The discontinuous matching of two planar linear foci can have three nested crossing limit cycles

Publ. Mat. (2014), 221–253 Proceedings of New Trends in Dynamical Systems. Salou, 2012. DOI: 10.5565/PUBLMAT Extra14 13 THE DISCONTINUOUS MATCHING OF TWO PLANAR LINEAR FOCI CAN HAVE THREE NESTED CROSSING LIMIT CYCLES Emilio Freire, Enrique Ponce, and Francisco Torres Abstract: The existence and stability of limit cycles in discontinuous piecewise linear systems obtained by the aggregation of two linear systems of focus type and having only one equilibrium point is considered. By using an adequate canonical form with five parameters, a thorough study of some Poincar´e maps is performed. Different bifurcations which are responsible for the appearance of crossing limit cycles are detected and parameter regions with none, one, two and three crossing limit cycles are found. In particular, a first analytical proof of the existence, for certain differential systems in the considered family, of at least three homotopic crossing limit cycles surrounding the equilibrium point, is included. This fact has recently been numerically discovered in a particular example by S.-M. Huan and X.-S. Yang [13]. 2010 Mathematics Subject Classification: 34C05, 34C07, 37G15. Key words: Discontinuous piecewise linear systems, limit cycles, bifurcations. 1. Introduction and preliminary results Piecewise linear (PWL) systems are the natural extension of linear systems in order to capture nonlinear behavior and they are extensively used in model biological processes [4] as well as electronics and mechanical devices, see for instance [1], [5] and [18]. Even for planar cases, the qualitative analysis of different possible behavior is a rather involved task, being the detection of limit cycles an important, non-trivial problem. The simplest family of PWL systems is the planar case constituted by two linear systems matched along a straight line. The case of planar PWL systems with continuous matching along the straight line was carefully studied in [8], where some canonical forms with only four parameters were obtained. In particular, by using these canonical forms, This work was partially supported by a MINECO/FEDER grant number MTM2009- 07849 and the grant EXC/P08-FQM-03770 of Junta de Andaluc´ıa. 222 E. Freire, E. Ponce, F. Torres the existence at most of one isolated invariant closed curve (a limit cycle or an isolated homoclinic loop) was shown. Always within the planar case, the discontinuous matching of two linear systems has recently deserved the attention by researchers. After the Filippov extension of the concept of solution to the discontinuity line [7], many works dealing with discontinuous PWL systems have appeared. In [10] and [11], PWL systems with Z2symmetry were investigated. In [17] particular Filippov systems were both linear dynamics are of saddle type were studied. The Hopf bifurcation in non-smooth systems was studied in [12], [3] and [19]. In fact, in [12] authors conjectured that the maximum number of limit cycles for the class of systems under study is exactly two. However, in [13] it was numerically obtained the existence of three limit cycles for a planar PWL system, but no analytical proof was provided. Recently, a computer assisted proof has been available, see [15], and based on such a proof it is shown in [2] that the three limit cycles can be obtained through a boundary focus bifurcation. In this paper, as one of the goals to be achieved, we will give the lacking analytical proof. Even we conjecture that three is indeed the maximum number of limit cycles to be found, it is still needed a proof of this fact. As claimed in [13], the study of discontinuous PWL systems is a difficult task because of the absence of a canonical form able to cope with a sufficiently broad class of systems. To fill this gap, in [9] a Li´enard-like canonical form with seven parameters was obtained; under the assumption of both linear focus dynamics, a simpler canonical form with only five parameters was also achieved. Even for the two foci cases, the number of qualitatively different phase portraits is great, so the study made in [9] was restricted to the case without equilibria in both open half planes. Particular instances of this last situation previously appeared in [16]. This