The planar discontinuous piecewise linear refracting systems have at most one limit cycle
Introduction
In the qualitative theory of the differential systems in the plane one of the most important problems is the determination and distribution of limit cycles, which is known as the famous Hilbert’s 16-th problem [1], [2] and its weak form [3], [4], [5], [6].
Since many real world differential systems involve a discontinuity or a sudden change [7], in recent years there is a growing interest in the following planar piecewise smooth vector fields where the discontinuity boundary divides the plane into two regions . The equilibrium of are called real or virtual if or , respectively.
Clearly the orbits are well defined in both zones . While if an orbit arrives the discontinuous boundary , different things can occur.
Definition 1 Let . Then we can classify into the following three open regions: crossing region , see Fig. 1.1. attracting region , see Fig. 1.2; escaping region , see Fig. 1.3.
The boundaries of the above three regions are called tangential point, that is . If an isolated periodic orbit of systems (1) has sliding points, then it is called a sliding limit cycle, otherwise we call it a crossing limit cycle.
The most simplest piecewise smooth differential systems are the piecewise linear differential systems with a straight line of separation. Without loss of generality we can assume that the separating straight line is , then we have where the dot denotes the derivative with respect to the time . We call systems (2) with (resp. ) the left (resp. right) subsystems for convenience.
In 2012 Freire, Ponce and Torres [8] reduced the study of the planar piecewise linear differential systems (2) to the following Liénard canonical forms where and denote the traces and determinants of the left and right subsystems, respectively.
If , then systems (3) become continuous differential systems. In 1990 Lum and Chua [9] did the following conjecture:
Conjecture 2 Planar continuous piecewise linear differential systems (3) have at most one limit cycle.
In 1998 Freire et al. [10] proved Conjecture 2 by qualitative analysis. Recently, Li and Llibre [11] provided the global phase portraits in the Poincaré disc of the planar continuous piecewise linear differential systems (3).
For the discontinuous systems (3) most of the known results [12], [13], [14], [15], [16], [17], [18], [19] are concerned with the lower bounds of the number of limit cycles. According to the equilibrium of left and right subsystems (3), we can classify systems (3) into six types, see Table 1. And for instance, when in the table we intersect the column with the row , and we obtain the number 2, this means that discontinuous systems (3) having in a linear saddle and in a node the maximum number of limit cycles that we know for such systems is 2. The case is the same as the case , which already appears with a in the table.
From Table 1, it appeared the following conjecture:
Conjecture 3 Planar discontinuous piecewise linear differential systems (3) have at most three crossing limit cycles.
As far as we known Conjecture 3 is still open and there were only several partial results for this conjecture. Llibre, Novaes and Teixeira [20], [21] proved that discontinuous systems (3) have at most two crossing limit cycles when . Giannakopoulos and Pliete [22] showed that discontinuous systems (3) with a symmetry have at most two crossing limit cycles. In [23] it is proved that if one of the subsystems (3) has a center then the maximum number of crossing limit cycles is two, and that this upper bound is sharp.
Definition 4 If for all , then systems (1) are known as refracting systems.
It is obvious that the whole discontinuous line of a refracting system is a crossing region. There are several papers classifying the generic equilibrium of the refracting systems, for dimension two see [24]; for dimension three see [25]; for dimension four see [26] and for arbitrary dimension see [27].
If then systems (3) become planar discontinuous piecewise linear refracting systems and have been studied in several papers [14], [15], [18], [19], [28]. All the previous results shown that the planar discontinuous piecewise linear refracting systems (3) of types have at most one limit cycle, see Table 2. More precisely we have
- •
see Theorems 3.4 and 3.5 of [14].
- •
see Theorem 3.1 of [15].
- •
see Theorem 1 of [28], or Theorem 3.1 of [19].
- •
see Theorem 3.1 of [18].
The dynamics of the planar discontinuous piecewise linear differential systems (3) are determined by We define the modal parameters where . Then the planar discontinuous piecewise linear refracting systems (3) can be written into the following normal forms where and For a proof of these normal forms see [12].
Remark 5 Systems (4) for have a focus; for and have a node; for and have a saddle; and for have an improper node.
