Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 19, pp. 1–14.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH AN

ALGEBRAIC LINE OF SEPARATION

ARMENGOL GASULL, JOAN TORREGROSA, XIANG ZHANG

Abstract. We study the number of limit cycles of planar piecewise linear
differential systems separated by a branch of an algebraic curve. We show that

for each n ∈ N there exist piecewise linear differential systems separated by

an algebraic curve of degree n having [n/2] hyperbolic limit cycles. Moreover,
when n = 2, 3, we study in more detail the problem, considering a perturbation

of a center and constructing examples with 4 and 5 limit cycles, respectively.

These results follow by proving that the set of functions generating the first
order averaged function associated to the problem is an extended complete

Chebyshev system in a suitable interval.

1. Introduction and statement of main results

For planar polynomial differential systems there still stands the unsolved Hil-
bert’s 16th problem, whose second part asks for the maximum number of limit
cycles and their distribution in terms of their degree, see for instance [9, 11]. As we
know, planar linear differential systems have no limit cycle, whereas the situation
is different for piecewise linear differential systems. For them, in the case when
they are defined in two zones separated by a straight line, it is known that there
are examples with 3 limit cycles, see [4, 7, 8, 13]. It is yet an open problem to know
the maximum number of limit cycles that this type of systems can have.

We address to a very related question. Let PLn be the set of planar piecewise
linear differential systems with two zones separated by a branch of an algebraic
curve of degree n, Cn, where in each of the two zones the differential system is
linear. As usual, limit cycles will be periodic solutions isolated in the set of all
periodic solutions. Here, the definition of solution in the two zones scenario is the
one given in [5, 6, 15]. We remark that in this setting there are crossing and sliding
limit cycles. In this paper, when we refer to limit cycles we will mean the ones
of crossing type, even when we do not explicitly mention it. Recall that this type
of periodic solutions cut the discontinuity curve only at finitely many points, both
vector fields are transversal to this curve at each of these points and, moreover,
both vectors fields point towards the same component of R2 \ Cn at each of them.

2010 Mathematics Subject Classification. 34C25, 34C07, 37G15.
Key words and phrases. Piecewise linear differential system; algebraic separation;

limit cycle; ECT-system.
c©2020 Texas State University.

Submitted April 23, 2018. Published February 14, 2020.

1



2 A. GASULL, J. TORREGROSA, X. ZHANG EJDE-2020/19

Let Ln ∈ N ∪ {∞} be the maximum number of limit cycles that systems inside
class PLn can have. Our first main result, proved in Section 2, implies that Ln
tends to infinity with n.

Theorem 1.1. For n ∈ N, there exists a polynomial fn(x) of degree n and a
piecewise linear differential system with two zones separated by the curve y = fn(x),
such that the corresponding piecewise linear differential system has at least [n/2]
hyperbolic limit cycles, where [·] denotes the integer part function. As a consequence,
Ln ≥ [n/2].

The above theorem does not aim to give the optimal lower bound for Ln, but
simply to prove that Ln goes to infinity with n. As far as we know, this result is
new.

It is worthwhile to comment here that there are already results proving that
systems with two zones can have an arbitrary large number of limit cycles (or even
infinitely many), see [3, 16, 18], but in all these previous works the separation curve
is neither polynomial, nor analytic.

Using similar arguments as in the proof of Theorem 1.1 we can get the next
result.

Proposition 1.2. There exists an analytic function f(x) and a piecewise linear
differential system with two zones separated by the curve y = f(x), such that the
corresponding piecewise linear differential system has infinitely many limit cycles.

Notice that Theorem 1.1 gives no information when n = 1, although it is known
that L1 ≥ 3. Next, we study in more detail the cases where the separation curve is
a branch Cn of a quadratic curve (n = 2) or a cubic curve (n = 3). This curve is
contained in the set {(x, y) : Fn(x, y) = 0}, for some irreducible polynomial Fn, of
degree n.