paper can be properly considered as a continuation of the work in [9]. Thus, we consider PWL systems where both dynamics are of focus type and the system has only one real equilibrium, procrastinating the cases with two real equilibria for a future work. We investigate the dynamical behavior of this family by studying mainly the existence and stability of the so called crossing limit cycles. We will provide, following a bifurcational approach, a particular mechanism to explain the existence of three limit cycles in a certain open set of the parameter space. Thus, we confirm analytically the phenomenon numerically observed in [13] for a specific system. Furthermore, we show that the non-genericity of the example in [13], where the two involved vector fields vanish at the same point, is not needed at all. Three Limit Cycles Assembling Two Linear Foci 223 In the study of planar piecewise linear systems with two zones separated by a straight line, we can assume without loss of generality that the discontinuity line is Σ = {(x, y) : x= 0}. If we define now the open sets S−={(x, y) : x < 0}, S+={(x, y) : x > 0}, the systems to be studied can be written as (1) ˙x =F(x) =    F+(x)=F+ 1(x), F+ 2(x)T=A+x+b+,if x∈S+, F−(x)=F− 1(x), F− 2(x)T=A−x+b−,if x∈S−. Here x= (x, y)T∈R2,A+= (a+ ij) and A−= (a− ij ) are 2 ×2 constant matrices and b+= (b+ 1, b+ 2)T,b−= (b− 1, b− 2)Tare constant vectors of R2. The restriction of the system to each half-plane is a linear system and so, orbits are well defined while they evolve without touching the y-axis. However, we must adopt some criterion to define the orbits arriving at the discontinuity line. When a solution arrives at the point (0, y) belonging to Σ, different things can occur. If F+ 1(0, y)F− 1(0, y)>0 then both vector fields are transversal to the discontinuity line and their normal components have the same sign. In this case, we will assume that orbits are concatenated in the natural way. We say that this point is a crossing point, so that the crossing set Σcis defined as follows (2) Σc={(0, y) : F+ 1(0, y)F− 1(0, y) = (a+ 12y+b+ 1)(a− 12y+b− 1)>0}. If F+ 1(0, y)F− 1(0, y)⩽0 then we say that this point is a sliding point. The sliding set Σs, which is the complementary set in Σ of the crossing set, is defined as, (3) Σs={(0, y) : F+ 1(0, y)F− 1(0, y) = (a+ 12y+b+ 1)(a− 12y+b− 1)⩽0}. As usual, when a orbit arrives at the sliding set we will adopt the Filippov convex method, see [14]. Since we are interested in the study of non-trivial periodic orbits, we look for posible periodic orbits not totally contained in either S+or S−. If these orbits have sliding points they will be called sliding periodic orbits. Otherwise, we speak of crossing periodic orbits, whose study is the main goal of this paper. When a+ 12a− 12 ⩽0, it is easy to see from (2) that the crossing set, if it exists, is an open interval of the y-axis, unbounded for a+ 12a− 12 = 0 and bounded for a+ 12a− 12 <0.In any case, the x-component of both vector fields has constant sign at the crossing set and so elementary qualitative arguments preclude the existence of crossing periodic orbits. This case 224 E. Freire, E. Ponce, F. Torres will not be considered in the rest of the paper and we assume in the sequel a+ 12a− 12 >0. As it was shown in Proposition 3.1 of [9], discontinuous planar piecewise linear systems (1) under the assumption a+ 12a− 12 >0 can be transformed by means of a certain homeomorphism into the Li´enard canonical form, (4) ˙x =T−−1 D−0x−0 a−if x∈S−, ˙x =T+−1 D+0x−−b a+if x∈S+, where T+,T−denote the traces and D+,D−denote the determinants of matrices A+and A−, respectively. It is worth noting that the discontinuity line remains invariant under the above homeomorphism. Moreover, crossing and sliding sets, tangency points and boundary equilibria of the original system are transformed by the quoted homeomorphism into sets and points of the same type for system (4). More precisely, such a homeomorphism induces a topological equivalence between systems (1) and (4) for all their orbits not intersecting the sliding set or for the ones having at most the tangency points in common with the sliding set. Also, the attractive or repulsive character of the sliding set is preserved, see [9]. As mentioned before, we assume that both dynamics are