Section snippets
Statements of the main results
It follows from Table 2 that the upper bounds for the maximum number of limit cycles of the planar discontinuous piecewise linear refracting systems (3) of type focus-node or focus–focus are still unknown. In the present paper we investigated the number of limit cycles for the above two remaining unsolved types. We shall use the normal forms (4) instead of (3) because the former one has only four parameters. Without loss of generality we can assume that the left subsystem of (4) has a focus.
Preliminary results
We recall the following results on the existence and uniqueness of limit cycles for planar discontinuous piecewise linear differential systems without sliding regions proved in [30].
Consider the following piecewise linear differential systems
Definition 12 We say that a point is a monodromic singularity of systems (13) if either is a tangential point, or a singularity of one of the subsystems of (13), and there exists a neighborhood of such
Proof of Theorem 6
It is obvious that if the right subsystems of (8) have a real node, then refracting systems (8) cannot have limit cycles. Thus a necessary condition for the existence of limit cycles of systems (8) is that .
We divide the proof of Theorem 6 into two cases.
Case 1: . Then the left subsystems of (8) have an virtual focus when , and an equilibrium on when . Doing the change of variables then the refracting systems (8) become
Proof of Theorem 7
For a refracting system (9) the left Poincaré map is also given by Lemma 15, and the right Poincaré map can be stated as follows, see Proposition 7 of [12].
Lemma 17 Assume that and . The parameter representation of the right Poincaré map of systems (9) is where . Moreover, ; ; . ; . has as an asymptote.
Proof (i) From (16) we have A
Proof of Theorem 10
In this section we only consider the unsolved case: . Without loss of generality we can assume that , otherwise doing the change of variables , we change into the former one. From (7) we know that the number of crossing limit cycles of refracting systems (10) are in correspondence with the negative zeros of the function .
We define the function According to statement (III) of Proposition 14, it is
CRediT authorship contribution statement
Shimin Li: Theoretically analysis the proof of Theorems 6 and 7, Writing - original draft. Changjian Liu: Theoretically analysis the proof of Theorem 10. Jaume Llibre: Revise manuscript according to reviewers’ suggestions.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We thank the reviewers for their valuable comments and suggestions that helped us to improve the presentation of our results.
The first author is partially supported by the National Natural Science Foundations of China (No. 12071091) and the Natural Science Foundation of Guangdong Province (No. 2019A1515011885).
The second author is partially supported by the National Natural Science Foundations of China (No. 11771315).
The third author is partially supported by the MINECO grants MTM2013-40998-P
References (31)
- et al.
Hilbert’s 16th problem for classical Liénard equations of even degree
J. Differential Equations
(2008) - et al.
Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis
J. Differential Equations
(2018) - et al.
Phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line
J. Differential Equations
(2019) - et al.
On hopf bifurcation in non-smooth planar systems
J. Differential Equations
(2010) - et al.
Existence of limit cycles in general planar piecewise linear systems of saddle-saddle dynamics
Nonlinear Anal.
(2013) - et al.
On the number of limit cycles in general planar piecewise linear systems of node-node types
J. Math. Anal. Appl.
(2014) - et al.
The number and stability of limit cycles for planar piecewise linear systems of node-saddle type
J. Math. Anal. Appl.
(2019) - et al.
Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type
Nonlinear Anal. Hybrid Syst.
(2019) - et al.
Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center
J. Math. Anal. Appl.
(2018) - et al.
Generic bifurcation of refracted systems
Adv. Math.
(2013)
The boundary focus-saddle bifurcation in planar piecewise linear systems. Application to the analysis of memristor oscillators
Nonlinear Anal. Ser. B: Real World Appl.
Hilbert’s 16th problem and bifurcations of planar polynomial vector fields
Internat. J. Bifur. Chaos Appl. Sci. Engrg.
New lower bounds for the Hilbert numbers using reversible centers
Nonlinearity
The cyclicity of period annuli of degenerate quadratic hamiltonian systems with elliptic segement loops
Ergodic Theory Dynam. Systems
The infinitestimal 16th Hilbert problem in the quadratic case
Invent. math.
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