For tackling this problem, and to fix a generic situation, we treat the case in which
Cn is tangent to the x-axis at the origin, and it is locally convex, see Figure 3. The
region above Cn is denoted by Ω+

n and the one below is denoted by Ω−n . To improve
the lower bounds for Ln, n = 2, 3, we consider the piecewise linear differential
system

(
ẋ
ẏ

)
=



(
y + ε(a+0 + a+1 x+ a+2 y)

−x+ ε(b+0 + b+1 x+ b+2 y)

)
, for (x, y) ∈ Ω+

n ,(
y + ε(a−0 + a−1 x+ a−2 y)

−x+ ε(b−0 + b−1 x+ b−2 y)

)
, for (x, y) ∈ Ω−n ,

(1.1)

which is a perturbation of a global linear center. As we will see in Proposition 3.2,
given any Cn as above, if we denote by MCn

the first order Melnikov function
associated to this vector field, it is a linear combination, with free parameters, of 6
functions. The expression of MCn

is already obtained in [12].
Our main result is based on proving that the functions defining MCn form an

extended complete Chebyshev system (ECT-system). In Section 3 we recall this
concept and its utility in our context.

Theorem 1.3. Let the separation curve Cn be tangent to the x-axis at the origin
and locally convex. Let MCn

be the first order Melnikov function associated to (1.1).



EJDE-2020/19 PIECEWISE LINEAR SYSTEMS WITH ALGEBRAIC SEPARATION 3

(a) If n = 2 and C2 is a symmetric conic with respect to the y-axis then the
functions defining MC2 form an ECT-system formed by 4 functions in the
whole interval of definition of MC2 . Then L2 ≥ 3.

(b) There exist C2 non symmetric conics with respect to the y-axis, being an
ellipse, a parabola or a hyperbola, such that the corresponding functions
defining MC2

form an ECT-system formed by 5 functions in some interval
(0, δ], for δ small enough. Consequently L2 ≥ 4.

(c) There exists C3, non symmetric cubic with respect to the y-axis, such that
the functions defining MC3

form an ECT-system formed by 6 functions in
some interval (0, δ], for δ small enough. Then L3 ≥ 5.

The case in which the separation curve is the symmetric parabola, F2(x, y) =
y−x2, is also studied in [14]. In that paper the authors also prove that 3 limit cycles
appear. They use that the 4 functions defining the first order Melnikov function
are linearly independent. Recently, in [2], a second order study has been developed
and applied to the symmetric cubic curve, F3(x, y) = y−x3. In that work the lower
bound for L3 has been increased to 7.

2. Proof of main results

Proof of Theorem 1.1. Let fn(x, ε) = εTn(x), with ε > 0 small enough, where
Tn(x) is the n-th Chebyshev polynomial of the first kind, i.e. for |x| ≤ 1,

Tn(x) = cos(n arccosx),

and for |x| > 1, its analytic extension. It is known that Tn(x) has degree n and all
its roots are in [−1, 1]. Now y = fn(x, ε) separates the (x, y) plane in two zones,
denoted by Ω+ when y ≥ fn(x, ε) and Ω− when y ≤ fn(x, ε).

Inside Ω+ and Ω− we consider respectively the linear differential system

(ẋ, ẏ) =

{
(x− 4y − 2, x/2− y), in Ω+,

(−y + 1, x), in Ω−,
(2.1)

which form a piecewise linear differential system in the whole plane. Some easy
calculations show that system (2.1) has the first integrals

H+(x, y) = 8y + x2 − 4xy + 8y2 and H−(x, y) = −2y + x2 + y2,

in Ω+ and Ω−, respectively.
Notice that the level curves H+(x, y) = a2 and H−(x, y) = b2, with a > 0, b > 0,

are ellipses and circles, respectively. Their intersection points with the x-axis are
both symmetric with respect to the origin. So the piecewise linear differential
system (2.1) with the separation curve y − fn(x, 0) = 0 has a center at the origin.
We can check that the ellipses H+(x, y) = a2 have positive slopes at the intersection
points with y = 0, whereas the circles H−(x, y) = b2 are perpendicular to y = 0.

Recall that the Chebyshev polynomial Tn(x) has exactly n zeros, all located in
the interval [−1, 1]. Moreover, for n even, the polynomial Tn(x) is even, and its
zeros are symmetric with respect to the origin. For n odd, the polynomial Tn(x) is
odd, and its n− 1 zeros except x = 0 are also symmetric with respect to the origin.

We claim that for ε > 0 small enough, the piecewise linear differential system
(2.1) with the separation curve y − fn(x, ε) = 0 has at least [n/2] hyperbolic limit
cycles. Moreover, when n is even, the system has exactly [n/2] limit cycles, all



4 A. GASULL, J. TORREGROSA, X. ZHANG EJDE-2020/19

them hyperbolic. Figure 1 illustrates the separation curve and the limit cycles for
n even.

Figure 1. Separation curve defined by a Chebyshev polynomial
and the limit cycles for n = 10.