of focus type, that is T2−4D < 0 in the two zones. Then, we know that matrices A± have the eigenvalues λ±=α±±iω±with ω±>0, so that in the canonical form (4) we have T±= 2α±and D±= (α±)2+ (ω±)2. By introducing the parameters γR=α+ ω+, γL=α− ω−, aR=a+ ω+, aL=a− ω−, and doing the change of variables (different for each half-plane) (x, y, t)→x ω(x), y, t ω(x),,where ω(x) = (ω−if x < 0, ω+if x > 0, it was proved in Proposition 4.1 of [9] that canonical form (4) reduces to (5) ˙x =2γL−1 1 + γ2 L0x−0 aLif x∈S−, ˙x =2γR−1 1 + γ2 R0x−−b aRif x∈S+, Three Limit Cycles Assembling Two Linear Foci 225 where for the sake of convenience we will put aL= (1 + γ2 L)xL, aR= (1 + γ2 R)xR. The canonical form (5) is invariant under the transformations shown in Table 1so that, after applying if needed symmetries Π1or Π2, it is not restrictive to assume b⩾0. Π1Π2Π3 x→x x → −x x → −x y→ −y y →y+b y → −y−b t→ −t t → −t t →t γL→ −γLγL→ −γRγL→γR γR→ −γRγR→ −γLγR→γL xL→xLxL→ −xRxL→ −xR b→ −b b → −b b →b xR→xRxR→ −xLxR→ −xL Table 1. Symmetries of canonical form (5). When b > 0, the sliding set of our canonical form is given by (6) Σs={(0, y) : 0 ⩽y⩽b}, shrinking to the origin for b= 0. Accordingly, the crossing set is the open set Σc= Σ\Σs. In the case b > 0, the convex method of Filippov leads to the sliding vector field, (7) ˙x= 0,˙y=g(y) = aL−aR by−aL,0⩽y⩽b, to be considered only on the sliding set. Note that from [14], as the x-components of both vector fields at the sliding set Σspoints outwards from Σs, the sliding set is unstable in the normal direction, so that the sliding set is usually called escaping sliding set. For the points where one of the right hand-sides in (5) vanishes, we distinguish as usual between real and virtual equilibrium points. More precisely, eL= (xL,2γLxL) is a real equilibrium point when xL⩽0, being virtual when xL>0. In the critical case xL= 0 we speak of a 226 E. Freire, E. Ponce, F. Torres boundary equilibrium point. Note that the origin is a visible tangency point for xL<0 (invisible, for xL>0) for the left vector field, see [14]. Similarly, eR= (xR,2γRxR) is a real equilibrium when xR⩾0 (boundary equilibrium point if xR= 0) and virtual when xR<0.Moreover, the right vector field has a visible tangency point for xR>0 (invisible, for xR<0) at the point (0, b). Note that for b= 0 the origin is a double tangency point, provided that xLxR6= 0. Points (0, y)∈Σswith g(y) = 0 in (7) act as well in some sense as equilibria of system (1) and they are called pseudo-equilibria for system (5). From (7) we deduce that our system has a pseudo-equilibrium point only when aRaL<0, and then it is given by (0,¯yp) with ¯yp=aLb aL−aR . Note that at the pseudo-equilibrium point (0,¯yp) the two vector fields of (5) become anticollinear. Next, in Section 2our main results are presented. 2. Statement of main results We concentrate our attention in system (5) with b⩾0 assuming that there exists only one equilibrium point in the interior of the zones S−,S+, so that we must have xRxL>0.The case when the system has no real equilibrium points in the interior of the open regions S−,S+was studied in [9], where it was shown the possible existence of two crossing limit cycles. Without loss of generality, we can assume xR<0 and xL<0,that is, the only equilibrium is located in the left zone, otherwise it suffices to apply the transformation Π3of Table 1. Next, we give our first result on limit cycles which is related to the case when the sliding set reduces to the origin. Theorem 1 (singleton sliding set, b= 0).Assuming in system (5)the conditions b= 0,xR<0and xL<0,the following statements hold. (a) If γRγL⩾0and γR+γL6= 0, then there are no crossing periodic orbits. (b) If γR=γL= 0, then every orbit is a periodic orbit and the configuration is a global center. (c) If γRγL<0and γL(γR+γL)⩾0, then there are no crossing periodic orbits. (d) If γRγL<0and γL(γR+γL)<0, then there is only one crossing periodic orbit which is stable for γL>0and unstable for γL<0. Three Limit Cycles Assembling Two Linear Foci 227 Since the equilibrium point is in the interior of the left zone, its stability is clearly determined by the sign of parameter γL.Later on, we will establish that the stability of the point at infinity, that is the equator of the corresponding Poincar´e sphere, is determined by