Crossing limit cycles of nonsmooth Liénard systems and applications
aa r X i v : . [ m a t h . D S ] O c t Crossing limit cycles of nonsmooth Li´enard systems andapplications ∗ Tao Li , Hebai Chen , Xingwu Chen †
1. Department of Mathematics, Sichuan University,Chengdu, Sichuan 610064, P. R. China2. School of Mathematics and Statistics, Central South University,Changsha, Hunan 410083, P. R. China
Abstract
Continuing the investigation for the number of crossing limit cycles of nonsmooth Li´enardsystems in [Nonlinearity (2008), 2121-2142] for the case of a unique equilibrium, in thispaper we consider the case of any number of equilibria. We give results about the existenceand uniqueness of crossing limit cycles, which hold not only for a unique equilibrium butalso for multiple equilibria. Moreover, we find a sufficient condition for the nonexistence ofcrossing limit cycles. Finally, applying our results we prove the uniqueness of crossing limitcycles for planar piecewise linear systems with a line of discontinuity and without slidingsets. Keywords: discontinuity, Li´enard systems, limit cycles, piecewise linear systems
A class of important dynamical systems is the Li´enard system, which is originated from physicsand then is applied to engineering, biology, chemistry and more fields. As usual, the Li´enardsystem can be written as ( ˙ x = F ( x ) − y, ˙ y = g ( x ) , (1.1)where F ( x ) := R x f ( s ) ds , f ( x ) and g ( x ) are two scalar functions. For system (1.1), a mainand challenging subject is to study the existence, uniqueness and number of limit cycles, i.e.,isolated periodic orbits in the phase space. The 13th Smale’s problem provided in [30] is aboutthe maximum number of limit cycles in the polynomial system of form ˙ x = F ( x ) − y, ˙ y = x ,i.e., the famous 16th Hilbert’s problem restricted to the polynomial system of form (1.1) with g ( x ) = x , and is still open. When system (1.1) is smooth, the investigation of the existence,uniqueness and number of limit cycles has a long history and many excellent results are obtainedas given in journal papers [9, 11–13, 25] and text book [34]. ∗ Supported by NSFC 11871355 and 11801079, Graduate Student’s Research and Innovation Fund of SichuanUniversity 2018YJSY047. † Author to whom any correspondence should be addressed. Email address: [email protected] (X.Chen). n the other hand, many models in practical problems are established by system (1.1) in anonsmooth form such as ( ˙ x = ax + bx − y, ˙ y = x − sgn( x ) , (1.2)which is the limit case of a smooth oscillator (see [5, 6]). Another system ˙ x = T x − y, ˙ y = Dx − ( − D · X ref if x < , − D · X ref if x > T, D, X ref are parameters.Motivated by practical problems, many mathematicians and engineers have started to investi-gate the existence, uniqueness and number of limit cycles for the nonsmooth Li´enard system(1.1). Usually, the nonsmoothness leads to much difficulty in the analysis of nonsmooth Li´enardsystems and much less results are obtained(see,e.g., [7, 23, 26, 28]), compared with the smoothcase.Consider the Li´enard system (1.1) with nonsmooth functions f ( x ) := (cid:26) f + ( x ) if x > ,f − ( x ) if x < , g ( x ) := (cid:26) g + ( x ) if x > ,g − ( x ) if x < , where f ± ( x ) , g ± ( x ) : R → R are smooth functions. Hence, the nonsmoothness or even disconti-nuity of system (1.1) occurs only on y -axis, the switching line . Let F ± ( x ) := R x f ± ( s ) ds . Werewrite the nonsmooth Li´enard system (1.1) as( ˙ x, ˙ y ) = ( (cid:0) F + ( x ) − y, g + ( x ) (cid:1) if x > , (cid:0) F − ( x ) − y, g − ( x ) (cid:1) if x < . (1.4)For convenience, we call the subsystem in x > x < right system and left system of(1.4), respectively. Since F ± (0) = 0, we always define the x -component of the vector field of(1.4) as − y on y -axis, i.e., ˙ x = − y . If g + (0) = g − (0), the y -component of the vector field of(1.4) is defined as g + (0) on y -axis, i.e., ˙ y = g + (0). Then (1.4) is continuous. If g + (0) = g − (0),the y -component of the vector field of (1.4) has a jump discontinuity on y -axis, which impliesthat (1.4) is a discontinuous system. Thus we define the solution of (1.4) passing through apoint in y -axis by the Filippov convention(see [14, 24]). In fact, observe that the switching lineconsists of the origin O and two crossing sets , i.e., the positive y -axis and the negative y -axis.Let q be a point in y -axis. If q belongs to the positive (resp. negative) y -axis, then the solutionof (1.4) passing through q crosses y -axis at q from right (resp. left) to left (resp. right). If q liesat O , it is proved in [28, Proposition 1] that q is a boundary equilibrium when g + (0) g − (0) = 0,a pseudo-equilibrium when g + (0) g − (0) < g + (0) g − (0) >
0. Thus anequilibrium of (1.4) has three types, including boundary equilibria and pseudo-equilibria lyingin x = 0, regular equilibria lying in x = 0. In the third case, q is also called a parabolic fold-foldpoint (see [3]) and, more precisely, both orbits of the left and right systems passing through q are quadratically tangent to the y -axis from the left half plane when g + (0) > , g − (0) > g + (0) < , g − (0) < x ≥ x ≤
0, orpresents intersections with both x > x <
0. Since the former one can be determined byone of the right and left systems, our attention is paid to the latter one, i.e., crossing limit cycle (see [24]). For (1.4), there exist a few results on the existence, uniqueness and number of crossinglimit cycles, such as [7, 23, 26, 28]. In [28], a necessary condition of the existence of crossing limitcycles and a sufficient condition for the uniqueness are given. In [23], a sufficient condition ofthe existence and uniqueness of crossing limit cycles is presented. The number of crossing limitcycles is studied in [7, 26]. Unfortunately, we observe that almost in all publications (1.4) isrequired to satisfy g + ( x ) > x > , g − ( x ) < x < . (1.5)Condition (1.5) implies that the origin O is the unique equilibrium of (1.4) and, hence, anycrossing limit cycle surrounds O as indicated in [28]. However, in many practical problemssystem (1.4) may have multiple equilibria that lie in the left and right half planes, such assystems (1.2) and (1.3), and then a crossing limit cycle can surround multiple equilibria. Onthe existence, uniqueness and number of crossing limit cycles of (1.4) with multiple equilibria,as far as we know, the results are lacking and there are only a few results restricted to concretemodels (see e.g., [6]). For smooth Li´enard systems with multiple equilibria, some results of limitcycles have been given in [11, 12, 33, 34].The goal of this paper is to study the existence, nonexistence and uniqueness of crossing limitcycles for the nonsmooth Li´enard system (1.4) with any number of equilibria. We particularlyemphasize the case of multiple equilibria. In order to state our main results, we give some basichypotheses for system (1.4) as the following (H1) - (H3) . (H1) There exists a constant x e ≥ g + ( x )( x − x e ) > x > x = x e . (H2) f + ( x ) > x > f − ( x ) < x < p = p ( x ) := ( F + ( x ) if x ≥ ,F − ( x ) if x < . (1.6)Clearly, p ( x ) is continuous and p (0) = 0 due to F ± (0) = 0. Moreover, p ′ ( x ) = f + ( x ) > x > p ′ ( x ) = f − ( x ) < x < (H2) . Thus p ( x ) is strictly increasing for x > x <
0, implying p ( x ) ≥ x ∈ R and p ( x ) has a strictly increasinginverse function x + ( p ) : [0 , p + ) → [0 , + ∞ ) (1.7)and a strictly decreasing inverse function x − ( p ) : [0 , p − ) → ( −∞ , , (1.8)where p ( x ) → p ± ( x → ±∞ ). It follows from the monotonicity that p + (resp. p − ) is either aconstant or infinity. 3 H3)
There exist the limits lim p → + g ± ( x ± ( p )) f ± ( x ± ( p )) = η ± (1.9)and −∞ < η + ≤ η − < + ∞ . Moreover, g + /f + | x = x + ( p ) < g − /f − | x = x − ( p ) for all sufficientlysmall p > η + = η − .Under hypothesis (H1) , in the right half plane system (1.4) has no equilibria if x e = 0 anda unique equilibrium if x e >
0. Moreover, the unique equilibrium in the right half plane lies at( x e , F + ( x e )). Let E be the point lying at ( x e , F + ( x e )) if x e > O if x e = 0. The numberof equilibria in the left half plane is not determined by (H1) - (H3) . In other word, system(1.4) in the left half plane may have any number of equilibria. On the y -axis, system (1.4)has no equilibria if g + (0) g − (0) > O if g + (0) g − (0) ≤ f + ( x ) = 1 , f − ( x ) = − , g + ( x ) = x − a and g − ( x ) = 2 x + a with a ≥ (H1) - (H3) .In the following, we state our main results. Theorem 1.1.