We now prove this claim. Set m = [(n − 2)/2] and let ±x0, . . . ,±xm be the
2(m+ 1) not null zeros of fn(x, ε). Then

xk = cos
(2k + 1

2n
π
)
, k = 0, 1, . . . ,m.

Clearly, for each k ∈ {0, 1, . . . ,m},
Γk :={(x, y) : H+(x, y) = H+(P±k), y ≥ 0}

∪ {(x, y) : H−(x, y) = H−(P±k), y ≤ 0},
is a periodic orbit of system (2.1), where P±k = (±xk, 0) for k ∈ {0, 1, . . . ,m}.

We now prove that Γk is a hyperbolic limit cycle of the piecewise linear differ-
ential system (2.1). For doing so, we compute the derivative of the Poincaré map
associated to the periodic orbit Γk. We take the transversal sections of the flows
of the two linear differential systems defined in (2.1) near Pk and P−k as two small
enough segments Σk and Σ−k in y = fn(x, ε) centered at Pk and P−k, respectively.
Notice that for ε > 0 suitably small, both segments Σk and Σ−k are transversal to
the flows of the two linear systems. Denote by Π−k the Poincaré map from Σ−k to

Σk along the flow of the linear system in Ω−, and by Π+
k the Poincaré map from

Σk to Σ−k along the flow of the linear system in Ω+. Then the composition

Πk = Π+
k ◦Π−k

is the Poincaré map of the piecewise linear differential system (2.1) near the periodic
orbit Γk defined on Σ−k.

For computing the derivative of the Poincaré map Πk, we use the next general
formula (see [1])

Π′(0) =
〈X(0), (γ′0(0))⊥〉
〈X(T ), (γ′1(0))⊥〉

exp
(∫ T

0

divX(ϕ(t)) dt
)
, (2.2)



EJDE-2020/19 PIECEWISE LINEAR SYSTEMS WITH ALGEBRAIC SEPARATION 5

where γ0(s) and γ1(s) are the local expressions of two transversal sections Σ0 and
Σ1 to a class C1 vector field X, T is the flying time from P = γ0(0) to Π(P ) = γ1(0),
and Π : Σ0 → Σ1 is the map induced by the flow ϕ of X, see Figure 2. Here, as
usual, 〈·, ·〉 is the inner product of two vectors and (u, v)⊥ = (−v, u).

Σ0 Σ1

P
Π(P )ϕ(t)

Figure 2. Map Π

Denote by X+ and X− the two vector fields associated to systems defined in
(2.1), respectively. We parameterize the two segments Σk and Σ−k by (x, fn(x, ε))
with x ∈ (xk − δ, xk + δ) and x ∈ (−xk − δ,−xk + δ), respectively, where δ > 0
is suitably small. Since the divergences of the two linear systems defined in (2.1)
identically vanish, by using (2.2) we have

(Π+
k )′(xk) =

〈X+(xk, 0), (1, f ′n(xk, ε))
⊥〉

〈X+(−xk, 0), (1, f ′n(−xk, ε))⊥〉

=
〈(xk − 2, xk/2), (εT ′n(xk),−1)〉

〈(−xk − 2,−xk/2), (εT ′n(−xk),−1)〉

=
(xk − 2)εT ′n(xk)− xk/2
−(xk + 2)εT ′n(−xk) + xk/2

,

and

(Π−k )′(−xk) =
〈X−(−xk, 0), (1, f ′n(−xk, ε))⊥〉
〈X−(xk, 0), (1, f ′n(xk, ε))⊥〉

=
〈(1,−xk), (εT ′n(−xk),−1)〉
〈(1, xk), (εT ′n(xk),−1)〉

=
εT ′n(−xk) + xk
εT ′n(xk)− xk

.

It follows that

Π′k(x−k) = (Π+
k )′(xk)(Π−k )′(x−k)

=
ε(xk − 2)T ′n(xk)− xk/2
−ε(xk + 2)T ′n(−xk) + xk/2

· εT
′
n(−xk) + xk

εT ′n(xk)− xk

= 1− ε
(
T ′n(xk)

(
2− 5

xk

)
− T ′n(−xk)

(
2 +

5

xk

))
+O(ε2).

By using the expression of xk, one gets that for k ∈ {0, 1, . . . ,m},

T ′n(xk) =
n sin(n arccosxk)√

1− x2k
=

(−1)kn√
1− x2k

,

T ′n(−xk) =
n sin(n arccos(−xk))√

1− x2k
=

(−1)k(−1)n+1n√
1− x2k

.