the sign of γL+γR. Hence, roughly speaking, Theorem 1states that we have one limit cycle on the condition that the origin and the infinity have different stabilities. For b6= 0, the sliding set is a segment with two tangency points at its endpoints: a visible tangency at the origin for the left vector field and a invisible tangency at the point (0, b) for the right vector field. However, this sliding segment is stable for b < 0 and unstable for b > 0. Such a change of stability is in fact the bifurcation phenomenon called V I1 in [14]. When b > 0, the sliding vector field (7) for xL<0 and xR<0 must be taken into account. Then we have ˙y > 0 in the sliding dynamics and there are no pseudo-equilibrium points. Since the sliding set is repulsive, the only orbit arriving at the sliding set is the tangent one while for reversing time all the sliding orbits leave the sliding set at the origin. Then, in order to establish the existence of sliding periodic orbits we only have to study the orbits through the origin. According to [14], note that it is possible the existence of invariant closed curves backwards in time which are not invariant forward in time; in this case, we speak of backward-invariant closed curves. Next, let us consider the existence of sliding periodic orbits forward and backward in time. In forward time, the tangent orbit at the origin enters in the sliding set Σsand slides along Σstowards the point (0, b). Next, it evolves in the left zone around the equilibrium point eL, either finally arriving at eLor coming back to the line x= 0 at a point (0, y1) with y1⩽0. If y1<0, then from the point (0, y1), the orbit never returns to the sliding set, so that we have no sliding periodic orbits. If y1= 0, we have a sliding periodic orbit. Thus, the only sliding periodic orbit in forward time appears when the orbit starting at the point (0, b) passes through the origin. To analyze the possible sliding periodic orbits backward in time, we can discard the case γL>0, for then when an orbit abandon the sliding set at the origin never returns to the sliding set, approaching the equilibrium point eL. Thus, assume to fix ideas that γL⩽0 and let us introduce the point ˆ p= (0,ˆy) with ˆy⩾0 such that the orbit forwards in time of the left vector field starting at the point ˆ ppasses through the origin. Also, we define the point p∗= (0, y∗) with y∗< b such that the orbit forward in time of the right vector field starting at the point p∗ 228 E. Freire, E. Ponce, F. Torres passes through the point ˆ p. Different cases appear by considering the position of the points ˆ pand p∗. If 0 ⩽ˆy⩽bthen the point ˆ pbelong to the sliding set. The case ˆy= 0 corresponds with a linear center in the left zone, that is γL= 0, and we trivially see that the system has no sliding periodic orbits. Assume ˆy > 0, then, always backward in time, the orbit slides along the line x= 0 to the origin. Consequently, we detect by going backward in time a closed sliding orbit which only uses the left zone and surrounds the equilibrium eL, see Figure 1. Note that if ˆy=b, then the limit cycle uses the full sliding set and thus we have the only sliding periodic orbit that can also be detected forward in time. y b ˆy eLx y eLx ˆy y∗ b Figure 1. Left panel: The sliding orbit lives in only one zone when ˆy⩽b. Right panel: Sliding orbit living in the two zones for when ˆy > b. The arrow indicates the sense for the forward time. If ˆy > b, then the point ˆ pbelongs to the crossing set. Next, by going backward in time, the orbit through the point ˆ pwill enter in the region S+surrounding the tangency point (0, b) to come back again to the line x= 0 at a point (0, y∗) with y∗< b, see Figure 1. If y∗⩾0,then the orbit slides back to the origin forming a unstable sliding limit cycle that uses both regions SLand SR. If y∗<0,then the backward orbits starting at the sliding set never returns to it and there are no sliding limit cycles. Remark 1.In the particular case where ˆy > b and the orbit starting at the origin arrives backward in time again to the origin, that is y∗= 0, Three Limit Cycles Assembling Two Linear Foci 229 we have a backward-invariant closed curve which only shares the origin with the sliding set, see Figure 2. This orbit is called crossing critical cycle, see Section 4.1.4 in [14], and represents the boundary between the sliding periodic orbits and the crossing periodic orbits. y x b eL ˆy Figure 2. The crossing critical cycle described in Remark 1. The above discussion shows that if there exists a sliding periodic orbit for system (5), then it is unique. In fact we can state the following proposition. Proposition 1 (sliding periodic orbits).Assume that xL<0,xR<0, b > 0in system (5). When γL⩾0the system has no sliding periodic orbits; if γL<0and regarding the values ˆy > 0and y∗< b above defined, the following statements hold. (a) If ˆy < b then there is one sliding orbit backward in time which lives in the left zone and it is unstable. (b) If ˆy=bthen there is one sliding orbit both backward and forward in time, which lives in the left zone and it is unstable. (c) If ˆy > b then the following cases arise. (i) If 0< y∗< b then there is one sliding periodic orbit backward in time which lives in the two zones and it is unstable. (ii) If y∗= 0 then there is one unstable crossing critical cycle which is unstable. (iii) If y∗<0then there are no sliding periodic orbit. 236 E. Freire, E. Ponce, F. Torres for appropriate values of parameters γ,σand ¯x. Its solutions have the form, (9) x(t)−¯x y(t)−¯y=eγt cos t+γsin t−sin t (1 + γ2) sin tcos t−γsin tx(0) −¯x y(0) −¯y, where ¯y= 2γ¯x+σ. We also introduce the auxiliary function, see [1] ϕγ(t) = 1 −eγt(cos t−γsin t). Function ϕγhas the following symmetry properties ϕ−γ(t) = ϕγ(−t), ϕ−γ(t) = ϕγ(−t),∀γ, t ∈R, and the graph shown in Figure 5. Note that function ϕγhas relative maxima at t=±πand for γ > 0 a first positive zero for a certain value ˆ t∈(π, 2π). −2π−ππ2π ˆ t Figure 5. The graph of function ϕγ(·) for a positive value of γ. Next, we introduce some Poincar´e maps which are our main tool in looking for crossing periodic orbits. For system (5), under the focus hypotheses, the direction of the flow at the line x= 0 assures that orbits starting at points (0, y) with y > 0 and big enough go into the zone S− until it again reaches Σ at a point (0, y1) with y1⩽0 after a time tL. Thus, we define a left Poincar´e map PLas y1=PL(y)⩽0. In order to integrate system (5) in the left zone we must put σ= 0 and ¯x=xLin (9). If γL= 0,then from (9) we explicitly obtain the map PL, (10) PL(y) = −y, y ⩾0. Three Limit Cycles Assembling Two Linear Foci 237 If xL<0,then the point (0,0) is a visible left tangency point and from (9) we obtain a parametric representation of the left Poincar´e map PL, (11) y=e−γLtLϕγL(tL) sin tL xL, PL(y) = −eγLtLϕ−γL(tL) sin tL xL, where tL∈(π, ˆ tL),being ϕ|γL|(ˆ tL) = 0.In this case the orbits must go around the real equilibrium eLlocated in S−. Direct computations from (11) show that the two first derivatives of the left Poincar´e map when γL6= 0 are given by P′ L(y) = −ϕγL(tL) ϕ−γL(tL)=y PL(y)e2γLtL<0,(12) P′′ L(y) = 2x2 L(1 + γ2 L)sinh γLtL−γLsin tL P3 L(y)e3γLtL.(13) Integrating the system in the zone S+from the point (0, z) with z < b, after a time tRwe arrive at the point (0, z1) with z1> b, and so we can define the right Poincar´e map PRas z1=PR(z)> b. In the trivial case γR= 0,from (9) we explicitly obtain (14) PR(z) = −z+ 2b, z ⩽b. If xR<0,the point (0, b) is a right invisible tangency point and the parametric representation of the right Poincar´e map is given by, (15) z=b+e−γRtRϕγR(tR) sin tR xR, PR(z) = b−eγRtRϕ−γR(tR) sin tR xR where tR∈(0, π). Analogously, the two first derivatives of the right Poincar´e map when γR6= 0 are given by P′ R(z) = z−b PR(z)−be2γRtR<0,(16) P′′ R(z) = 2x2 R(1 + γ2 R)sinh γRtR−γRsin tR (PR(z)−b)3e3γRtR.(17) Finally, as the main tool in looking for crossing periodic orbits, we define the Poincar´e map Pas the composition P=PR◦PL,to be defined for points (0, y) with y > 0 and big enough. Next, we establish some properties of Poincar´e maps when system (5) has only one equilibria which is located in the zone x < 0,that is when xL<0 and xR<0. We begin by considering the left Poincar´e map PL. 238 E. Freire, E. Ponce, F. Torres Proposition 3 (The map PL).Assume that xL<0in system (5). Then the following statements are true for the left Poincar´e map PL. (a) If γL= 0, then PL(y) = −yfor all y⩾0. If γL6= 0, then PLis defined in (11)for π < t ⩽ˆ tLbeing ϕ|γL|(ˆ tL) = 0. (b) If γL<0, then we define ˆy=y0(ˆ tL)>0, and we have PL(ˆy) = 0, PL: [ˆy, ∞)→(−∞,0] and lim y→ˆy+P′ L(y) = −∞. (c) If γL>0, then we define ˆy1=y1(ˆ tL)<0,and we have PL(0) = ˆy1, PL: [0,∞)→(−∞,ˆy1]and lim y→0+P′ L(y) = 0. (d) If γL6= 0, then we have P′ L(y)<0for all ywhere map PLis defined, lim y→∞ P′ L(y) = −eγLπ,sign P′′ L(y) = −sign γL, and map PLhas the asymptote ALgiven by AL(y) = −eγLπy+ 2xLγL(1 + eγLπ) with sign (AL(y)−PL(y)) = sign(γL). Proof: Statement (a) follows from (10) and (11). To prove statement (b) assume γL<0,then π < tL<ˆ tLand from (12) we get lim y→ˆy+P′ L(y) = lim tL→ˆ t+ L P′ L(y) = −∞. The proof of statement (c) is analogous to the proof of statement (b). If γL6= 0,then lim y→∞ P′ L(y) = lim tL→π− P′ L(y) = −eγLπ. Since sign (sinh γLtL−γLsin tL)=sign γL,from (13) it is easy to see that sign P′′ L(y)=−sign γL.The computation of the asymptote is straightforward and the proof is finished. Here, we see that the proof of Proposition 1directly follows from the properties of left Poincar´e map given in Proposition 3and the discussion done above the statement of Proposition 1. The following proposition about the right Poincar´e map is similar and it is given without proof. Proposition 4 (The map PR).Assuming xR<0in system (5), the following statements hold for the right Poincar´e map PR. Three Limit Cycles Assembling Two Linear Foci 239 (a) If γR= 0 then, PR(y) = −y+ 2bfor all y⩽b. (b)If γR6= 0 then the right Poincar´e map PRgiven in (15)is defined for all y⩽b, where t∈(0, π); in particular we have PR(b) = b. Its two first derivatives satisfy P′ R(b) = −1, P′ R(y)<0,lim y→−∞ P′ R(y) = −eγRπ and its second derivative satisfies sign P′′ R(y) = sign γR. (c) The map P−1 Rhas the asymptote AR−1given by AR−1(y) = −e−γRπy+ (b+ 2xRγR)(1 + e−γRπ) and satisfies sign P−1 R(y)−AR−1(y)= sign (γR). As a consequence of the above results we obtain some properties for the complete Poincar´e map. In Figure 6the left and right Poincar´e maps are shown for different values of parameters γLand γR. PLˆyy γL<0 γL>0γL= 0 PR y γR<0 γR>0 γR= 0 Figure 6. The left Poincar´e map for different values of parameter γLand the right Poincar´e map for different values of parameter γRand b= 0. Proposition 5 (The full Poincar´e map).Assuming xL<0and xR<0 in system (5), the following statements hold. (a) The Poincar´e map Pis well defined for y⩾yP⩾0where, as defined in Proposition 3(b), yP= ˆyfor γL<0and yP= 0 when γL⩾0. (b)If γR= 0, then P(y) = −PL(y) + 2b. If in addition γL= 0,then P(y) = y+ 2b. (c) If γL= 0, then P(y) = PR(−y)and sign P′′(y) = −sign γR. 240 E. Freire, E. Ponce, F. Torres (d) For all y > yP,the derivative of map Pis given by P′(y) = y PL(y) PL(y)−b P(y)−be2(γRtR+γLtL) and satisfies lim y→∞ P′(y) = e(γR+γL)π,lim y→y+ P P′(y) =      ∞,if γL<0, 1,if γL= 0, 0,if γL>0. Proof: Statements (a), (b), and (c) directly follows from Propositions 3 and 4. Statement (d) can be easily deduced from Propositions 3,4and the chain rule. Remark 2.It is worth noting that from Proposition 5(d), it is direct to deduce the stability of the point at infinity. More precisely, if γR+γL<0, then the point at infinity is unstable while is stable for γR+γL>0. 4. Proof of main results First, we review a necessary condition for the existence of crossing periodic orbits. A crossing periodic orbit Γ has exactly two points at the y-axis, namely the two points (0, yL) and (0, yU) with yL<0⩽b < yU=yL+h, with h > 0. We can define, by removing the two crossing points, the left open arc ΓL= Γ ∩S−,the right open arc ΓR= Γ ∩S+, and the oriented segments, IL={(x, y) : x= 0, y = (1 −µ)yL+µyU,0⩽µ⩽1}, and IR={(x, y) : x= 0, y =µyL+ (1 −µ)yU,0⩽µ⩽1}. Since ΓL∪ILis a closed Jordan curve, its interior ΩL= int{ΓL∪IL} and the value σL= area (ΩL) are well defined. Analogously, it can be considered ΩR= int {ΓR∪IR}and σR= area (ΩR). Then, according to Proposition 3.6 of [9], if system (5) has a crossing periodic orbit passing through the points (0, yL) and (0, yL+h) where h > 0,then (18) 2γLσL+ 2γRσR+bh = 0. Remark 3.We remark that in the two cases (i) γL>0, γR>0, b > 0; and (ii) γLγR⩾0, γL+γR6= 0, b= 0,the equality (18) cannot be fulfilled and so, system (5) cannot have crossing limit cycles. In the following we will tackle the proofs of the different assertions given in Section 2. Next we consider the proof of Theorem 1. Three Limit Cycles Assembling Two Linear Foci 