and we must prove that p converges to S. Indeed, without

loss of generality, take p = p6 ∈ Σ5,6 \ S since all points in R5 reaches Σ5,6. Observe that the trajectory
ϕ6

t through p6 = (x6, y6, z6) writes ϕ6
t (p6) = (−t +x6, t +y6, t + z6). Once y6 > 0 and t +y6 is increasing

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 569

Fig. 3. The trajectories going to the 3D triangular center.

in t we get ϕ6
t (p6) ∩ Σ6,7 = ∅. Consequently, each point starting in Σ5,6 reaches Σ2,6 in a finite time in

sewings points and enters in the region R2 where the dynamics is governed by the flow associated to this
region, namely, ϕ2

t .
On the region R2, the flow starting on a point p2 = (x2, y2, 0) ∈ Σ2,6 is given by ϕ2

t (p2) = (t+x2, −t+
y2, 3t), where x2 < 0 and y2 > 0. Note that the trajectory through p2 meets Σ1,2 when t + x2 = 0, i.e.
for t = −x2 > 0. On the other hand, if t = y2, then ϕ2

t=y2
(p2) ∈ Σ2,3. Consequently, the plane

Γ2 = {(x, y, z) ∈ R
3 |y = −x, x < 0, y > 0} separates the region R2 into two disjoint parts, one of them

having the set S as it ω-limit and the other one reaching the set Σ2,3.
Those points which reach Σ2,3 have a behaviour similar to the points of Σ5,6. In fact, all they go to

Σ3,4 and the plane Γ4 = {(x, y, z) ∈ R
3 |z = −y, y < 0, z > 0} separates the region R4 into two disjoint

parts, one of them having the set S as it ω-limit and the other one reaching the set Σ4,8.
Again, those points which reach Σ4,8 has a behaviour similar to the points of Σ5,6 and Σ2,3. In fact,

all they go to Σ5,8 and the plane Γ5 = {(x, y, z) ∈ R
3 |z = −x, z < 0, x > 0} separates the region R5 into

two disjoint parts, one of them having the set S as it ω-limit and the other one reaching the set Σ5,6.
Observe that each one of the planes Γ2, Γ4 and Γ5 guides half of the points in Σ2,6, Σ3,4 and Σ5,8,

respectively, until S. In this way from the domain Σ5,6 of the complete first return map, just the points
of the region Φ1 = {(0, y, z) ∈ Σ5,6 |0 < y < −z/8} return to Σ5,6. In fact, saturating by the flow the
straight line r1 = {(0, y, z) ∈ Σ5,6 |y = −z/8} we obtain that it goes to the straight line r2 = Σ5,8 ∩ Γ5.

Repeating the previous argument, we get that the complete first return map ϕZ0 satisfies
ϕ2

Z0
(Φ1\Φ2) → S, where Φ2 = {(0, y, z) ∈ Σ5,6 |0 < y < −z/64}. Recursively, ϕn

Z0
(Φn−1\Φn) → S,

where Φn = {(0, y, z) ∈ Σ5,6 |0 < y < −z/8n}. Therefore, the result follows. �

Corollary 1 Given a point in q ∈ R
3 it is possible to determine the number of times that the trajectory

through q hits a connect component of Σ before reaches S.

Proof. We write the proof for one connect component ofΣ , e.g.Σ5,6. The proof for the others components
is similar. Given a point q ∈ R

3, let q1 = (0, y1, z1) the intersection of its trajectory with Σ5,6. Following
the final steps of the proof of the previous proposition we can see that if |y1|/|z1| > 1/8 then the trajectory

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570 T. CARVALHO ET AL.

by q converges to S before it returns to Σ5,6. Repeating the same argument, if 1/8 > |y1|/|z1| > 1/64
then the trajectory through q converges to S after hits Σ5,6 once.