6 A. GASULL, J. TORREGROSA, X. ZHANG EJDE-2020/19

As a consequence,

Π′k(x−k) = 1 + ε
n(−1)k

(
2
(
(−1)n+1 − 1

)
xk + 5

(
(−1)n+1 + 1

) )
xk
√

1− x2k
+O(ε2)

=


1− ε 4n(−1)

k√
1−x2

k

+O(ε2), for n even,

1 + ε 10n(−1)k

xk

√
1−x2

k

+O(ε2), for n odd.

The above computations imply:

• For each k ∈ {0, 1, . . . ,m} and ε > 0 suitably small, Γk is a hyperbolic limit
cycle of system (2.1).
• For n even, Γ2s are stable, and Γ2`+1 are unstable for s, ` ∈ Z and 0 ≤

2s, 2`+ 1 ≤ m.
• For n odd, Γ2s are unstable, and Γ2`+1 are stable for s, ` ∈ Z and 0 ≤

2s, 2`+ 1 ≤ m.

These facts prove the first part of the claim.

Let us prove the second part, i.e. that when n is even and ε > 0 is small enough,
system (2.1), with the separation curve y − fn(x, ε) = 0, has exactly [n/2] limit
cycles.

It suffices to prove that for n even system (2.1) has no more limit cycles than
the [n/2] ones given above. Let Γ∗ be a limit cycle of the considered piecewise
linear system. Then, Γ∗ intersects the separation curve y = εTn(x) at exactly two
points, provided ε > 0 suitably small. Denote these two points by (v, εTn(v)) and
(u, εTn(u)) with v < u. We know that Γ∗ is given by one piece of the level curve
H+(x, y) = a20 and one piece of H−(x, y) = b20, for some a0, b0 ∈ R. We define

W+(x, ε) := x2 + (8− 4x)εTn(x) + 8ε2T 2
n(x),

W−(x, ε) := x2 − 2εTn(x) + ε2T 2
n(x).

With the above definitions, we have

W+(u, ε) = W+(v, ε) and W−(u, ε) = W−(v, ε).

On the other hand, since H−(x, y) = b20 and y = εTn(x) are both symmetric with
respect to the y-axis, we must have v = −u. This implies

0 = W+(u, ε)−W+(−u, ε) = −4uε(Tn(u) + Tn(−u)) = −8uεTn(u).

Hence u is a zero of Tn and Γ∗ is one of the hyperbolic limit cycles Γk’s, as we
wanted to prove. This completes the proof of the theorem.

Proof of Proposition 1.2. We choose the separation curve as y = ε cosx sepa-
rating the (x, y) plane in two zones with Ω+ := {(x, y)| y ≥ ε cosx} and Ω− :=
{(x, y)| y ≤ ε cosx}. The piecewise linear differential system with two zones sepa-
rated by y = ε cosx is again (2.1).

Notice that at the intersection points on the x-axis of y = ε cosx with the level
curves H+(x, y) = a2 and H−(x, y) = b2, the slopes of the two curves H+(x, y) = a2

and H−(x, y) = b2 at the intersection points are

dy

dx
=

x

2x− 4
,

dy

dx
= x,

whose absolute values are always larger than a positive constant. Using this fact
together with similar arguments as in the proof of Theorem 1.1 we get that for



EJDE-2020/19 PIECEWISE LINEAR SYSTEMS WITH ALGEBRAIC SEPARATION 7

ε > 0 small enough the piecewise linear differential system has infinitely many limit
cycles, which pass through the points (−π/2− kπ, π/2 + kπ) for k ∈ N. Thus, the
proposition follows.

3. Preliminary results and proof of Theorem 1.3.

Recall that given n + 1 smooth functions g0, . . . , gn on an open interval I, it
is said that (g0, . . . , gn) is an ECT-system on I if for all k = 0, 1, 2, . . . , n, any
nontrivial linear combination

γ0g0(x) + · · ·+ γkgk(x) (3.1)

with γj ∈ R has at most k isolated zeros on I counting multiplicities. We remark
that it is easy to see that given a linear combination of n+ 1 functions, as in (3.1)
with k = n, there always exist γj such that it has at least n zeros in I. When
the functions form an ECT-system, we can moreover ensure that these n zeros are
simple. When the function (3.1) corresponds to a first order Melnikov function, the
condition that its zeros are simple is precisely the one needed to ensure that they
give rise to limit cycles of the perturbed system.

A very useful characterization of ECT-systems is the following proposition (see
[10]).

