Abstract
In this note we consider the classical Massera theorem, which proves the uniqueness of a periodic solution for the Liénard equation
and investigate the problem of the existence of such a periodic solution when f is monotone increasing for
MSC: 34C05, 34C25.
Keywords:
Liénard equation; limit cycles; Massera theorem1 Preliminaries and wellknown results
The problem of the existence and uniqueness of a periodic solution for the Liénard equation
has been widely investigated in the literature. Among the uniqueness results, a special place is, without any doubt, taken by the classical Massera theorem. This is due to the simple geometrical ideas and the fact that this result, despite several efforts, is in most cases no more valid for the generalized Liénard equation
For related results concerning the more general equation
we refer to [1].
Equation (1.1) is equivalent to the phaseplane system
and to the Liénard system
We present the classical Massera theorem which is a milestone among the results of limit cycles uniqueness for system (1.2).
Theorem 1.1 (Massera [2])
System (1.2) has at most one limit cycle which is stable, and hence equation (1.1) has at most one nontrivial periodic solution which is stable, provided thatfis continuous and
1.
2.
The theorem of Massera improved a previous result due to Sansone [3] in which there was the additional assumption
We observe that in his paper Massera was proving the uniqueness of limit cycles, regardless of the existence, because only the monotonicity properties and the continuity were required. It is well known that in order to fulfill the necessary conditions for the existence of limit cycles, the only two cases to be considered are the following:
A
B
This is because if we consider the distance from the origin, namely
Hence the distance from the origin increases if f is negative and decreases if f is positive, and a necessary condition for the existence of limit cycles is that
As for [A], in a very recent paper [5], the monotonicity assumptions, which are required in the whole line, were relaxed
to the interval
Theorem 1.2 ([5])
The Liénard system (1.3) has exactly one limit cycle, which is stable, provided thatfis continuous and
1. if
2. if
wherea, bare the zeros of
Let us consider case [B]. Now the existence of the limit cycle is no more granted, and, as far as we know, this case was not explicitly investigated in the literature.
The aim of this note is to discuss such a situation. In the next section, we present
some sufficient conditions for the existence, and hence the uniqueness, of the limit
cycle as well as a necessary condition. Such a condition shows that actually there
are situations in which there is no limit cycle despite the monotonicity properties
and the fact that
2 Existence of the limit cycle
We consider the case
B
The case ‘
At first we work in the phaseplane
and we give a constructive result based on the methods used in [8].
Theorem 2.1The phaseplane system (1.2) has exactly one limit cycle, which is stable, provided thatfis continuous and
1.
2.
3.
4.
Proof We just notice that the continuity of f guarantees the property of uniqueness for the solutions to the Cauchy problem associated
to system (1.2), and therefore the trajectories of such a system cannot intersect.
Consider the 0isocline of system (1.2), that is, the points in the phaseplane were
For these points, we have
and that
In
This means that in
Therefore, if
Figure 1. Trajectory
We observe that if we keep all the sign assumptions and the limits at infinity, but not the monotonicity, clearly the uniqueness is no more granted, but the existence result still holds, and one can relax the assumption
with
as in [8].
Hence we have the following corollary.
Corollary 2.2The phaseplane system (1.2) has at least a stable limit cycle provided thatfis continuous and
1.
2.
As already mentioned, the first result is based on the direct construction of a winding trajectory.
A more powerful approach can be adopted in the Liénard plane, and it is based on the
results presented in [9], where the problem of intersection with the vertical isocline
where
where as usual
The following result holds.
Theorem 2.3The Liénard system (1.3) has exactly one limit cycle, which is stable, provided thatfis continuous and
1.
2.
3.
4.
Proof As already mentioned, the continuity of f guarantees the property of uniqueness for the solutions to the Cauchy problem associated to system (1.3), and therefore the trajectories of such a system cannot intersect.
As the proof is based on different applications of both necessary and sufficient conditions
for the intersection of a semitrajectory with the vertical isocline
As well as in the phaseplane, for any point
and
With respect to the intersection of
and
As in [9], we define
and, finally, we introduce the function
For this particular case, we have the following result.
Theorem 2.4 ([[9], Theorem 2.5.])
If, for every
Theorem 2.5 ([[9], Theorem 2.3.])
If there is a constant
then, for every
Let us see that hypothesis (2.15) implies that there exists
In our assumptions, for
A straightforward calculation yields
therefore, choosing
Then, for
At last
Then limit (2.22) is verified provided that
ϵ and η being arbitrary, the existence of
As the necessary condition for the intersection of the negative semitrajectory with
the vertical isocline is denied, there exists a point
Figure 2. Trajectory
If we relax the monotonicity assumptions, we lose uniqueness, but existence still holds. Hence we have the following corollary.
Corollary 2.6The Liénard system (1.3) has at least a stable limit cycle provided thatfis continuous and
1.
2.
Now we consider the case of nonexistence of limit cycles. We consider again the results
of [9] and, in particular, the case in which the semitrajectory
Theorem 2.7The Liénard system (1.3) has no limit cycles provided thatfis continuous and
1.
2.
3.
4.
Proof At first we need the following results.
Theorem 2.8 ([[9], Theorem 2.6.])
If there is a constant
then, for every
Theorem 2.9 ([[9], Theorem 2.2])
If, for every
As in the previous theorem, the assumptions (2.28) and (2.29) become respectively
where α is the nonnull zero of F.
With the same calculation, which we omit for the sake of simplicity, we get that if
assumptions hold, we can provide a trajectory which is unwinding and goes to infinity
without intersecting the curve
Figure 3. Trajectory
Now assume, by contradiction, that there exists a limit cycle. The monotonicity property
of the Massera theorem guarantees that such a limit cycle is unique and stable. Therefore
there are trajectories surrounding the limit cycle which are winding and going to
such a cycle. But with the same calculation already applied, which we omit for the
sake of simplicity, if assumptions (2.26) and (2.27) hold, we can provide a large
trajectory which is unwinding and goes to infinity without intersecting the curve
But as already mentioned, the monotonicity properties of the Massera theorem give the uniqueness, and hence in this situation we can conclude that there is no limit cycle and all trajectories are unwinding. □
We observe that there are cases which still are not investigated. For instance, if
we consider
we can produce a trajectory which lies entirely below the curve
In order to describe in full the possible configurations, we consider the following example.
Example 2.10 Let f be continuous, monotone decreasing for
Let
Define
If
This is because if
We can describe the movement of such a cycle. At
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally in writing this paper.
Acknowledgements
We thank the anonymous referees for their valuable suggestions and Dr. Francesco Della Santa for the figures.
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