In general, if 1/8n−1 > |y1|/|z1| > 1/8n then the trajectory through q converges to S before it hits
Σ5,6 a number n − 1 of times. �

3.2. Asymptotic stability at the origin of a perturbation of Z0

In this subsection, we exhibit a perturbation Z0
ε of the system Z0 for which the behaviour of the trajectories

of the model presented in Bernard and el Kharroubi (1991) can be topologically reproduced. We stress
that the model introduced in Bernard and el Kharroubi (1991) does not present asymptotical stability at
the origin once the trajectories scape from the origin and goes to infinity (which happens in case ε < 0
in Lemma 1). Besides, if ε > 0 in Lemma 1, we prove that the Z0

ε is asymptotically stable at origin.
Consider initially the smooth function F0

ε : R → R, such that

F0
ε (x) = εx.

Let be Z0 given by (1.1) and

Z0
ε (x, y, z) = X0

i,ε(x, y, z), if (x, y, z) ∈ Ri, i = 1, 2, . . . , 8, (3.2)

where

X0
2,ε = X2 +

(
0, 2 ∂F0

ε (z)
∂z , 0

)
, (3.3)

and X0
i,ε(x, y, z) = Xi(x, y, z) for i = 1, 3, 4, 5, 6, 7, 8 are given by (1.2). Associated to the perturbation

(3.3) we have the normalized sliding vector fields given by

Z0,s
1,4,ε(x, y, z) = Zs

1,4(x, y, z) = (1, 0, −1)

Z0,s
1,5,ε(x, y, z) = Zs

1,5(x, y, z) = (−1, 1, 0)

Z0,s
1,2,ε(x, y, z) =

(
0, −1 + ∂F0

ε (z)
∂z , 1

)
.

(3.4)

So, we get that dynamics is governed by the flow φ
Z0,s

1,2,ε
of Z0

1,2,ε given in (3.4) is parameterized by

φ
Z0,s

2,ε
(t, p) = (x, −t + F0

ε (t) + y, t + z), (3.5)

where p = (x, y, z).
In the following, we present the result that treat of the stability of Z0

ε :

Lemma 1 Consider the perturbation Z0
ε , given in (3.2) of Z0. We get

(a) if ε > 0 then the origin is asymptotically stable;

(b) if ε < 0 then the origin is asymptotically unstable.

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 571

Proof. First of all, we prove that S is invariant and a global attractor for Z0
ε . In fact, we have that the

sliding vector fields Z0,s
1,4,ε and Z0,s

1,5,ε coincides with the unperturbed case. Given p ∈ Σ12, by (3.5) we
get that trajectories of Z0

ε intercepts the axis z (frontier of Σ12) and then follows the trajectory of Z0,s
1,4,ε,

that coincides with Z0,s
1,4. Therefore, we conclude that S is invariant for Z0

ε . The proof that S is a global
attractor of Z0

ε is similar to the proof of Proposition B and will be omitted.
As S is invariant and a global attractor of Z0

ε , to characterize the stability is necessary and sufficient
to study the stability of the first return map ϕZ0

ε
= φ

Z0,s
1,2,ε

◦ φ
Z0,s

1,5,ε
◦ φ

Z0,s
1,4,ε

.

We consider the transverse section on S, the domain of ϕZ0
ε
, given by S2 = {(0, 0, z); z > 0}.

So, we have φ
Z0,s

1,2,ε
◦ φ

Z0,s
1,5,ε

◦ φ
Z0,s

1,4,ε
(t, (0, 0, z0)) = φ

Z0,s
1,2,ε

(t, (0, z0, 0)). The first return map is given

by the third coordinate of the point in S2 provided by the intersection of the graph of function G0(z) =
−z + F0

ε (z) + z0 with the half straight line S2. Therefore, ϕZ0
ε

satisfies

z − ϕZ0
ε
(z) − F0

ε (ϕZ0
ε
(z)) = 0.