Proposition 3.1. (g0, . . . , gn) is an ECT-system on I if and only if for all k =
0, 1, . . . n, W [g0, . . . , gk](x) 6= 0 for all x ∈ I, where

Wk = W [g0, . . . , gk](x)
.
= det

(
g
(i)
j (x)

)
0≤i,j≤k

is the Wronskian of (g0, . . . , gk) at x ∈ I.

For the sake of completeness, and following [12, 17], we include the main steps
of the proof of the following proposition, that uses the averaging method to obtain
the Melnikov function associated to (1.1).

Proposition 3.2. Let Cn be the separation curve tangent to the x-axis at the origin
and locally convex given in system (1.1). Then its first order Melnikov function MCn

is

MCn(r) =
1

r

5∑
k=0

γkgk(r), (3.2)

where γk, k = 0, . . . , 5, are arbitrary real parameters and
√
x2 + y2 = r > 0 param-

eterizes the closed orbits of the unperturbed system (ε = 0). Moreover,

g0(r) = 1, g1(r) = θ1(r)− θ2(r), g2(r) =
1

r

(
f1(r)− f2(r)

)
,

g3(r) = f21 (r)− f22 (r), g4(r) =
1

r

(√
1− f21 (r)−

√
1− f22 (r)

)
,

g5(r) = f1(r)
√

1− f21 (r)− f2(r)
√

1− f22 (r),

and fj(r) = cos(θj(r)), where the functions θj(r), j = 1, 2, are given by the cuts of
Cn and x2 + y2 = r2 shown in Figure 3. The function MCn

is defined when both
functions θj(r) are well-defined.



8 A. GASULL, J. TORREGROSA, X. ZHANG EJDE-2020/19

Proof. To apply the averaging method we consider polar coordinates, (x, y) =
(r cos θ, r sin θ), and system (1.1) writes as

dr

dθ
=

εA±(r, θ)

−1 + εB±(r, θ)/r
= −A±(r, θ)ε+O(ε2), for ±Gn(r, θ) ≥ 0,

where Gn(r, θ) is the expression of the separation curve Fn(x, y) in polar coordi-
nates. Notice that Gn ≥ 0 and Gn ≤ 0 corresponds to Ω+

n and Ω−n , respectively,
and

−A±(r, θ) = α±0 r + α±1 cos θ + α±2 sin θ + α±3 r cos(2θ) + α±4 r sin(2θ),

with α±0 = −(a±1 + b±2 )/2, α±1 = −a±0 , α
±
2 = −b±0 , α±3 = −(a±1 − b±2 )/2 and

α±4 = (a±2 + b±1 )/2.
Observe also that the parameters α±i can be chosen arbitrarily, because of the

arbitrariness of a±i and b±j . The above non-autonomous differential equation is well

defined on any interval r ∈ [r0, β] where r0 > 0 is a fixed and small enough number
and β depends on each specific Fn.

x2 + y2 = r2

Cn

θ2

θ1

(0, 0)

r

Figure 3. Definition of functions θ1(r) and θ2(r) in Proposition 3.2.

The first order averaged function (or Melnikov function) is (see [12])

MCn(r) = −
∫ θ2(r)

θ1(r)

A+(r, s) ds−
∫ θ1(r)+2π

θ2(r)

A−(r, s) ds

= 2α−0 πr − (α+
0 + α−0 )

(
θ1(r)− θ2(r)

)
r

− (α+
1 − α

−
1 )
(

sin(θ1(r))− sin(θ2(r))
)

+ (α+
2 − α

−
2 )
(

cos(θ1(r))

− cos(θ2(r))
)
− α+

3 − α
−
3

2

(
sin(2θ1(r))− sin(2θ2(r))

)
r

+
α+
4 − α

−
4

2

(
cos(2θ1(r))− cos(2θ2(r))

)
r.



EJDE-2020/19 PIECEWISE LINEAR SYSTEMS WITH ALGEBRAIC SEPARATION 9

Writing the above expression in terms of fj(r) = cos(θj(r)) we arrive to (3.2). �

Lemma 3.3. (a) When Cn is a symmetric curve with respect to the y-axis, that is
when Fn(−x, y) = Fn(x, y), the function MCn

reduces to

MCn
(r) =

1

r

3∑
k=0

βkhk(r), (3.3)

h0(r) = 1, h1(r) = θ1(r), h2(r) =
f1(r)

r
, h3(r) = f1(r)

√
1− f21 (r),

where βk, k = 0, . . . , 3, are arbitrary real constants.
(b) When n = 2 one of the functions g2, g3, g4, and g5 can be removed in the

expression (3.2) of MCn
. In particular, when the coefficient of xy in F2 is not zero

then the function g5 can be always removed.