If system (1.4) with (H1)-(H3) has a crossing periodic orbit Γ , then (i) Γ surrounds O and E counterclockwise; (ii) the equations F − ( x − ) = F + ( x + ) , g − ( x − ) f − ( x − ) = g + ( x + ) f + ( x + ) (1.10) have at least one solution ( x − , x + ) = ( x −∗ , x + ∗ ) with x −∗ < < x + ∗ satisfying that Γ transversallyintersects both lines x = x −∗ and x = x + ∗ . Theorem 1.1 provides two necessary conditions for the existence of crossing periodic orbitsfor system (1.4). These two necessary conditions help us to determine the configuration ofcrossing periodic orbits, that is, any crossing periodic orbit neither lie in each side of the line x = x e nor lie in the strip ˆ x −∗ < < ˆ x + ∗ , where (ˆ x −∗ , ˆ x + ∗ ) is the solution of (1.10) satisfyingthat ˆ x −∗ < < ˆ x + ∗ is the narrowest strip. On the other hand, Theorem 1.1 can be regarded asa generalization of [28, Theorem 2] from one equilibrium to multiple equilibria. It is requiredin [28] that system (1.4) satisfies the condition (1.5), implying that O is the unique equilibriumof (1.4) and any crossing periodic orbit surrounds O . However, in this paper we replace (1.5) bythe weaker hypothesis (H1) , which allows (1.4) to have multiple equilibria and crossing limitcycles surrounding multiple equilibria. For example, by Theorem 1.1 the crossing periodic orbitΓ surrounds at least two equilibria ( O and E ) when x e > g + (0) g − (0) ≤
0. Therefore, ourresult holds not only for a unique equilibrium, but also for multiple equilibria.
Theorem 1.2.
For system (1.4) with (H1)-(H3) , assume that the equations in (1.10) have aunique solution ( x − , x + ) = ( x −∗ , x + ∗ ) with x −∗ < < x + ∗ and one of the following hypothesesholds: (H4) x + ∗ ≥ x e and F + ( x ) f + ( x ) /g + ( x ) is increasing for x > x + ∗ ; H5) K − ( x − ( p )) < K + ( x + ( p )) for all p , p satisfying p > p ≥ F + ( x + ∗ ) , where K ± ( x ) := ( g ± ( x )) ′ f ± ( x ) − ( f ± ( x )) ′ g ± ( x )( f ± ( x )) and x ± ( p ) are given in (1.7) and (1.8) respectively.Then system (1.4) has at most one crossing periodic orbit, which is a stable and hyperboliccrossing limit cycle if it exists. Theorem 1.2 is a result about the uniqueness of crossing limit cycles for system (1.4). Inthe aspect of the number of equilibria, Theorem 1.2 can be regarded as a generalization of [28,Theorem 3] from one equilibrium to multiple equilibria because the weaker hypothesis (H1) than (1.5) allows (1.4) to have multiple equilibria as mentioned before. In the aspect of thesmoothness of systems, Theorem 1.2 can be regarded as a generalization of [11, Theorems 2.1and 2.2] from smooth Li´enard systems to nonsmooth ones.In the following theorem we give a sufficient condition for the existence of periodic annuli forsystem (1.4). It can be looked as a sufficient condition for the nonexistence of isolated crossingperiodic orbits, i.e., crossing limit cycles.
Theorem 1.3.
For system (1.4) with (H2) , assume that (1.9) holds and g + ( x ) f + ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = x + ( p ) ≡ g − ( x ) f − ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = x − ( p ) (1.11) for all p > . If system (1.4) has a crossing periodic orbit, then there exists a periodic annulusincluding this crossing periodic orbit, i.e., there is no crossing limit cycles. To apply our main results of system (1.4), we study the number of crossing limit cycles forthe piecewise linear system ˙ z = ( A + z + b + if x > ,A − z + b − if x < , (1.12)where z = ( x, y ) ⊤ ∈ R , A ± = (cid:18) a ± a ± a ± a ± (cid:19) ∈ R × , b ± = (cid:18) b ± b ± (cid:19) ∈ R . In this paper we always assume that system (1.12) is nondegenerate , i.e., det A ± = 0. System(1.12) has been widely used as a model in engineering, physics and biology (see [1, 17, 31]), andmany contributions have been made in recent years (see [4, 15, 16, 19, 21, 22, 32]). Although thetwo subsystems of (1.12) are linear, the switching of the vector fields in different regions leadsto great complexity and difficulty in the research on the number of crossing limit cycles. When(1.12) is continuous, it is proved in [19] that there exists at most one crossing limit cycle andthis number can be reached. When (1.12) is discontinuous, in many publications examples wereprovided for (1.12) to have three crossing limit cycles, such as [4,15,16,22]. However, the problem5f maximum number of crossing limit cycles for discontinuous system (1.12) is still open. On theother hand, we checked in all papers presenting examples with three crossing limit cycles andfound that all these systems have sliding sets , namely { (0 , y ) : ( a +12 y + b +1 )( a − y + b − ) < } 6 = ∅ .Therefore, a natural question is how about the maximum number of crossing limit cycles fordiscontinuous system (1.12) without sliding sets . This question was answered in [28, 29] for thecase that O is a Σ-monodromic singularity, i.e., all orbits in a small neighborhood of O of (1.12)turn around O , and the maximum number is 1. Besides, it was proved in [20] (resp. [18]) thatthe maximum number is also 1 for focus-saddle type (resp. for focus-focus type with partialregions of parameters). However, the problem of the maximum number of crossing limit cyclesfor general discontinuous (1.12) without sliding sets is still open. Applying our main results forsystem (1.4), we completely answer this open problem in the following theorem. Theorem 1.4.