Explicitly, we get thatϕZ0
ε
(0, 0, z) =

(
0, 0,

z

1 + ε

)
. Therefore, if ε > 0 then lim

n→∞(ϕn
Z0
ε
(0, 0, z)) = (0, 0, 0)

and if ε < 0 then lim
n→−∞(ϕn

Z0
ε
(0, 0, z)) = (0, 0, 0). �

Proof of Theorem D. The proof is straightforward using Lemma 1. �

4. Properties of a specific perturbation of System Z0

4.1. Analysis of the first return map of the perturbed vector field

Consider the C∞-functions Fi
ε(x) : R → R, where i = 1, 2, 3, such that

F1
ε (x) = εx(ε − x)(2ε − x) . . . (kε − x), (4.1)

F2
ε (x) = x sin(πε2/x), (4.2)

F3
ε (x) = ε(x − ε)k+1, (4.3)

with k ∈ N.

Lemma 2 Consider the function F1
ε (x) given by (4.1).

(a) If ε < 0 then F1
ε does not have roots in (0, +∞).

(b) If ε > 0 then F1
ε has exactly k roots in (0, +∞) and these roots are {ε, 2ε, . . . , kε}.

(c)
∂F1

ε

∂x
(jε) = εk+1(−1)j−1j!(k − j)! for j ∈ {1, 2, . . . , k}. It means that such partial derivative at jε is

positive for j odd and negative for j even.

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572 T. CARVALHO ET AL.

Proof. When x > 0, by a straightforward calculation F1
ε (x) = 0 if, and only if, (ε− x)(2ε− x) . . . (kε−

x) = 0. So, the roots of F1
ε (x) in (0, +∞) are ε, 2ε, . . . , kε. Moreover,

∂F1
ε

∂x
(x) = ε(ε − x)(2ε − x) . . . (kε − x) −

k∑
i=1,i �=j

εx(εi − x).

Therefore,

∂F1
ε

∂x
(jε) = ε(jε)(ε − jε)(2ε − jε) . . . ((j − 1)ε − jε)((j + 1)ε − jε)

((j + 1)ε − jε) . . . (kε − jε)

= εk+1(−1)j−1j!(k − j)!

This proves items (b) and (c). Item (a) follows immediately. �

Lemma 3 Consider the function F2
ε (x) given by expression (4.2). For ε �= 0 the function F2

ε has infinitely
many roots in (0, ε2), these roots are {ε2/2, ε2/3, . . . } and

∂F2
ε

∂x
(ε2/j) = ∓π j for j ∈ {2, 3, . . . }.

It means that such derivative at (ε2/j) is positive for j odd and negative for j even.

Proof. When x > 0, by a straightforward calculation Fi
ε(x) = 0 if, and only if, sin(πε2/x) = 0. So, the

roots of Fi
ε(x) in (0, ε2) are ε2/2, ε2/3, . . . . Moreover,

∂F2
ε

∂x
(x) = sin

(
πε2

x

)
− x cos

(
πε2

x

) (
πε2

x2

)
.

Therefore,
∂F2

ε

∂x
(ε2/j) = ∓π j. �

Lemma 4 Consider F3
ε given in (4.3). Then x = ε is a zero of multiplicity k of equation F3

ε (x) = 0.

Proof. In fact, by expression (4.3) we get that F3
ε (ε) = ∂F3

ε

∂x
(ε) = · · · = ∂kF3

ε

∂xk
(ε) = 0 and

∂k+1F3
ε

∂xk+1
(ε) =

(k + 1)!ε �= 0. �

Consider Z0 given by (1.1) and

Z1,2,3
ε (x, y, z) = X1,2,3

i,ε (x, y, z), if (x, y, z) ∈ Ri, i = 1, 2, . . . , 8, (4.4)

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 573

where

X1
2,ε = X2 +

(
0, 2

∂F1
ε

∂z
(z), 0

)
,

X2
2,ε = X2 +

(
0, 2

∂F2
ε

∂z
(z), 0

)
(4.5)

X3
2,ε = X2 +

(
0, 2

∂F3
ε

∂z
(z), 0

)

and X1,2,3
i,ε (x, y, z) = Xi(x, y, z) for i = 1, 3, 4, 5, 6, 7, 8 are given by (1.2). Associated to the perturbation

(4.5) we have the normalized sliding vector fields given by

Z1,2,3,s
1,4,ε (x, y, z) = Zs

1,4(x, y, z) = (1, 0, −1)

Z1,2,3,s
1,5,ε (x, y, z) = Zs

1,5(x, y, z) = (−1, 1, 0)

Z1,s
1,2,ε(x, y, z) =

(
0, −1 + ∂F1

ε

∂z
(z), 1

)
(4.6)

Z2,s
1,2,ε(x, y, z) =

(
0, −1 + ∂F2

ε

∂z
(z), 1

)

Z3,s
1,2,ε(x, y, z) =

(
0, −1 + ∂F3

ε

∂z
(z), 1

)
.

