1. Introduction
The study of limit cycles of differential systems in
(i.e., periodic orbits of a differential system in
isolated in the set of all periodic orbits of that system) goes back essentially to Poincaré [
1] at the end of the nineteenth century and their existence became important in application due to their relation with real world phenomena, see for instance the limit cycle of van der Pol equation [
2,
3], or the one of the Belousov–Zhavotinskii model [
4,
5].
Continuous piecewise linear differential systems separated by straight lines appear naturally in control theory (see, for instance, Refs. [
6,
7,
8,
9,
10,
11]). The easiest continuous piecewise linear differential systems are the ones formed by two linear differential systems separated by a straight line and for such systems it is well known that one is the upper bound on the number of limit cycles that they can have and that this upper bound is reached (see, for instance, Refs. [
12,
13,
14,
15] and the references therein).
The unique linear differential systems which are Hamiltonian are linear centers and linear saddles. In [
16] the authors obtained the maximum number of limit cycles of continuous and discontinuous piecewise differential systems formed by linear centers and separated by either one or two parallel straight lines. In the present paper we do a symmetric study for continuous and discontinuous piecewise differential systems formed by linear Hamiltonian saddles and separated by either one or two parallel straight lines.
In the following two theorems we prove that if the continuous differential systems are formed by linear Hamiltonian saddles, then if they are separated by either one straight line (Theorem 1) or by two parallel straight lines (Theorem 2), they do not have limit cycles.
Theorem 1. A continuous piecewise differential system separated by one straight line and formed by two linear Hamiltonian saddles does not have limit cycles.
Theorem 2. A continuous piecewise differential system separated by two parallel straight lines and formed by three linear Hamiltonian saddles does not have limit cycles.
The study of discontinuous piecewise linear differential systems separated by straight lines goes back to Andronov et al. [
17]. Nowadays, they have attracted the attention of many authors mainly because these systems appear in mechanics, electrical circuits, economy, etc. (see, for instance, the books [
18,
19], the surveys [
20,
21] and the references therein). To provide an explicit example, consider a planar Coulomb friction damping vibration system of the form
where
x is the displacement of the oscillator mass
m,
k is the stiffness coefficient of spring,
is the coefficient Coulomb friction,
g is the gravitational acceleration, and
is the sign function of relative sliding speed
for
and
. Note that this model is equivalent to
which is a discontinuous planar piecewise linear differential system separated by the straight line
.
In planar discontinuous piecewise linear differential systems, we can have two kinds of limit cycles: The sliding limit cycle and the crossing limit cycle. A sliding limit cycle contains a segment of the discontinuity lines, and a crossing limit cycle only contains isolated points of the discontinuity lines. In the present paper we only study crossing limit cycles, but for simplicity we shall call them limit cycles instead of crossing limit cycles.
The easiest discontinuous piecewise differential systems are the ones formed by two linear differential systems and separated by a straight line. There are examples of such systems with three limit cycles, but it is not known if three is the maximum number of limit cycles that such systems can exhibit (see [
22,
23,
24,
25,
26,
27]).
We first show, as for the continuous systems, that discontinuous piecewise differential systems formed by linear Hamiltonian saddles and separated by one straight line do not have limit cycles.
Theorem 3. A discontinuous piecewise differential system separated by one straight line and formed by two linear Hamiltonian saddles does not have limit cycles.
Theorem 3 is proven in
Section 5. It can be extended to discontinuous systems separated by two parallel straight lines but in this case the upper bound on the number of limit cycles is one and this upper bound is reached. Thus, we have proved the extension of the 16th Hilbert problem on the maximum number of limit cycles for the polynomial differential systems of a given degree to the class of discontinuous piecewise differential systems formed by three linear Hamiltonian saddles separated by two parallel straight lines.
Theorem 4. A discontinuous piecewise differential system separated by two parallel straight lines and formed by three linear Hamiltonian saddles can have at most one limit cycle. Moreover, there are systems in this class with one limit cycle, see Figure 1. Theorem 4 is proven in
Section 6. We remark that it is clear from the proof of Theorem 4 that there are systems in the statement of Theorem 4 that do not have limit cycles.
The paper is organized in such a way that in
Section 2, before the proof of the main theorems, we provide a normal form for an arbitrary differential system with linear Hamiltonian saddles.
3. Proof of Theorem 1
Take a continuous piecewise differential system separated by one straight line and formed by two linear Hamiltonian saddles. Without loss of generality and taking into account the symmetry of the problem we can assume that the straight line of continuity is
. It follows from Proposition 1 that we can assume that the systems in
and
are written in the form (
2).
We have system
in
with the first integral
and system
in
with the first integral
Since the piecewise differential system is continuous, both systems must coincide on and so , , and .
Note that if the continuous piecewise differential system has a limit cycle taking into account that the two differential systems are linear Hamiltonian saddles, this system must have a periodic orbit intersecting the discontinuity line
in exactly two points, namely
and
with
. Since
and
are two first integrals, we have that
that is
So all periodic orbits of these systems are in a continuum of periodic orbits yielding the non-existence of limit cycles. This completes the proof of the theorem.