If discontinuous system (1.12) has no sliding sets, then there exists at most onecrossing limit cycle and this number can be reached. Moreover, it is possible that the number ofequilibria surrounded by this crossing limit cycle is exactly k (1 ≤ k ≤ . The remainder of this paper is organized as follows. In Section 2 we give proofs of The-orems 1.1, 1.2 and 1.3 for nonsmooth Li´enard system (1.4). In Section 3 we apply the mainresults for system (1.4) to study the number of crossing limit cycles of system (1.12) and proveTheorem 1.4.
The purpose of this section is to provide the proofs of Theorems 1.1, 1.2 and 1.3. Firstly, wedescribe some geometrical properties of a crossing periodic orbit of system (1.4) with (H1) .Figure 1: An illustration of geometrical properties of Γ.
Lemma 2.1.
If system (1.4) with (H1) has a crossing periodic orbit Γ , then (i) Γ intersects the curve y = F + ( x ) ( resp. y = F − ( x )) at a unique point in x > resp. x < , denoted by A ( resp. C ) as in Figure 1 ; intersects any line x = l satisfying x C < l < x A at exactly two points, where x A and x C are the abscissas of A and C , respectively; (iii) Γ surrounds O and E counterclockwise.Proof. According to the third paragraph in Section 1, O is an equilibrium when g + (0) g − (0) ≤ g + (0) g − (0) >
0. This implies that Γ cannot pass through O . Moreover, ˙ x > <
0) for all ( x, y ) in the below (resp. above) of y = p ( x ), where p ( x )is defined in (1.6). Therefore, conclusions (i) and (ii) hold and Γ surrounds O counterclockwise.Let B (resp. D ) be the intersection of Γ and the positive y -axis (resp. the negative y -axis) andΓ := Γ AB ∪ Γ BC ∪ Γ CD ∪ Γ DA , see Figure 1. If x e = 0, conclusion (iii) obviously holds because E lies at O . If x e >
0, it follows from (H1) that the curve corresponding to Γ DA (resp. Γ AB )goes down for 0 < x < x e and goes up for x e < x < x A as t increases. Thus Γ also surrounds E counterclockwise, implying the conclusion (iii). Proof of Theorem 1.1.
Under hypotheses, if system (1.4) has a crossing periodic orbit Γ,then it satisfies the geometrical properties in Lemma 2.1 and we still use the denotations in theproof of Lemma 2.1, see Figure 2(a). Conclusion (i) is obtained directly from Lemma 2.1(iii). (cid:451)(cid:451) (a) (b)Figure 2: Γ in xy -plane and γ in py -plane.In order to prove conclusion (ii), we apply the change p = p ( x ) to system (1.4), where p ( x )is defined in (1.6). Thus the right and left systems of (1.4) are transformed into ( ˙ p = f + ( x + ( p ))( p − y ) , ˙ y = g + ( x + ( p )) , ( ˙ p = f − ( x − ( p ))( p − y ) , ˙ y = g − ( x − ( p )) , (2.1)respectively, from which we get dydp = ϕ + ( p ) p − y := g + ( x + ( p )) f + ( x + ( p ))( p − y ) , dydp = ϕ − ( p ) p − y := g − ( x − ( p )) f − ( x − ( p ))( p − y ) (2.2)for p >
0, respectively. Here x + ( p ) (resp. x − ( p )) given in (1.7) (resp. (1.8)) is the inversefunction of p = p ( x ) for x ≥ x < p = 0 due to the continuities of f ± , g ± at x = 0 and (H3) . Thus we always assumethat (2.1) and (2.2) are well defined for p ≥
0. Moreover, under the change p = p ( x ), the crossingperiodic orbit Γ becomes the orbit γ := γ AB ∪ γ BC ∪ γ CD ∪ γ DA in py -plane, see Figure 2(b),where γ DA ∪ γ AB and γ BC ∪ γ CD are the orbits of the first system and the second one in (2.1),respectively.Let ∆ + (resp. ∆ − ) be the region surrounded by y -axis and Γ DA ∪ Γ AB (resp. Γ BC ∪ Γ CD ),Ω + (resp. Ω − ) be the region surrounded by y -axis and γ DA ∪ γ AB (resp. γ BC ∪ γ CD ), S (Ω ± ) bethe areas of Ω ± . By Green’s formula we have0 = Z Γ DA ∪ Γ AB − g + ( x ) dx + ( F + ( x ) − y ) dy + Z Γ BC ∪ Γ CD − g − ( x ) dx + ( F − ( x ) − y ) dy = Z Z ∆ + f + ( x ) dxdy + Z Z ∆ − f − ( x ) dxdy − Z BD − g + ( x ) dx + ( F + ( x ) − y ) dy − Z DB − g − ( x ) dx + ( F − ( x ) − y ) dy = Z Z ∆ + f + ( x ) dxdy + Z Z ∆ − f − ( x ) dxdy = Z Z Ω + dpdy − Z Z Ω − dpdy = S (Ω + ) − S (Ω − ) . (2.3)Thus γ BC ∪ γ CD must cross γ DA ∪ γ AB at some p ∈ (0 , min { p A , p C } ), where p A and p C are theabscissas of points A and C in py -plane. Otherwise, from (H3) we get that γ BC (resp. γ DA )always lies below γ AB (resp. γ CD ). This means that S (Ω + ) − S (Ω − ) >
0, contradicting (2.3).Suppose that ϕ + ( p ) < ϕ − ( p ) for 0 < p < min { p A , p C } , then y AB ( p ) > y BC ( p ) and y CD ( p ) >y DA ( p ) for 0 < p < min { p A , p C } by applying the theory of differential inequalities to systems in(2.2), where y = y AB ( p ), y = y BC ( p ), y = y CD ( p ) and y = y DA ( p ) describe the orbits γ AB , γ BC , γ CD and γ DA , respectively. Thus γ BC ∪ γ CD does not cross γ DA ∪ γ AB , implying a contradiction.Consequently, ϕ + ( p ) = ϕ − ( p ) has at least one solution in 0 < p < min { p A , p C } , denoted by p ∗ . Choosing x + ∗ := x + ( p ∗ ) and x −∗ := x − ( p ∗ ), we finally obtain that the equations (1.10) haveat least one solution ( x − , x + ) = ( x −∗ , x + ∗ ) with x −∗ < < x + ∗ satisfying that Γ transversallyintersects both the verticals x = x ±∗ , i.e., conclusion (ii) is proved. Proof of Theorem 1.2.
The essential idea of this proof comes from [11, 28] and it is accom-plished by two steps. Assume that system (1.4) has a crossing periodic orbit Γ := ( x ( t ) , y ( t ))and define λ Γ := Z Γ − f − ( x ( t )) dt + Z Γ + f + ( x ( t )) dt, where Γ + := Γ ∩ { ( x, y ) : x ≥ } and Γ − := Γ ∩ { ( x, y ) : x ≤ } . In the first step, we prove λ Γ <
0, which implies that Γ is a stable and hyperbolic crossing limit cycle by [10, Theorem2.1]. In the second step, we prove that system (1.4) cannot have two stable and hyperbolic limitcycles in succession, which implies the uniqueness of crossing limit cycles associated with theresult of the first step.