Proof. (a) In this situation, f2 = −f1 and θ2 = π−θ1. By plugging these identities
in (3.2) we obtain (3.3).

(b) The two cuts near (0, 0) between F2(x, y) = 0 and x2 + y2 = r2 are the

points
(
rfk(r), r

√
1− f2k (r)

)
for k = 1, 2. Writing F2(x, y) =

∑2
i+j=0 di,jx

iyj and
imposing that

1

r2

(
F2

(
rf1(r), r

√
1− f2k (r)

)
− F2

(
rf2(r), r

√
1− f22 (r)

))
≡ 0

we obtain

d1,0g2(r) + d0,1g4(r) + (d2,0 − d0,2)g3(r) + d1,1g5(r) ≡ 0.

Since d1,1 6= 0, the function g5 can be expressed in terms of g2, g3 and g4. Thus
item (b) follows. �

Proof of Theorem 1.3 (a). For n = 2, by Lemma 3.3 we have to prove that the
4 functions that define (3.3) form an ECT-system in a suitable interval. Let us
prove first that, under our hypotheses it is not restrictive to write the quadratic
polynomial F2 that defines C2, as

F2(x, y) = x2 +Ky2 − y = 0. (3.4)

In general, F2(x, y) =
∑2
i+j=0 di,jx

iyj . Since it passes through the origin, d0,0 = 0.

Moreover, the symmetry condition F2(−x, y) = F2(x, y) implies that d1,0 = d1,1 =
0. Additionally d0,1 6= 0, because otherwise F2 would be a homogeneous conic.
Multiplying F2 by a constant, if necessary, we can assume that d0,1 = −1, and the
convexity condition forces that d2,0/d0,1 < 0. In short, F2(x, y) = L2x2+d0,2y

2−y.
Finally, changing the scale, that is taking x1 = λx and y1 = λy for a suitable λ, we
arrive again to a piecewise linear system as (1.1), but with F2 as in (3.4).

To study MC2
, when F2(x, y) = x2 + Ky2 − y, it is convenient to consider

separately three cases: K < 1, K = 1, and K > 1. For the first case, we write
K = 1− d2, for d > 0. Then we choose C2 to be a branch of the quadratic curve

F2(x, y) = x2 + (1− d2)y2 − y = 0,

which is either a parabola if d = 1, or an ellipse if 0 < d < 1, or a hyperbola if
d > 1. Notice that C2 at the origin satisfies y′(0) = 0, y′′(0) > 0, and it is locally
convex.



10 A. GASULL, J. TORREGROSA, X. ZHANG EJDE-2020/19

For this curve C2, we have that θ1(r) = arccos(f1(r)) and we can compute f1(r)

as follows. Using that (x, y) =
(
rf1(r), r

√
1− f21 (r)

)
, we obtain

F2

(
rf1(r), r

√
1− f21 (r)

)
= 0.

Thus, r
(
(1 − d2)r + d2rf21 (r) −

√
1− f21 (r)

)
= 0. Solving this last expression we

obtain

f1(r) =

√
2d4r2 − 2d2r2 − 1 +

√
4d2r2 + 1√

2d2r
.

To simplify the computations, we can use a new variable u instead of r,

r = Φ(u)
.
=

1− u2

4du
,

to avoid the inner square root in f1, because 4d2r2 + 1 = (1 + u2)2/(4u2). Now,
0 < u < 1 when C2 is a parabola (d = 1) or a branch of hyperbola (d > 1), and
u ∈ (ud, 1) if it is an ellipse (0 < d < 1), with ud = (1 − d)/(1 + d). With this
change, we get

F1(u)
.
= f1 (Φ(u)) =

∆(u, d)

d(u+ 1)

with

∆(u, d) =
√

((d+ 1)u+ d− 1)((d− 1)u+ d+ 1).

Notice that both factors inside the square root are positive for all values of d, but
restricted to the respective intervals of definition. When d = 1 this can be easily
seen by direct substitution. Otherwise, this becomes apparent by writing

((d+ 1)u+ d− 1)((d− 1)u+ d+ 1) = (d2 − 1)
(
u− 1− d

1 + d

)(
u− 1 + d

1− d

)
.