4. Proof of Theorem 2
Assume that we have a continuous piecewise differential system separated by two parallel straight lines and formed by three linear Hamiltonian saddles. Without loss of generality and due to the symmetry of the problem we can assume that the straight lines of discontinuity are
and
. It follows from Proposition 1 that we can assume that the systems in
,
and
are written as in (
2).
We have system (
6) with first integral (
7) in
, system (
8) with first integral (
9) in
, and system
with first integral
in
.
Since the piecewise differential system is continuous, systems (
6) and (
8) must coincide in
, and systems (
8) and (
11) must coincide in
. Doing so we obtain
Note that if the continuous piecewise differential system has a limit cycle taking into account that the two differential systems are linear Hamiltonian saddles, this system must have a periodic orbit intersecting each discontinuity line
in exactly two points, namely
,
,
and
, with
and
. Since
,
and
are three first integrals, we have that
The solutions
of these last systems satisfying the necessary condition
and
are
where
. Note that we only have two solutions taking the upper signs of
or the lower signs of
. Hence all periodic orbits of these systems are in a continuum of periodic orbits yielding the non existence of limit cycles. This completes the proof of the theorem.
5. Proof of Theorem 3
Assume that we have a discontinuous piecewise differential system separated by one straight line and formed by two linear Hamiltonian saddles. Without loss of generality and due to the symmetry of the problem we can assume that the straight line of continuity is
. It follows from Proposition 1 that we can assume that the systems in
and
are written in the form (
2).
We have system (
6) with first integral (
7) in
, and system (
8) with first integral (
9) in
.
Note that, if the discontinuous piecewise differential system has a limit cycle taking into account that the two differential systems are linear Hamiltonian saddles, this system must have a periodic orbit intersecting the discontinuity line
in exactly two points, namely
and
with
. Since
and
are two first integrals we have that (
10) must be satisfied, that is
The solutions
of (
14) satisfying the condition
either do not exist if
, or there is a continuum of solutions. So the periodic orbits of the discontinuous piecewise linear differential systems are in a continuum of periodic orbits, and consequently this differential system has no limit cycles. This completes the proof of the theorem.
6. Proof of Theorem 4
Take a discontinuous piecewise differential system separated by two parallel straight lines and formed by three linear Hamiltonian saddles. Without loss of generality and due to the symmetry of the problem we can assume without loss of generality that the straight lines of discontinuity are
and
. It follows from Proposition 1 that we can assume that the systems in
,
and
are written as in (
2).
We have system (
6) with first integral (
7) in
, system (
8) with first integral (
9) in
, and system (
11) with Hamiltonian (
12) in
. Note that if the discontinuous piecewise differential system has a limit cycle taking into account that the two differential systems are linear Hamiltonian saddles, this system must have a periodic orbit intersecting each discontinuity line
in exactly two points, namely
,
,
and
, with
and
. Since
,
and
are three first integrals, we have that system (
13) must be satisfied. Doing so we get
Assume first that
the solutions of Equation (
15) are
,
and
,
being
functions in the variables
and
, respectively. In this case all periodic orbits of the these systems are in a continuum of periodic orbits yielding the non-existence of limit cycles.
Assume now that
and
the solutions of Equation (
15) are
,
and
,
being
functions in the variable
, respectively. In this case all periodic orbits of these systems are in a continuum of periodic orbits yielding the non-existence of limit cycles.
Assume now that
and
, the solutions of Equation (
15) are
,
, and
,
being
functions in the variable
, respectively. In this case, all periodic orbits of the these systems are in a continuum of periodic orbits yielding the non-existence of limit cycles.
Finally, assume that
. The solution of the first and third equations are (
17) and (
16). Introducing these solutions into the second and fourth equations in (
15) we get
and
Taking
and solving in
we get
where
whenever
. The case with
yields
. Introducing this into
and solving in
, we obtain
, which is not possible. So, we assume that
. Now introducing
into the first equation in (
18) and solving in
we get
where
whenever
and if
then there is at most one solution
.
When
, since
there is at most one solution with
and
. In summary, an upper bound for the number of limit cycles is one.
To complete the proof of Theorem 4 we provide an example of a system in this class with one limit cycle. This will complete the proof of Theorem 4.
The Hamiltonians of the three linear Hamiltonian systems with a saddle are
where the Hamiltonian system in the half-plane
is
the Hamiltonian system in the strip
is
and the Hamiltonian system in the half-plane
is
These three linear differential systems are saddles because the determinants of their linear part are , and , respectively.
The discontinuous piecewise differential system formed by the three linear differential systems (
19), (
20) and (
21) in order to have one limit cycle intersecting the two discontinuous straight lines
at the points; these points must satisfy system (
13), and this system has a unique solution satisfying that
and
, namely
Drawing the corresponding limit cycle associated to this solution we obtain the limit cycle of
Figure 1.