241 Proof of Theorem 1:(a) See Remark 3. (b) From Proposition 5(b) we have P(y) = yfor y⩾0, and the statement follows. When the divergence are opposite, that is γRγL<0,then by using the transformation Π1if necessary, see Table 1, we see that it is enough to consider γL<0 and γR>0. Firstly, we will prove that Poincar´e map Phas at most a fixed point. Assuming γL<0,from Proposition 5(a), a value ˆy > 0 exists such that the Poincar´e map is defined for y⩾yP= ˆyand P(ˆy) = 0. For y⩾yP, we introduce the function ν(y) = γLtL+γRtR,with π < tL<ˆ tL and 0 < tR< π, where tLis the phase needed in (11) to obtain PL and tRis the corresponding phase in (15) to determine z=PL(y) and PR(z) = P(y). In this case, when yincreases tLbecomes smaller and tRbecomes bigger and so the function ν(y) is increasing. From Proposition 5(d), taking into account that b= 0 the derivative of the Poincar´e map at a fixed point ¯y, if any, reduces to P′(¯y) = e2ν(¯y). Suppose now that map Phas several fixed points and let ¯y1be the smallest one. Then it is easy to see that P′(¯y1)⩾1,and so ν(¯y1)⩾0. Taking the next fixed point ¯y2>¯y1, the only possibility is ν(¯y1) = 0,1 = P′(¯y1)< P ′(¯y2) = e2ν(¯y2) because two consecutive fixed points with derivatives greater than 1 are not allowed. In this situation, for points y∈(¯y1,¯y2) we have y > P(y) and ν(y)>0,so putting b= 0 in Proposition 5(d) we get P′(y)>1 for all y∈(¯y1,¯y2), which leads to a contradiction with the mean value theorem in such interval. Therefore, the map Phas at most only one fixed point. (c) As mentioned before, we can assume without loss of generality γL<0 and so γL+γR⩽0.Suppose also that the Poincar´e map has a fixed point ¯y, then we must have ν(¯y)⩾0.But in this case, we have ν(¯y) = γLtL+γRtR⩽γL(tL−tR)<0, because 0 < γR⩽−γLand tR< π < tL.We get a contradiction, so the map Phas no fixed points and there are no crossing periodic orbits. (d) It suffices to assume γL+γR>0. Since P(ˆy) = 0 and from Proposition 5(d) we have lim y→∞ P′(y)>1, by using the mean value theorem adequately, we deduce that P(y)> y when yis big enough. Consequently, from the intermediate value theorem there exists a value ¯ywith 242 E. Freire, E. Ponce, F. Torres P(¯y) = ¯yand P′(¯y)>1,i.e. system (5) has only one crossing limit cycle, which is unstable. In the following, the existence of periodic orbits in system (5) for b > 0 is considered. Note that the left Poincar´e map does not depends upon the parameter b, but both Poincar´e maps PRand P−1 Rdepend on the parameter bin a rather specific way. In fact, if we make explicit the dependence on bin the notation, from (14) and (15) we see that (19) P−1 R(y;b) = b+P−1 R(y−b; 0). In short, the graph of P−1 R(y;b) is just a translation of the graph of P−1 R(y; 0) by the vector (b, b), see Figure 7. P−1 R(y) y B γR<0 γR>0 γR= 0 Figure 7. The inverse half Poincar´e map P−1 Rfor b= 0 and several values of parameter γR. To obtain the map P−1 Rfor b > 0,it suffices to translate the graphs to the point B, which has the coordinates (b, b). If ¯yis a fixed point of the Poincar´e map P, then P(¯y)=PR(PL(¯y); b)= ¯y, and so PL(¯y) = P−1 R(¯y;b).Hence the existence of crossing periodic orbits is equivalent to the existence of zeroes with y > yPof the function (20) Ψ(y;b) = P−1 R(y;b)−PL(y), well defined when both y⩾yPand y⩾b, where yP⩾0 satisfies yP= ˆy for γL<0 and yP= 0 otherwise. Three Limit Cycles Assembling Two Linear Foci 243 If we use ′to denote the derivatives with respect to the variable y, the two first derivatives of the function Ψ when y6= ˆyread as Ψ′(y;b) = 1 (P′ R◦P−1 R)(y;b)−P′ L(y), Ψ′′(y;b) = −P′′ R◦P−1 R(y;b) P′ R◦P−1 R3(y;b)−P′′ L(y). (21) In the next lemma some properties of the function Ψ are introduced. Lemma 1. Given γR,γLand assuming aL<0,aR<0,b > 0for the function Ψdefined in (20)the following statements hold. (a) If Ψ(y;b)6= 0,then Ψ(y;b)(P(y)−y)>0. (b)If Ψ(¯y;b) = 0,then P(¯y) = ¯y. If in addition ¯y6= ˆyand Ψis decreasing (respectively increasing) with respect to variable yat the point ¯y, then ¯ycorresponds to a stable (respectively unstable) crossing periodic orbit. In particular, if Ψ′(¯y;b)<0, then ¯ycorresponds to a stable crossing periodic orbit (respectively, unstable for Ψ′(¯y;b)>0). Furthermore, we have (22) δ∞= lim y→∞ Ψ′(y;b) = e(γR+γL)π−1e−γRπ. (c) The function