A straightforward calculation shows that the trajectories φZi,s
1,2,ε

of Zi
1,2,ε, with i = 1, 2, 3, given in (4.6)

are parameterized by

φ
Z1,s

2,ε
(t, p) = (x, −t + F1

ε (t) + y, t + z)

φ
Z2,s

2,ε
(t, p) = (x, −t + F2

ε (t) + y, t + z) (4.7)

φ
Z3,s

2,ε
(t, p) = (x, −t + F3

ε (t) + y, t + z),

where p = (x, y, z).

Remark 2 Take Zε = Zi
ε, with i = 1, 2, 3. It is easy to see that Zε → Z0 when ε → 0, in the Cr-topology.

The next result takes into account the first return map of the perturbed sliding vector fields. However,
from now on we consider the region S2 as starting point instead of S1. Of course, in this case the return
map can be obtained in a totally analogous way.

Lemma 5 The first return map of ϕ
Z1,2
ε

: S2 → S2 satisfies the equation

z − ϕ
Z1,2
ε

(z) − F1,2
ε (ϕ

Z1,2
ε

(z)) = 0.

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574 T. CARVALHO ET AL.

Proof. Consider the initial condition p0 = (0, 0, z0) ∈ S2. The first return map is given by φ
Z1,2,s

1,2,ε
◦φ

Z1,2,s
1,5,ε

◦
φ

Z1,2,s
1,4,ε

. By expression (3.1) and (4.7), we get that φ
Z1,2,s

1,2,ε
◦φ

Z1,2,s
1,5,ε

◦φ
Z1,2,s

1,4,ε
(t, (0, 0, z0)) = φ

Z1,2,s
1,2,ε

(t, (0, z0, 0)).

So, as we get in the proof of Proposition 1, the first return map is given by the third coordinate of the
point in S2 given by the intersection of the graph of function G1,2(z) = −z + F1,2

ε (z) + z0 with the half
straight line S2. Therefore, ϕ

Z1,2
ε

satisfies

z − ϕ
Z1,2
ε

(z) − F1,2
ε (ϕ

Z1,2
ε

(z)) = 0. �

Lemma 6 The first return map ϕZ3
ε

associated to Z3
ε is expressed by

ϕZ3
ε
(z) = z + ε(z − ε)k+1.

Proof. As the previous case, we get that ϕZ3
ε
(0, 0, z) = P1 ◦ P3 ◦ P2(0, 0, z) = P1(0, z, 0) and the

application Π3 is given implicitly as one of the solutions z of

G3(z, ε) = 0, (4.8)

where G3(z, ε) = −z + F3
ε (z) + y. Note that putting y = ε then z = ε is a solution of (4.8). In other

words, P1(0, ε, 0) = (0, 0, ε). Considering the change of coordinates z = z − ε and y = y − ε then (4.8)
becomes

−z + εzk+1 + y = 0.

Therefore, we obtain

ϕZ3
ε
(0, 0, z) = z − ε(z − ε)k+1. �

4.2. Limit cycles of the perturbed vector fields Z1,2
ε

Proposition 1 Consider Z1,2
ε given by expression (4.4). Then Z1

ε has exactly k limit cycles and Z2
ε has

infinite many limit cycles, all of them situated in S. Moreover, these limit cycles are hyperbolic repeller
limit cycles if j is odd and a hyperbolic attractor limit cycles if j is even.