Step 1 . We prove λ Γ <
0. 8ollowing the denotations and geometric properties of Γ in Lemma 2.1 and Theorem 1.1, wefirstly prove p C > p A . In fact, under the assumption of theorem, ϕ + ( p ) = ϕ − ( p ) has a uniquesolution p ∗ ∈ (0 , min { p A , p C } ). Moreover, ϕ + ( p ) < ϕ − ( p ) for 0 < p < p ∗ and ϕ + ( p ) > ϕ − ( p ) for p > p ∗ . Applying the theory of differential inequalities to systems in (2.2), we obtain y AB ( p ) > y BC ( p ) , y CD ( p ) > y DA ( p ) for 0 < p < p ∗ . (2.4)From the proof of Theorem 1.1 γ BC ∪ γ CD must cross γ DA ∪ γ AB . Without loss of generality,assume that γ BC ∪ γ CD and γ DA ∪ γ AB have a crossing point at a value p for which γ AB crosses γ BC . Then, it follows from (2.4) and ϕ + ( p ) > ϕ − ( p ) for p > p ∗ that p ∗ ≤ p < min { p A , p C } and y AB ( p ) > y BC ( p ) for 0 < p < p ,y AB ( p ) < y BC ( p ) for p < p < min { p A , p C } . (2.5)Here the theory of differential inequalities is applied in the second inequality. Moreover, sincethe number of crossing points of γ BC ∪ γ CD and γ DA ∪ γ AB must be even from (2.4), we furtherobtain that there exists a value p with p ∗ ≤ p < min { p A , p C } such that γ CD crosses γ DA at p . Similarly, y CD ( p ) > y DA ( p ) for 0 < p < p ,y CD ( p ) < y DA ( p ) for p < p < min { p A , p C } . (2.6)Hence, combining with (2.5) and (2.6), we get p C ≥ p A .On the other hand, we have ϕ + ( p ) > ϕ − ( p ) > > p − y AB ( p ) > p − y BC ( p ) for p ∗ ≪ p < p A . Thus dy AB ( p ) − dy BC ( p ) dp = ϕ + ( p ) p − y AB ( p ) − ϕ − ( p ) p − y BC ( p ) < ϕ + ( p ) p − y BC ( p ) − ϕ − ( p ) p − y BC ( p )= ϕ + ( p ) − ϕ − ( p ) p − y BC ( p ) < p with p ∗ ≪ p < p A , so that y AB ( p ) − y BC ( p ) is strictly decreasing in p ∗ ≪ p < p A andthen p C = p A , i.e., p C > p A .Let M and N (resp. P and Q ) be the intersections of Γ and the vertical x = x −∗ (resp. x = x + ∗ ). Then we denote Γ byΓ = Γ DP ∪ Γ P A ∪ Γ AQ ∪ Γ QB ∪ Γ BN ∪ Γ NC ∪ Γ CM ∪ Γ MD as shown in Figure 3(a) and γ which corresponds with Γ under the change p = p ( x ) by γ = γ DP ∪ γ P A ∪ γ AQ ∪ γ QB ∪ γ BN ∪ γ NC ∪ γ CM ∪ γ MD as shown in Figure 3(b), where γ DP ∪ γ P A ∪ γ AQ ∪ γ QB and γ BN ∪ γ NC ∪ γ CM ∪ γ MD are theorbits of the first system and the second one in (2.1), respectively. Therefore, λ Γ = Z Γ DP ∪ Γ QB f + ( x ) dt + Z Γ BN ∪ Γ MD f − ( x ) dt + Z Γ PA ∪ Γ AQ f + ( x ) dt + Z Γ NC ∪ Γ CM f − ( x ) dt = Z γ DP ∪ γ QB dpp − y + Z γ BN ∪ γ MD dpp − y + Z γ PA ∪ γ AQ dpp − y + Z γ NC ∪ γ CM dpp − y (2.7)9a) (b)Figure 3: The eight orbit arcs of Γ in xy -plane and the ones of γ in py -plane.due to ˙ p = f ± ( x ± ( p ))( p − y ) in (2.1). For brevity, we neglect the variable t of x, y and p in (2.7)and the rest of this proof if confusion does not arise.Firstly, we prove J := Z γ DP ∪ γ QB dpp − y + Z γ BN ∪ γ MD dpp − y < . (2.8)Let y = y DP ( p ), y = y QB ( p ) , y = y BN ( p ) and y = y MD ( p ) for 0 < p ≤ p ∗ describe γ DP , γ QB , γ BN and γ MD , respectively. Then p − y DP ( p ) > , p − y MD ( p ) > , p − y BN ( p ) < , p − y QB ( p ) < . Moreover, from (2.5) and (2.6) we have y BN ( p ) < y QB ( p ) , y MD ( p ) > y DP ( p ) (2.9)for 0 < p < p ∗ . Hence, J = Z p ∗ dpp − y DP ( p ) + Z p ∗ dpp − y QB ( p ) + Z p ∗ dpp − y BN ( p ) + Z p ∗ dpp − y MD ( p )= Z p ∗ y DP ( p ) − y MD ( p )( p − y DP ( p ))( p − y MD ( p )) dp + Z p ∗ y BN ( p ) − y QB ( p )( p − y BN ( p ))( p − y QB ( p )) dp< , i.e., (2.8) holds.Secondly, we prove J := Z γ PA ∪ γ AQ dpp − y + Z γ NC ∪ γ CM dpp − y < , (2.10)implying λ Γ = J + J < µ := p A − p ∗ p C − p ∗ , η := ( p C − p A ) p ∗ p C − p ∗
10s in [11]. Clearly, 0 < µ < p C > p A > p ∗ and η = (1 − µ ) p ∗ . By the lineartransformation ( e p = µp + η := ψ ( p ) , e y = µy + η := φ ( y ) , the second system in (2.1) is transformed into ˙ e p = f − (cid:18) x − (cid:18) e p − ηµ (cid:19)(cid:19) ( e p − e y ) , ˙ e y = µg − (cid:18) x − (cid:18) e p − ηµ (cid:19)(cid:19) . (2.11)Denote the orbit of (2.11) corresponding with γ NC ∪ γ CM by e γ NC ∪ e γ CM . Since ψ ( p C ) = p A , φ ( p C ) = p A , ψ ( p ∗ ) = p ∗ and φ ( p ∗ ) = p ∗ , the orbit e γ NC ∪ e γ CM is from ( p ∗ , µy NC ( p ∗ ) + η ) to( p ∗ , µy CM ( p ∗ ) + η ) after passing through A , see Figure 3(b). Thus J = Z γ PA ∪ γ AQ dpp − y + Z e γ NC ∪ e γ CM d e p e p − e y = Z p A p ∗ dpp − y P A ( p ) + Z p ∗ p A dpp − y AQ ( p ) + Z p A p ∗ dpp − e y NC ( p ) + Z p ∗ p A dpp − e y CM ( p )= Z p A p ∗ y P A ( p ) − e y CM ( p )( p − y P A ( p ))( p − e y CM ( p )) dp + Z p A p ∗ e y NC ( p ) − y AQ ( p )( p − y AQ ( p ))( p − e y NC ( p )) dp, (2.12)where y = e y NC ( p ) and y = e y CM ( p ) describe e γ NC and e γ CM , respectively. In order to prove J <
0, from (2.12) it is sufficient to prove y P A ( p ) − e y CM ( p ) < , e y NC ( p ) − y AQ ( p ) < p ∗ ≤ p < p A because p − y P A ( p ) > , p − e y CM ( p ) > , p − y AQ ( p ) < , p − e y NC ( p ) < φ ( y ) − y = (1 − µ )( p ∗ − y ), p ∗ − y CM ( p ∗ ) > p ∗ − y NC ( p ∗ ) <