With this new variable, the Melnikov function M̃C2
(u)

.
= MC2

(Φ(u)) is written
as

M̃C2(u) =
1

Φ(u)

3∑
k=0

βkHk(u),

with

H0(u) = 1, H1(u) = arccos
( ∆(u, d)

d(1 + u)

)
,

H2(u) =
4u∆(u, d)

(1− u)(1 + u)2
, H3(u) =

(1− u)∆(u, d)

d2(1 + u)2
.

From these functions we can compute the ordered list of Wronskians,

W [H0](u) = 1,

W [H0, H1](u) = − 2

(1 + u)∆(u, d)
,

W [H0, H1, H2](u) =
16(d2(u+ 1)4 + 2(u− 1)4)

(u+ 1)5(u− 1)3∆2(u, d)
,

W [H0, H1, H2, H3](u) =
2048

(1 + u)6(u− 1)3∆3(u, d)
.



EJDE-2020/19 PIECEWISE LINEAR SYSTEMS WITH ALGEBRAIC SEPARATION 11

Thus, the proof in this case (K = 1 − d2) finishes because these Wronskians are
non-vanishing and, by Proposition 3.1, the above 4 functions form an ECT-system
in the respective intervals of definition.

For the third case, K > 1, we write K = 1 + d2, for d > 0. Then we choose C2

to be a piece of the quadratic curve

F2(x, y) = x2 + (1 + d2)y2 − y = 0,

which is always an ellipse. In this situation

f1(r) =

√
2d4r2 + 2d2r2 − 1 +

√
−4d2r2 + 1√

2d2r
,

and here, to simplify the computations, it is convenient to introduce the new vari-
able v as,

r = Ψ(v)
.
=

2v

v2 + 4d2
, 0 < v < 2d,

also to avoid the inner square root in f1. Doing similar computations, that we omit
for the safe of brevity, we also prove that the corresponding set with 4 functions is
an ECT in (0, 2d).

Finally, the case K = 1 is easier because f1(r) =
√

1− r2, and it is straight-
forward to prove that the functions hk(r), k = 0, 1, 2, 3, given in (3.3) form an
ECT-system in (0, 2d).

Hence for all K ∈ R it is possible to find a Melnikov function MC2
(r) of the

form (3.3) with 3 simple zeros in any given interval in the domain of definition of
the function. These zeros give rise to the 3 limit cycles stated in the theorem and
prove that L2 ≥ 3. This lower bound will be improved in the proof of item (b). �

Proof of Theorem 1.3 (b) and (c). We will study in terms of Cn, and for δ > 0
small enough, whether t he first 5 or all the 6 functions that define (3.2) form an
ECT-system for r ∈ (0, δ).

We will choose C3 to be a piece of the cubic curve

(x− y)2 + (1− d2)y2 + ax3 − y = 0, (3.5)

that passes trough the origin. Notice that when a = 0, it is either a parabola if
d = 1, or an ellipse if 0 < d < 1, or a hyperbola if d > 1. When a 6= 0 it is a
proper cubic. This curve always has at the origin a horizontal tangent point, it is
not symmetric with respect to the y-axis and it is locally convex.

As in the proof of item (a) of the theorem, the functions fk(r), k = 1, 2, that
appear in the Melnikov function given in Proposition 3.2, can be obtained by substi-

tuting (x, y) = (rf, r
√

1− f2 ) in (3.5). Then, they are the two solutions, f = fk(r)
for k = 1, 2, of equation

r2(d2 − 1)f2 − r(2rf + 1)
√

1− f2 + r2(2− d2) + ar3f3 = 0,

that satisfy f1(0) = 1 and f2(0) = −1, respectively. The series expansion of f1 is
written as

f1(r) =1− 1

2
r2 + (2− a)r3 − 1

8
(4a2 − 8d2 − 32a+ 57)r4

+ (3ad2 + 2a2 − 8d2 − 14a+ 24)r5

+
1

16
(48a2d2 − 40d4 − 384ad2 − 84a2 + 728d2 + 704a− 1249)r6 +O

(
r7
)
,



12 A. GASULL, J. TORREGROSA, X. ZHANG EJDE-2020/19

and it can be seen that f2(r) = −f1(−r).
Then we can obtain the corresponding series expansions of the six functions