Ψand its derivative with respect to the parameter b satisfy the inequalities Ψ(y;b)> b + Ψ(y; 0),∂Ψ(y;b) ∂b =∂P−1 R(y;b) ∂b >1. (d) If γLγR⩾0and y6= ˆy, then sign Ψ′′(y;b) = sign(γL+γR)and so function Ψhas at most two zeroes. Proof: (a) If Ψ(y;b)>0,then PL(y)< P−1 R(y;b) and applying to this inequality PR, which is decreasing, we get P(y)> y. Analogously, when Ψ(y;b)<0,we get P(y)< y. (b) If Ψ(¯y;b) = 0,then it directly follows P(¯y) = ¯y. If the function Ψ is decreasing (respectively, increasing) then h(p) = P(p)−pis also decreasing (respectively, increasing) and ¯pcorresponds to a stable (respectively unstable) crossing periodic orbit. The other assertions of this statement are direct. (c) From (19) we have Ψ(y;b) = b+P−1 R(y−b, 0) −PL(y).Since the map P−1 Ris decreasing and b > 0, we have Ψ(y;b)> b +P−1 R(y, 0) −PL(y) = b+ Ψ(y; 0), 244 E. Freire, E. Ponce, F. Torres and the derivative with respect to the parameter bsatisfies, ∂Ψ(y;b) ∂b = 1 −(P−1 R)′(y−b; 0) >1. (d) From Proposition 3(d) we have sign P′′ L(y) = −sign γLfor y6= ˆy. From Proposition 4(b) we also have sign P′′ R(y) = sign γR, and having in mind that P′ R(y)<0,the conclusion follows from (21). Of course, the function Ψ depends on the values γRand γL. When necessary, we will introduce the explicit dependence on the parameter γR, keeping fixed the value of γLif the contrary it is not stated, so that we can use the notation Ψ(y;b;γR). As the function Ψ is real analytic, we can assure that, in every compact interval where the function Ψ has opposite signs at its endpoints, its number of zeroes is finite and so, there exists at least one zero where the function Ψ is strictly monotone, either increasing or decreasing. More precisely, if Ψ(y1;¯ b, ¯γR)Ψ(y2;¯ b, ¯γR)<0, then there exists at least a value ¯ywith y1<¯y < y2, such that Ψ(¯y;¯ b, ¯γR) = 0, and Ψ is a monotone function in the variable yat the point ¯y. Therefore, if Ψ(y1;¯ b, ¯γR)>Ψ(y2;¯ b, ¯γR) we can assure the existence of a stable crossing limit cycle, and when Ψ(y1;¯ b, ¯γR)<Ψ(y2;¯ b, ¯γR) the existence of an unstable crossing limit cycle. Here, we state an important remark about persistence of crossing periodic orbits. Remark 4.Assume for system (5) the existence of a crossing periodic orbit corresponding to the condition Ψ(¯y,¯ b, ¯γR) = 0 with ¯y6= ˆyand that for certain values α > 0 and β > 0 the inequality Ψ(¯y−α;¯ b, ¯γR)Ψ(¯y+ β;¯ b, ¯γR)<0 holds. By the continuity of the function Ψ with respect to the parameters band γR, we must have for |ε1|and |ε2|small enough sign Ψ(¯y−α;¯ b+ε1,¯γR+ε2) = sign Ψ(¯y−α;¯ b, ¯γR) and sign Ψ(¯y+β;¯ b+ε1,¯γR+ε2) = sign Ψ(¯y+β;¯ b, ¯γR). Then, there exists at least a value ε3such that Ψ(¯y+ε3;¯ b+ε1,¯γR+ε2) = 0, and with the function Ψ(y;¯ b+ε1,¯γR+ε2) having the same type of monotonicity at the points ¯yand ¯y+ε3. Hence, the assumed crossing periodic orbit persists with the same stability under a sufficiently small perturbation in parameters band γR. Next, we will analyze the persistence of the crossing critical cycle, that is the persistence of the zeroes Ψ(ˆy;b) = 0.First we present a technical lemma. Three Limit Cycles Assembling Two Linear Foci 245 Lemma 2. If xR<0then for every value yR>0there exists a unique value 0< bR< yRsuch that P−1 R(yR;bR) = 0. Proof: Let us introduce the function ϕ(b) = P−1 R(yR;γR, b), defined for every value of γRand 0 ⩽b⩽yR. By considering the endpoints of the domain of the function ϕwe have ϕ(0) <0 and ϕ(yR) = yR> 0, so there exists a value bRwith 0 < bR< yRsuch that ϕ(bR) = P−1 R(yR;γR, bR) = 0. Since funtion ϕis increasing, see Lemma 1(c), we conclude the uniqueness of the value bR. PL, P −1 R y bbCC yR ˆyˆy+ε P−1 R(y;bCC ) P−1 R(y;b) PL(y) Figure 8. The map PLfor γL<0 and the map P−1 R in a neighborhood of b=bCC. For b=bCC we have PL(ˆy) = P−1 R(ˆy) = 0 and there exists a crossing critical cycle. For bCC −ε < b < bCC we have Ψ(ˆy;b)<0, while Ψ(ˆy+ε;b)>0 and one unstable crossing limit cycle exists. Proof of Proposition 2:From Proposition 3(b) there exists a value ˆy > 0, which does not depend on parameter b, with PL(ˆy) = 0. By Lemma 2, we can select a unique value bCC =bCC(γR) with 0 < bCC (γR)<ˆysuch that P−1 R(ˆy, γR, bCC (γR)) = 0. Taking γR= 0 in (15) it is direct to obtain bCC (0) = ˆy/2. 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