0. Thus e y CM ( p ∗ ) − y CM ( p ∗ ) = φ ( y CM ( p ∗ )) − y CM ( p ∗ ) = (1 − µ )( p ∗ − y CM ( p ∗ )) > , e y NC ( p ∗ ) − y NC ( p ∗ ) = φ ( y NC ( p ∗ )) − y NC ( p ∗ ) = (1 − µ )( p ∗ − y NC ( p ∗ )) < < µ <
1. Using (2.9) and (2.14), we get y P A ( p ∗ ) = y DP ( p ∗ ) ≤ y MD ( p ∗ ) = y CM ( p ∗ ) < e y CM ( p ∗ ) ,y AQ ( p ∗ ) = y QB ( p ∗ ) ≥ y BN ( p ∗ ) = y NC ( p ∗ ) > e y NC ( p ∗ ) , (2.15)i.e., (2.13) holds for p = p ∗ .To prove (2.13) for p ∗ < p < p A , we consider system (2.11) without tildes and the firstsystem in (2.2) for p ∗ < p < p A . We can rewrite them as dydp = h ( p, y ) := ϕ − (( p − η ) /µ )( p − η ) /µ · p − ηp − y (2.16)and dydp = H ( p, y ) := ϕ + ( p ) p · pp − y , (2.17)11espectively. We only prove the first inequality in (2.13) when (H4) or (H5) holds, and thesecond one can be treated analogously. Thus p − y > (H4) . Then ϕ + ( p ) /p is decreasing in p ∗ < p < p A .Moreover, since 0 < µ < η = (1 − µ ) p ∗ >
0, we have ( p − η ) /µ > p for p > p ∗ . Thus ϕ − (( p − η ) /µ )( p − η ) /µ < ϕ + (( p − η ) /µ )( p − η ) /µ ≤ ϕ + ( p ) p (2.18)due to ϕ − ( p ) < ϕ + ( p ) for p > p ∗ . According to (H4) , we get x + ∗ ≥ x e , so that g + ( x ) /f + ( x ) > x > x + ∗ by (H1) and (H2) , i.e., ϕ + ( p ) > p ∗ < p < p A . Hence, for p ∗ < p < p A and p − y > H ( p, y ) > h ( p, y ) < ϕ + ( p ) p · p − ηp − y = H ( p, y ) · p − ηp < H ( p, y ) , (2.19)where (2.18) and the fact that p > η > y P A ( p ∗ ) < e y CM ( p ∗ ) as in (2.15), if thereexists ¯ p with p ∗ < ¯ p < p A such that y P A (¯ p ) = e y CM (¯ p ), we obtain from (2.19) that y P A ( p ) > e y CM ( p ) for ¯ p < p ≤ p A by applying the theory of differential inequalities to systems (2.16) and(2.17). This contradicts the fact that y P A ( p A ) = e y CM ( p A ), and consequently, y P A ( p ) < e y CM ( p )for p ∗ < p < p A , i.e., the first inequality of (2.13) holds under (H4) .Now assume that system (1.4) satisfies (H5) . Considering the function G ( p ) := µϕ − (cid:18) p − ηµ (cid:19) − ϕ + ( p )for p ∗ < p < p A , we obtain G ′ ( p ) = K − (cid:18) x − (cid:18) p − ηµ (cid:19)(cid:19) − K + ( x + ( p )) < (H5) and ( p − η ) /µ > p . Moreover, since η = (1 − µ ) p ∗ and ϕ + ( p ∗ ) = ϕ − ( p ∗ ), G ( p ∗ ) = µϕ − (cid:18) p ∗ − ηµ (cid:19) − ϕ + ( p ∗ ) = µϕ − ( p ∗ ) − ϕ + ( p ∗ ) = ( µ − ϕ + ( p ∗ ) . Combining with ϕ + ( p ∗ ) = g + ( x + ∗ ) /f + ( x + ∗ ) and 0 < µ <
1, we further have G ( p ∗ ) ≤ x + ∗ ≥ x e and G ( p ∗ ) > < x + ∗ < x e by (H1) and (H2) . In the first case, G ( p ) < h ( p, y ) − H ( p, y ) = G ( p ) p − y < p ∗ < p < p A and p − y >
0, implying that y P A ( p ) − e y CM ( p ) < p ∗ < p < p A by a same analysis with the last paragraph. In the second case, G ( p ) has at most one zeropoint in p ∗ < p < p A . When G ( p ) has no zero points, G ( p ) > p ∗ < p < p A , so that h ( p, y ) − H ( p, y ) > p − y >
0. By the theory of differential inequalities, it directlyfollows from y P A ( p ∗ ) < e y CM ( p ∗ ) that y P A ( p ) − e y CM ( p ) < p ∗ < p < p A . When G ( p ) has azero point, denoted by q , we get h ( p, y ) − H ( p, y ) > p ∗ < p < q and h ( p, y ) − H ( p, y ) < q < p < p A . Thus y P A ( p ) − e y CM ( p ) < p ∗ < p ≤ q and, by a same analysis with the lastparagraph y P A ( p ) − e y CM ( p ) < q < p < p A . In conclusion, the first inequality of (2.13) alsoholds under (H5) . 12 tep 2 . We prove the uniqueness of crossing periodic orbits.
Assume that system (1.4) has two adjacent crossing periodic orbits Γ and Γ . By Theo-rem 1.1, both Γ and Γ surround O and E . Moreover, it follows from Step 1 and [10, Theorem2.1] that both Γ and Γ are stable and hyperbolic crossing limit cycles. Let A be the openregion surrounded by Γ and Γ . Consider the α -limit set L of the orbit of (1.4) with someinitial value ( x , y ) ∈ A having Γ as the ω -limit set. Similar to the smooth case [11], by thePoincar´e-Bendixson Theorem in nonsmooth dynamical systems (see [2]) and the special struc-ture of (1.4), L must consist of an equilibrium (¯ x, p (¯ x )) ∈ A and an unstable homoclinic orbitto (¯ x, p (¯ x )) which cuts the switching line x -axis. On the other hand, since (H1) holds and allcrossing periodic orbits surround E , we get ¯ x <
0, i.e., (¯ x, p (¯ x )) is an equilibrium of the leftsystem. Thus, from (H2) we have div( F − ( x ) − y, g − ( x )) | x =¯ x = f − (¯ x ) <
0. This contradictsthat L is unstable by [8, Theorem 1], which not only holds for hyperbolic saddles, in fact, butalso holds for semi-hyperbolic ones. That is, L cannot be α -limit set of the orbit of (1.4) withthe initial value ( x , y ). Finally, we conclude that (1.4) has at most one crossing periodic orbit.Combining with Steps 1 and 2, we complete the proof, that is, system (1.4) has at most onecrossing periodic orbit, which is a stable and hyperbolic crossing limit cycle if it exists. Proof of Theorem 1.3.