Proof. The proof that S is invariant and a global attractor for Z1,2
ε is analogous that we done in Proposition

1 and we omit it.
So, if there exists limit cycles, then they are situated at S. Moreover, when we restrict the flow of

Z1,2
ε to S, by Proposition B, the fixed points of the first return map ϕ

Z1,2
ε

= φ
Z1,2,s

1,2,ε
◦ φ

Z1,2,s
1,5,ε

◦ φ
Z1,2,s

1,4,ε
provide

us the periodic orbits of systems Z1,2
ε . Since these fixed points are isolated (limit cycles), the stability

of each one is given by the stability of the fixed point of the difeomorphism ϕ
Z1,2
ε

. In both cases, we
consider the transverse section on S, the domain of ϕ

Z1,2
ε

, given by S2 = {(0, 0, z); z > 0}.
By Lemma 5, we get that the fixed points of the first return maps are given by the zeros of the

function F1,2
ε (z). Considering the first perturbation Z1

ε , by Lemma 2, we conclude that these roots in
S2 for ε > 0, are given by ε, 2ε, . . . , kε. Therefore, we get that the fixed points of ϕZ1

ε
are given by

p1 = (0, 0, ε), p2 = (0, 0, 2ε), . . . , pk = (0, 0, kε) with pi ∈ S2, i = 1, . . . , k.

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 575

So, the stability of these fixed points are given by item (c) of Lemma 2 and provides that pj is a
hyperbolic repeller fixed point for j odd and a hyperbolic attractor fixed point for j even. In this way, we
conclude that the orbits γ 1

j , passing through pj of Z1
ε are hyperbolic repeller limit cycles if j is odd and

a hyperbolic attractor limit cycles if j is even.
In an analogous way, considering the second perturbation Z2

ε , the fixed points of ϕZ2
ε
, are given by

the zeros of F2
ε . Therefore, by Lemma 3, we get that periodic orbits of Z2

ε are given by the trajectories γ 2
j

passing through the points qj = (0, 0, ε2/j), j ∈ {2, 3, . . . }. As the previous case, the stability is given by
the Lemma 3. Moreover, qj is a hyperbolic repeller fixed point for j odd and a hyperbolic attractor fixed
point for j even. Therefore, γ 2

j are hyperbolic repeller limit cycles if j is odd and a hyperbolic attractor
limit cycles if j is even. �

Remark 3 (Asymptotically stability of limit cycles) One should observe that the results of the current
section also guarantee the asymptotical stability of some limit cycles. We remember that these limit
cycles are obtained through the perturbation of the center contained in S, stated in Proposition A. Indeed,
Proposition 1 assures that, by using Lemmas 4 and 5, the limit cycles coming from the perturbation of
such a center are hyperbolic. Moreover, if j is even, they attract every trajectories which are sufficiently
close to them, i.e. those limit cycles are asymptotical stable.

5. Proof of the main results of the article

5.1. Proof of Theorem A and Corollary B

Proof of Theorem A: Item (a): It follows from Remark 2.
Item (b): It follows from Proposition 1.
Item (c): It follows from Proposition B and a remark inside the proof of Proposition 1. �

Proof of Corollary B: Suppose that the codimension of the origin of Z0, given by (1.1) is m < ∞.
Then, in a neighbourhood of Z0 there are PSVFs of m distinct topological types. This is a contradiction
due Theorem A. So, the codimension of this singularity is infinite. �

5.2. Proof of Theorem C

Let K ⊂ R
3 be a compact set around the origin. Consider the restrictions to K of Z0 and of its

perturbations. Since it will not cause confusion, we denote Z0|K = Z0 and Zε|K = Zε.
Let Z3

ε be the perturbation of Z0 given by (4.4). By Lemma 6 we get that the first return map is given
by

ϕZ3
ε
(0, 0, z) = z − ε(z − ε)k+1.

We define the application

F3,j
ε (z) = ε(z − ε)j

(
z − ε

2

)
. . .

(
z − ε

k − j

)
.

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576 T. CARVALHO ET AL.

