The planar discontinuous piecewise linear refracting systems have at most one limit cycle

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Abstract

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.

The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has 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 X(q)=X(q)if h(q)<0,X+(q)if h(q)>0,where the discontinuity boundary Σ={qR2:h(q)=0} divides the plane R2 into two regions Σ±={qR2:±h(q)>0}. The equilibrium p± of X± are called real or virtual if p±Σ± or p±Σ, respectively.

Clearly the orbits are well defined in both zones Σ±. While if an orbit arrives the discontinuous boundary Σ, different things can occur.

Definition 1

Let X±h(q)=h(q),X±(q). Then we can classify Σ into the following three open regions:

  • (i)

    crossing region Σc={qΣ:X+h(q)Xh(q)>0}, see Fig. 1.1.

  • (ii)

    attracting region Σa={qΣ:X+h(q)>0,Xh(q)<0}, see Fig. 1.2;

  • (iii)

    escaping region Σe={qΣ:X+h(q)<0,Xh(q)>0}, see Fig. 1.3.

The boundaries Σt of the above three regions are called Σtangential point, that is Σt={qΣ:X+h(q)Xh(q)=0}. 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 x=0, then we have ẋẏ=a1,1a1,2a2,1a2,2xy+b1b2if x<0,a1,1+a1,2+a2,1+a2,2+xy+b1+b2+if x>0,where the dot denotes the derivative with respect to the time t. We call systems (2) with x<0 (resp. x>0) 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 ẋẏ=T1D0xy0aif x<0,T+1D+0xyba+if x>0,where T± and D± denote the traces and determinants of the left and right subsystems, respectively.

If b=0,a=a+, 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 S with the row N, and we obtain the number 2, this means that discontinuous systems (3) having in x<0 a linear saddle and in x>0 a node the maximum number of limit cycles that we know for such systems is 2. The case NS is the same as the case SN, which already appears with a 2 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 a+a=0. Giannakopoulos and Pliete [22] showed that discontinuous systems (3) with a Z2 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 X+h(q)=Xh(q) for all qΣ, then systems (1) are known as refracting systems.

It is obvious that the whole discontinuous line Σ{(0,0)} 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 b=0 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 SS,NN,FS,SN have at most one limit cycle, see Table 2. More precisely we have

  • SS see Theorems 3.4 and 3.5 of [14].

  • NN see Theorem 3.1 of [15].

  • FS see Theorem 1 of [28], or Theorem 3.1 of [19].

  • SN see Theorem 3.1 of [18].

The dynamics of the planar discontinuous piecewise linear differential systems (3) are determined by Δ±=(T±)24D±.We define the modal parameters m{R,L}=iif Δ±<0,0if Δ±=0,1if Δ±>0,where i2=1. Then the planar discontinuous piecewise linear refracting systems (3)|b=0 can be written into the following normal forms ẋẏ=2γL1γL2mL20xy0αLif x<0,2γR1γR2mR20xy0αRif x>0,where α{R,L}=2a±|Δ±|if Δ±0,a±if Δ±=0,and γ{R,L}=T±|Δ±|if Δ±0,T±2if Δ±=0.For a proof of these normal forms see [12].

Remark 5

Systems (4) for m=i have a focus; for m=1 and |γ|1 have a node; for m=1 and |γ|<1 have a saddle; and for m=0 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 ẊẎ=μ1μ210XY+μ00if Y<0,μ1+μ2+10XY+μ0+0if Y>0.

Definition 12

We say that a point pΣ={Y=0} is a Σmonodromic singularity of systems (13) if either p is a tangential point, or a singularity of one of the subsystems of (13), and there exists a neighborhood of p 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 αR<0.

We divide the proof of Theorem 6 into two cases.

Case 1: αL0. Then the left subsystems of (8) have an virtual focus when αL>0, and an equilibrium on Σ when αL=0. Doing the change of variables X=2γLxy,Y=x,if x<0, orX=2γRxy,Y=x,if x>0,then the refracting systems (8) become ẊẎ=2γL(γL2+1

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 αR<0 and γR<0. The parameter representation of the right Poincaré map PR(z) of systems (9) is z=αReγRt1+γRtγR2t,PR(z)=αReγRt1γRtγR2t,where t0. Moreover,

  • (i)

    limzPR(z)=z2=αRγR; limz0PR(z)=1; limzPR(z)=0.

  • (ii)

    PR(z)<0; PR(z)<0.

  • (iii)

    PR(z) has z=z2 as an asymptote.

Proof

(i) From (16) we have limzPR(z)=limt+αReγRt1γRtγR2t=αRγR.A

Proof of Theorem 10

In this section we only consider the unsolved case: γLγR<0,αL<0<αR. Without loss of generality we can assume that γL<0,γR>0, otherwise doing the change of variables X=x,Y=y,T=t, we change γL>0,γR<0 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 PL1(z)PR(z).

We define the function f(γ,t)=eγtφγ(t)=eγtcost+γsint.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

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