As in the proof of Theorem 1.1, by the change p = p ( x ) we cantransform the right and left systems of (1.4) into the systems in (2.1) and then get differentialequations in (2.2) for p >
0. The differential equations in (2.2) can be continuously extendedto p = 0 by defining ϕ ± (0) = η ± due to (1.9). Additionally, from (1.11) we have η + = η − and ϕ + ( p ) ≡ ϕ − ( p ) for all p ≥
0, implying that the two equations in (2.2) coincide for p ≥
0. Hence,any orbit of the first differential equations in (2.2) going from a point in the negative y -axis toa point in the positive y -axis corresponds with a crossing periodic orbit of (1.4). Conversely,the existence of crossing periodic orbits of (1.4) ensures that the first differential equations in(2.2) have an orbit going from a point in the negative y -axis to a point in the positive y -axis.Consequently, if (1.4) has a crossing periodic orbit Γ, then all orbits in the neighborhood of Γare crossing periodic orbits by the continuous dependence of solutions on initial values, i.e., theproof is completed. In this section we apply Theorems 1.1, 1.2 and 1.3 to study the number of crossing limit cyclesfor discontinuous system (1.12). In particular, the proof of Theorem 1.4 will be presented.According to [17], system (1.12) has no crossing limit cycles for a +12 a − ≤ x -component of both vector fields has same sign on crossing sets. For a +12 a − >
0, it is provedin [17, Proposition 3.1] that (1.12) is C -homeomorphic to the Li´enard canonical form (cid:18) ˙ x ˙ y (cid:19) = (cid:18) t R − d R (cid:19) (cid:18) xy (cid:19) − (cid:18) − ba R (cid:19) if x > , (cid:18) t L − d L (cid:19) (cid:18) xy (cid:19) − (cid:18) a L (cid:19) if x < , (3.1)13here t { R,L } and d { R,L } are the traces and determinants of A ± , b = a − a +12 b +1 − b − , a L = a − b − − a − b − , a R = a − a +12 ( a +12 b +2 − a +22 b +1 ) . Although (1.12) and (3.1) are not Σ-equivalent, there exists a topological equivalence for alltheir orbits without sliding segments as indicated in [17]. This means that crossing limit cyclesof (1.12) are transformed into crossing limit cycles of (3.1) in a homeomorphic way. Therefore,in order to study the existence, uniqueness and number of crossing limit cycles of (1.12), we onlyneed to consider (3.1).
Theorem 3.1.
Assume that system (3.1) satisfies b = 0 , t L < , t R > , d L > , d R > . (3.2)(i) If a R /t R > a L /t L , then a necessary condition for the existence of crossing periodic orbits is d R /t R > d L /t L . In addition, if there exists a crossing periodic orbit, then it is unique andstable. (ii) If a R /t R < a L /t L , then a necessary condition for the existence of crossing periodic orbitsis d R /t R < d L /t L . In addition, if there exists a crossing periodic orbit, then it is uniqueand unstable. (iii) If a R /t R = a L /t L , then a necessary condition for the existence of crossing periodic orbitsis d R /t R = d L /t L . In addition, if there exists a crossing periodic orbit, then there exists aperiodic annulus including this crossing periodic orbit.Proof. Since b = 0, system (3.1) is exactly the nonsmooth Li´enard system (1.4) satisfying F + ( x ) = t R x, f + ( x ) = t R , g + ( x ) = d R x − a R ,F − ( x ) = t L x, f − ( x ) = t L , g − ( x ) = d L x − a L . (3.3)Clearly, it follows from (3.3) and d R > (H1) holds by choosing x e := 0 if a R ≤ x e := a R /d R if a R >
0. Moreover, (H2) holds because of (3.3) and t R > > t L in (3.2). Bythe definitions of x + ( p ) and x − ( p ) given below (H2) , we get x + ( p ) = p/t R and x − ( p ) = p/t L .Moreover, for system (3.1) the equations (1.10) become t L x − = t R x + , d L x − − a L t L = d R x + − a R t R . (3.4)If a R /t R > a L /t L , thenlim p → + g + ( x + ( p )) f + ( x + ( p )) = − a R t R < − a L t L = lim p → + g − ( x − ( p )) f − ( x − ( p )) , i.e., (H3) holds. Thus, by Theorem 1.1 a necessary condition for the existence of crossingperiodic orbits is that the equations (3.4) have solutions with x − < < x + , which is equivalent14o d R /t R > d L /t L because a R /t R > a L /t L and t R >
0. On the other hand, if (3.1) has a crossingperiodic orbit, then K − ( x − ( p )) = d L t L < d R t R = K + ( x + ( p ))for all p , p satisfying p > p >
0, i.e., (H5) holds, where K ± ( x ± ( p )) are defined in (H5) . ByTheorem 1.2, (3.1) has a unique crossing periodic orbit, which is stable. Thus conclusion (i) isproved.If a R /t R < a L /t L , applying the changes ( t, x, y ) → ( − t, − x, y ) and( t L , d L , a L , t R , d R , a R ) → ( − t R , d R , − a R , − t L , d L , − a L ) (3.5)to (3.1) we observe that the form of (3.1) is invariant. Thus conclusion (ii) is directly obtainedfrom conclusion (i).If a R /t R = a L /t L , thenlim p → + g + ( x + ( p )) f + ( x + ( p )) = − a R t R = − a L t L = lim p → + g − ( x − ( p )) f − ( x − ( p )) . Define Λ( p ) := g + ( x + ( p )) f + ( x + ( p )) − g − ( x − ( p )) f − ( x − ( p ))for p >
0. For system (3.1), we getΛ( p ) = (cid:18) d R t R p − a R t R (cid:19) − (cid:18) d L t L p − a L t L (cid:19) = (cid:18) d R t R − d L t L (cid:19) p. When d R /t R < d L /t L , we have Λ( p ) < p >
0, i.e., (H3) holds. Moreover, (0 ,
0) is the uniquesolution of equations (3.4). Hence, (3.1) has no crossing periodic orbits by Theorem 1.1. When d R /t R > d L /t L , by the changes ( t, x, y ) → ( − t, − x, y ) and (3.5), the nonexistence of crossingperiodic orbits is directly obtained from the case of d R /t R < d L /t L . Thus, d R /t R = d L /t L is anecessary condition for the existence of crossing periodic orbits. Then, if there exists a crossingperiodic orbit, we have Λ( p ) ≡ p >
0, i.e., condition (1.11) of Theorem 1.3 is satisfied.Therefore, there exists a periodic annulus including this crossing periodic orbit by Theorem 1.3.Conclusion (iii) is proved.We remark that a similar result to Theorem 3.1 is given in [28, Theorem 4] for system (3.1)satisfying (3.2) and a L > > a R , which is not required in our Theorem 3.1. So Theorem 3.1generalizes [28, Theorem 4] and this generalization is crucial for us to prove Theorem 1.4. Wewill see this in the proof of Theorem 1.4 later. Lemma 3.1.