Consider V a neighbourhood of Z0 and ε > 0 sufficiently small such that Z3,j
ε ∈ V , where

Z3,j
ε (x, y, z) = X3,j

i,ε (x, y, z), if (x, y, z) ∈ Ri, i = 1, 2, . . . , 8, with

X3,j
2,ε = X2 +

(
0, 2

∂F3,j
ε

∂z
(z), 0

)
, (5.1)

and X3,j
i,ε (x, y, z) = Xi(x, y, z) for i = 1, 3, 4, 5, 6, 7, 8 are given by (1.2). Associated to the family (5.1)

we have the normalized sliding vector fields given by

Z3,j,s
1,4,ε(x, y, z) = Zs

1,4(x, y, z) = (1, 0, −1)

Z3,j,s
1,5,ε(x, y, z) = Zs

1,5(x, y, z) = (−1, 1, 0)

Z3,j,s
1,2,ε(x, y, z) =

(
0, −1 + ∂F3,j

ε

∂z
(z), 1

)
.

A straightforward calculation shows that the trajectories φ
Z

3,j,s
1,2,ε

of Z3,j
1,2,ε are parameterized by

φ
Z

3,j,s
1,2,ε

(t, p) = (x, −t + F3,j
ε (t) + y, t + z), (5.2)

where p = (x, y, z).
The next lemma provides the expression of the first return map for Z3,j

ε

Lemma 7 The first return map of Z3,j
ε is given by

ϕ
Z

3,j
ε

(0, 0, z) = z − F3,j
ε (z).

Proof. The proof is analogous to that one presented in Lemma 6 and will be omitted. �

Proof of Theorem C. Consider in Ω r the Cr-topology with r ≥ k + 1. By Lemma 7, the expression of
the first return map of Z3,j

ε is given by

ϕ
Z

3,j
ε

(0, 0, z) = z − F3,j
ε (z)

and therefore the limit cycles are given by the solution of

F3,j
ε (z) = 0.

So we conclude that z1 = ε is a zero of multiplicity j and z2 = ε/2, . . . , zk−j = ε/(k − j) are simple
zeros of this equation, where j = 0, 1, . . . , k − 1. In this way, we obtain that for each j the perturbation
Z3,j

ε presenting exactly k − j limit cycles.
From Proposition 21 of Buzzi et al. (2014), two PSVFs are equivalent if and only if the first return

maps are conjugated. It is easy to see that the first return maps are conjugated if and only if the number
of zeros are equal and with the same multiplicity (k − j simple zeros and one zero of multiplicity j).
So, in a neighbourhood (in the topology Ck+1) of Z3,j

ε we get exactly k distinctly topologically types of
PSVFs, when j = 0, 1, . . . , k − 1.

Therefore, Z3
ε has codimension k. �

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BIRTH OF LIMIT CYCLES FROM A 3D TRIANGULAR CENTER 577

6. Conclusion

In this article, we consider a 3D model, given by (1.1), where eight smooth vector fields are combined,
in the Filippov’s sense, in order to create a center with a triangular shape at the sliding region S, where
the sliding vector field is defined. Moreover, S is a global attractor for the trajectories of (1.1).

An important remark is that all smooth vector fields considered, and also the sliding vector fields, are
‘constant’ and even in this case the PSVF Z0 given by (1.1) has infinite codimension. This fact reveals
the complexity involved in the analysis of piecewise smooth dynamical systems.

Acknowledgements

First of all we would like to thank the anonymous referees for so many constructive suggestions.

Funding

São Paulo Research Foundation (FAPESP) (#2014/02134-7 to T.C.) and the CAPES (88881.030454/2013
-01 to T.C.) from the program CSF-PVE and (1576689 to T.C.) from the program PNPD ; São
Paulo Research Foundation (FAPESP) ( #2013/25828-1, #2014/18508-3 to R.D.E.); Goiás Research
Foundation (FAPEG), PROCAD/CAPES ( 88881.0 68462/2014-01, #2012/10 26 7000 803 to D.J.T.);
CNPq/Brazil (478230/2013-3 and 443302/2014-6 to T.C. and D.J.T.); UFG as a part of project numbers
35796, 35798 and 040393.

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