defined in (3.2),

g0(r) =1,

g1(r) =− π + 2r −
(

2d2 + 4a− 31

3

)
r3 +

(
4d4 − 2a2d2 + 24ad2

− 57d2 − 3a2 − 22a+
1543

20

)
r5 +O

(
r7
)
,

g2(r) =
2

r
− r +

(
2d2 − 57

4

)
r3 −

(
5d4 − 91d2 +

1249

8

)
r5 +O

(
r7
)
,

g3(r) =− 4r +
(

12d2 − 26
)
r3 −

(
40d4 − 230d2 +

347

2

)
r5 +O

(
r7
)
,

g4(r) =8r3 −
(

32d2 − 92
)
r5 +O

(
r7
)
,

g5(r) =2r −
(

2d2 + 4a− 9
)
r3 +

(
4d4 − 2a2d2 + 24ad2 − 53d2

− 7a2 + 2a+
163

4

)
r5 +O

(
r7
)
.

Next we compute all the ordered Wronskians,

W0(r) =1,

W1(r) =2−
(
6d2 + 12a− 31

)
r2 +O

(
r4
)
,

W2(r) =8r−3 −
(
48d2 − 248

)
r−1 +O(r),

W3(r) =
(
1536d2 − 1024

)
r−3 +

(
−30720d4 + 111360d2 − 18432

)
r−1

+O(r),

W4(r) = −
(
983040d2 + 15532032

)
r−2

+ 33030144d4 + 509214720d2 − 1666842624 +O
(
r2
)
,

W5(r) =37748736a
(

70d4a5 + 105a3d6 + 35ad8 + 420a6d2 − 1050d4a4

− 1120d6a2 − 140d8 − 4970a5d2 + 5145d4a3 + 3640d6a

+ 630a6 + 25116d2a4 − 7406a2d4 − 3500d6 − 8820a5

− 73160a3d2 − 10349d4a+ 50582a4 + 119172a2d2

+ 24036d4 − 141180a3 − 67274ad2 + 204636a2 − 27252d2

− 173848a+ 91336
)

+O(r2).

Notice that since the coefficients of r−1 and r−3 in W3 have no common roots as
polynomials in d, the Wronskian W3 is clearly non-vanishing in a neighborhood of
the origin. Then, when a = 0, for all d, the first 5 Wronskians do not vanish in an
open neighborhood of the origin. Moreover, by item (b) of Lemma 3.3, since F2 has
the monomial xy, we already knew that g5 is a linear combination of the previous
4 functions and 5 is the maximum number of Wronskians that can be non-zero.

On the other hand, when a 6= 0 and for most values of d (it suffices that the
huge polynomial that gives the first order terms of W5(r) does not vanish) we know
that the six functions gk, k = 0, 1, . . . , 5, form an ECT-system in a small enough
neighborhood of r = 0, as we wanted to prove.



EJDE-2020/19 PIECEWISE LINEAR SYSTEMS WITH ALGEBRAIC SEPARATION 13

In particular, when a = 0 (respectively, a 6= 0) it is possible for all d (respectively,
for most d) to find a Melnikov functionMCn(r) of the form (3.2) with 4 (respectively,
5) simple zeros in any small enough given interval (0, ρ). Moreover, the functions
gi and Wi, i = 0, . . . , 5 have a convergent series in a small enough neighborhood
of the origin and the size of such interval is completely independent of the interval
[r0, β] in Proposition 3.2. Hence, we can choose adequately the value of r0 in such a
way that the desired zeros of MCn

(r) are in (r0, ρ). Consequently, the simple zeros
of MCn(r) give rise to the limit cycles stated in the theorem. �

Acknowledgments. This work has received funding from the Agencia Estatal de
Investigación (Grant MTM2016-77278-P (FEDER)), the Agència de Gestió d’A-
juts Universitaris i de Recerca (Grant 2017 SGR 1617), the NNSF China (Grants
11671254 and 11871334) and the European Unions Horizon 2020 research and in-
novation programme (Grant Dynamics-H2020-MSCA-RISE-2017-777911).

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Armengol Gasull

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra,

Barcelona, Catalonia, Spain
Email address: gasull@mat.uab.cat

Joan Torregrosa

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra,
Barcelona, Catalonia, Spain

Email address: torre@mat.uab.cat

Xiang Zhang

School of Mathematical Sciences, Key Laboratory of Scientific and Engineering Com-

puting (Ministry of Education), Shanghai Jiao Tong University, Shanghai 200240, China
Email address: xzhang@sjtu.edu.cn


	1. Introduction and statement of main results
	2. Proof of main results
	Proof of Theorem ??
	Proof of Proposition ??.

	3. Preliminary results and proof of Theorem ??.
	Acknowledgments

	References