Assume that b = 0 , d L d R = 0 in system (3.1) . Then there exist no crossing limitcycles if t L t R ≥ .Proof. If t L t R ≥ t L + t R = 0, the result of no crossing limit cycles is obtained directlyfrom [17, Proposition 3.7]. If t L t R ≥ t L + t R = 0, i.e., t L = t R = 0, the equilibrium of the15eft (resp. right) system of (3.1) is either a center when d L > d R >
0) or a weak saddle(the sum of two eigenvalues is zero) when d L < d R < Proof of Theorem 1.4.
As indicated in the second paragraph of this section, we can equiv-alently consider system (3.1) to investigate the existence, uniqueness and number of crossinglimit cycles of discontinuous system (1.12). Furthermore, it is easy to verify that (1.12) has nosliding sets if and only if (3.1) has no ones and that (1.12) is nondegenerate if and only if (3.1)is nondegenerate. Therefore, we only need to consider nondegenerate (3.1) without sliding sets.By the nonexistence of sliding sets and nondegeneracy, (3.1) satisfies b = 0 and d R d L = 0.Totally there are 7 cases(C1) a L = a R = 0 , (C2) a L > ≥ a R , (C3) a L < ≤ a R , (C4) a L > , a R > a L = 0 , a R < , (C6) a L = 0 , a R > , (C7) a L < , a R < . By the change( x, y, t, t L , d L , a L , t R , d R , a R ) → ( − x, − y, t, t R , d R , − a R , t L , d L , − a L ) , (C5), (C6) and (C7) are transformed into (C2), (C3) and (C4), respectively. Thus, we only needto consider (C1), · · · , (C4).Assume that (3.1) satisfies (C1). Then (3.1) is continuous, where the definition of continuityis given below (1.4). It is proved in [19, Corollary 3] that continuous (1.12) has at most onecrossing limit cycle, so does (3.1).Assume that (3.1) satisfies (C2). When a R = 0 and t R − d R ≥
0, the equilibrium of theright system lies in the switching line y -axis and it is neither focus nor center, implying that(3.1) cannot have crossing limit cycles. When either a R = 0 , t R − d R < a L > > a R , theorigin O is a Σ-monodromic singularity (see [29]), i.e., all orbits in a small neighborhood of O turn around O . Thus (3.1) also has at most one crossing limit cycle by [29, Theorem 1.1].Assume that (3.1) satisfies (C3). When d L < d R <
0, at least one of equilibria of the leftand right systems is a saddle. Moreover, this saddle lies in x > x ≤ d R > , d L > t L t R ≥
0, (3.1) also has no crossing limit cycles by Lemma 3.1. When d R > , d L > t L < < t R , (3.1) satisfies condition (3.2) in Theorem 3.1. Thus (3.1) has at most one crossinglimit cycle by Theorem 3.1. When d R > , d L > t R < < t L , by the change( x, y, t, t L , d L , a L , t R , d R , a R ) → ( x, − y, − t, − t L , d L , a L , − t R , d R , a R )we obtain the uniqueness of crossing limit cycles from the case d R > , d L > , t L < < t R .16ssume that (3.1) satisfies (C4). Applying the change( t, x, y ) → ( ( t/a R , x/a R , y ) for x > , ( t/a L , x/a L , y ) for x ≤ (cid:18) ˙ x ˙ y (cid:19) = (cid:18) t R /a R − d R /a R (cid:19) (cid:18) xy (cid:19) − (cid:18) (cid:19) if x > , (cid:18) t L /a L − d L /a L (cid:19) (cid:18) xy (cid:19) − (cid:18) (cid:19) if x < . (3.7)Observing that (3.7) is continuous, we know that (3.7) has at most one crossing limit cycleby [19, Corollary 3] again. Finally, we conclude that (3.1) has at most one crossing limit cyclebecause (3.6) is a homeomorphism.In conclusion, nondegenerate and discontinuous (1.12) without sliding sets has at most onecrossing limit cycle. Next, we show the reachability of this number and the location of crossinglimit cycles by considering the following nondegenerate and discontinuous system as an example, (cid:18) ˙ x ˙ y (cid:19) = (cid:18) −
12 0 (cid:19) (cid:18) xy (cid:19) − (cid:18) (cid:19) if x > , (cid:18) − −
15 0 (cid:19) (cid:18) xy (cid:19) − (cid:18) χ (cid:19) if x < , (3.8)where χ ∈ { ǫ, , } and − ≪ ǫ < E := (1 ,
2) is an unstable focus of (3.8) and O is avisible tangency point, i.e., the orbit of the right system passing through O is tangent to x = 0at O from the right side. Then the forward orbit of the right system starting from (0 , y ) with y ≤ y -axis at a point (0 , y ) with y > t + >
0. Hence we define the right Poincar´e map P R ( y ) := y . As completedin [15, 17], the parametric representation of P R is given by y ( t + ) = e − t + − cos t + sin t + + 1 , y ( t + ) = − e t + − cos t + sin t + + 1for t + ∈ ( π, ˆ t + ] and P R (0) = − e ˆ t + sin ˆ t + > , lim y →−∞ P ′ R ( y ) = − e π , (3.9)where ˆ t + ∈ ( π, π ) satisfies y (ˆ t + ) = 0, i.e., the time for the orbit passing from (0 ,
0) to (0 , P R (0))in the right half plane.For the left system, its equilibrium E := ( χ, − χ ) is a stable focus. Moreover, O is aboundary equilibrium if χ = 0 and an invisible (resp. a visible) tangency point if χ = 1 (resp. ǫ ), i.e., the orbit of the left system passing through O is tangent to x = 0 at O from right (resp.left) side. Then there exists ˆ z ≥ , z ) with z ≥ ˆ z evolves in the left half plane and reaches again y -axis at a point of form170 , z ) with z ≤ t − ≥
0. Choosing ˆ z as the minimum one, we define theleft Poincar´e map P L ( z ) := z . From [15, 17] again, the expression of P L is given by z ( t − ) = χ e t − − cos t − − t − sin t − ! , z ( t − ) = − χ e − t − − cos t − + 2 sin t − sin t − ! if χ = ǫ, P L ( z ) = − e − π z if χ = 0, where t − ∈ ( π, ˆ t − ] (resp. [0 , π )) for χ = ǫ (resp. 1) and ˆ t − ∈ ( π, π ) satisfies z (ˆ t − ) = 0.In addition, we haveˆ z = ( χ = ǫ, ǫe t − sin ˆ t − if χ = ǫ, lim z → + ∞ P ′ L ( z ) = − e − π . (3.10)Let P ( y ) := P L ( P R ( y )). From the first equality of (3.9) and (3.10), we have ˆ z < P R (0)for any χ and − ≪ ǫ <
0. Thus P ( y ) is well defined for y ≤ P (0) < P R (0) > , P L (ˆ z ) = z (ˆ t − ) = 0. On the other hand, from the second equality of (3.9) and(3.10) we have lim y →−∞ P ′ ( y ) = lim y →−∞ P ′ L ( P R ( y )) · P ′ R ( y ) = e − π < , implying that P ( y ) > y for y closed to −∞ . Therefore, P ( y ) has a fixed point in y < χ , i.e., system (3.8) has a crossing periodic orbit. Since (3.8) has no sliding sets, then thiscrossing periodic orbit is a crossing limit cycle by the first part of this proof, i.e., the reachabilityis proved.Notice that for system (3.8), E is a boundary equilibrium if χ = 0 and a regular equilibriumif χ = ǫ , but it is not an equilibrium if χ = 1. Moreover, O is a regular point if χ = 1, aboundary equilibrium if χ = 0 and a pseudo-equilibrium if χ = ǫ . Thus system (3.8) exactlyhas one equilibrium E if χ = 1, two equilibria O ( E ) and E if χ = 0, and three equilibria E , O and E if χ = ǫ , which eventually imply that the number of equilibria surrounded by thiscrossing limit cycle is exactly 1 (resp. 2 ,
3) if χ = 1 (resp. 0 , ǫ ). The proof is completed. References [1] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk,
Piecewise-Smooth Dynamical Systems: Theoryand Applications , Applied Mathematical Sciences, Vol.163, Springer Verlag, London, 2008.[2] C. A. Buzzi, T. Carvalho, R. D. Euz´ebio, On Poincar´e-Bendixson Theorem and non-trivial minimal sets inplanar nonsmooth vector fields,
Publ. Mat. (2018), 113-131.[3] C. A. Buzzi, J. C. R. Medrado, M. A. Teixeira, Generic bifurcation of refracted systems, Adv. Math. (2013), 653-666.[4] C. A. Buzzi, C. Pessoa, J. Torregrosa, Piecewise linear perturbations of a linear center,
Discrete Contin.Dyn. Syst. (2013), 3915-3936.[5] Q. Cao, M. Wiercigroch, E. E. Pavlovskaia, C. Grebogi, J. M. T. Thompson, Archetypal oscillator for smoothand discontinuous dynamics, Phys. Rev. E (2006), 046218.[6] H. Chen, Global analysis on discontinuous limit cycle case of a smooth oscillator, Int. J. Bifur. Chaos (2016), 1650061.
7] H. Chen, M. Han, Y. Xia, Limit cycles of a Li´enard system with symmetry allowing for discontinuity,
J.Math. Anal. Appl. (2018), 799-816.[8] S. Chen, Z. Du, Stability and perturbations of Homoclinic loops in a class of piecewise smooth systems,
Int.J. Bifur. Chaos (2015), 1550114.[9] C. J. Christopher, S. Lynch, Small-amplitude limit cycle bifurcations for Li´enard systems with quadratic orcubic dumping or restoring forces, Nonlinearity (1999), 1099-1112.[10] Z. Du, Y. Li, W. Zhang, Bifurcation of periodic orbits in a class of planar Filippov systems, Nonlin. Anal. (2008), 3610-3628.[11] F. Dumortier, C. Li, On the uniqueness of limit cycles surrounding one or more singularities for Li´enardequations, Nonlinearity (1996), 1489-1500.[12] F. Dumortier, C. Li, Quadratic Li´enard equations with quadratic damping, J. Differential Equations (1997), 41-59.[13] F. Dumortier, D. Panazzolo, R. Roussarie, More limit cycles than expected in Li´enard equations,
Proceedingof Amer. Math. Soc. (2007), 1895-1904.[14] A. F. Filippov,
Differential Equation with Discontinuous Righthand Sides , Kluwer Academic Publishers,Dordrecht, 1988.[15] E. Freire, E. Ponce, F. Torres, The discontinuous matching of two planar linear foci can have three nestedcrossing limit cycles,
Publ. Mat.
Vol. extra(2014), 221-253.[16] E. Freire, E. Ponce, F. Torres, A general mechanism to generate three limit cycles in planar Filippov systemswith two zones,
Nonlinear Dyn. (2014), 251-263.[17] E. Freire, E. Ponce, F. Torres, Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn.Syst. (2012), 181-211.[18] E. Freire, E. Ponce, F. Torres, Planar Filippov systems with maximal crossing set and piecewise linear focusdynamics, Progrss and Challenges in Dynamical Systems , 221-232, Springer Proc. Math. Stat., 54, Springer,Heidelberg, 2013.[19] E. Freire, E. Ponce, F. Rodrigo, F. Torres, Bifurcation sets of continuous piecewise linear systems with twozones,
Int. J. Bifur. Chaos (1998), 2073-2097.[20] E. Poncea, J. Rosa, E. Velab, The boundary focus-saddle bifurcation in planar piecewise linear systems.Application to the analysis of memristor oscillators, Nonlinear Anal.: Real World Appl. (2018), 495-514.[21] F. Giannakopoulos, K. Pliete, Planar systems of piecewise linear differential equations with a line of discon-tinuity, Nonlinearity (2001), 1611-1632.[22] S. Huan, X. Yang, On the number of limit cycles in general planar piecewise linear systems, Discrete Contin.Dyn. Syst. (2012), 2147-2164.[23] F. Jiang, J. Shi, Q. Wang, J. Sun, On the existence and uniqueness of a limit cycle for a Li´enard systemwith a discontinuous line, Commun. Pure Appl. Anal. (2016), 2509-2526.[24] Y. A. Kuznetsov, S. Rinaldi, A. Gragnani, One-parameter bifurcations in planar Filippov systems, Int. J.Bifur. Chaos (2003), 2157-2188.[25] C. Li, J. Llibre, Uniqueness of limit cycles for Li´enard differential equations of degree four, J. DifferentialEquations (2012), 3142-3162.[26] J. Llibre, A. C. Mereu, Limit cycles for discontinuous generalized Li´enard polynomial differential equations,
Electron. J. Differential Equations (2013), 1-8.[27] J. Llibre, D. D. Novaes, M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynam-ical systems,
Nonlinear Dyn. (2015), 1159-1175.[28] J. Llibre, E. Ponce, F. Torres, On the existence and uniqueness of limit cycles in Li´enard differential equationsallowing discontinuities, Nonlinearity (2008), 2121-2142.[29] J. Medradoa, J. Torregrosa, Uniqueness of limit cycles for sewing planar piecewise linear systems, J. Math.Anal. Appl. (2015), 529-544.[30] S. Smale, Mathematical problems for the next century,
Math. Intell. (1998), 7-15.[31] A. Tonnelier, W. Gerstner, Piecewise-linear differential equations and integrate-and-fire neurons: insightsfrom two-dimensional membrane models, Phys. Rev. E (2003), 021908.[32] J. Wang, C. Huang, L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear systemof saddle-focus type, Nonlinear Anal.: Hybrid Syst. (2019), 162-178.
33] D. Xiao, Z. Zhang, On the uniqueness and nonexistence of limit cycles for predator-prey systems,
Nonlinearity (2003), 1185-1201.[34] Z. Zhang, T. Ding, W. Huang, Z. Dong, Qualitative Theory of Differential Equations , Science Publisher,1985 (in Chinese); Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1992., Science Publisher,1985 (in Chinese); Transl. Math. Monogr., vol. 101, Amer. Math. Soc., Providence, RI, 1992.