On the existence and uniqueness of limit cycles in Liénard differential equations allowing discontinuities

ON THE EXISTENCE AND UNIQUENESS OF LIMIT CYCLES IN LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES Abstract. In this paper we study the non–existence and the uniqueness of limit cycles for the Li´enard differential system of the form ¨x−f(x) ˙x+g(x) = 0 where the functions fand gsatisfy xf(x)>0 and xg(x)>0 for x6= 0 but they can be discontinuous at x= 0. In particular our results allow first to prove the non–existence of limit cycles under suitable assumptions, and second to prove the existence and uniqueness of a limit cycle in a class of discontinuous Li´enard systems which are relevant in engineering applications. 1. Introduction and statement of the main results One of the main problems in the qualitative theory of planar differential equations is to know the existence of limit cycles and its number. This problem restricted to polynomial differential equations is the well known 16–th Hilbert’s problem [7]. Since Hilbert’s problem turned out a strongly difficult one Smale [12] particularized it to Li´enard differential systems in his list of problems for the present century. For Li´enard systems there are many results about the non–existence, existence and uniqueness of limit cycles, see for instance [3, 5, 9, 13, 14]. In this paper we provide a new contribution to this subject which can be also applied to Li´enard differential systems with some kind of discontinuities. We consider for x∈[a, b], where −∞ < a < 0< b < ∞, the Li´enard differential equation (1) x′′ −f(x)x′+g(x) = 0, 1991 Mathematics Subject Classification. 58F21, 34C05, 58F14. Key words and phrases. Li´enard differential systems, uniqueness, non–existence, limit cycles, discontinuities. ∗The first author is partially supported by a MEC/FEDER grant number MTM2005- 06098-C02-01 and by a CICYT grant number 2005SGR 00550. The second and third authors are partially supported by a MEC/FEDER grant number MTM2006-00847. E. Ponce acknowledges the hospitality and support from Centre de Recerca Matem`atica, Bellaterra, Barcelona, Spain, from January to July 2007. 1 2 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES where the functions fand gare given by (2) f(x) = f1(x) if x < 0, f2(x) if x > 0,g(x) = g1(x) if x < 0, g2(x) if x > 0, being f1,g1continuously differentiable in [a, 0], and f2,g2continuously differentiable in [0, b]. Note that the functions fand gare not defined at x= 0 so that, if we eventually define f(0) and g(0), they are allowed to have a jump discontinuity at the origin. By using the classical Li´enard plane we can obtain the equivalent differential system (3) x′=F(x)−y, y′=g(x),where F(x) = Zx 0 f(s)ds, and it is understood that F(0) = 0,while g(0) is not defined by now. This system has associated the vector field (4) X(x) = X1(x) if x≤0, X2(x) if x≥0,where Xi(x) = F(x)−y gi(x), with x= (x, y)Tand standing i= 1 for x≤0, and i= 2 for x≥0. The ambiguity in the definition of X(x) on x= 0 will be clarified later on. Since the system can be discontinuous we must adopt some criterion in order to define solutions starting at or passing through the allowed discontinuity line x= 0. Typically this is done by using the so called Filippov approach, see for instance [10]. However here only the vertical component of the vector field (4) could be discontinuous at the y-axis, while its horizontal component turns out to be continuous. In fact, we have x′=−yon x= 0. Thus if we consider for instance orbits starting at points with x < 0,then these orbits are well defined whenever they do not touch the y-axis but they can arrive at this straight line (obviously only at points (0, y) with y≤0) by extending g(x) as if g(0) were equal to g1(0).Now starting from the point (0, y) with y < 0 we assume that g(0) = g2(0) and we continue the orbit inside x > 0 using system (3). From the above paragraph and using the standard terminology of planar Filippov systems [10], the crossing set of the discontinuity line of system (3) includes the negative y-axis. Similar arguments for x > 0 imply that the crossing set is the y-axis without the origin. In [10] the origin is then called a singular isolated sliding point. In short, except orbits arriving at the origin and assuming that the system is actually discontinuous, it is natural to allow concatenation of solutions in an obvious way so that the system has no sliding (Filippov) solutions. The only possible singular point may be the origin, where each vector field can LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES 3 Figure 1. The three main cases for the local phase plane at the origin when it is not a boundary equilibrium point: regular point, pseudo-saddle and pseudo-focus. either vanish or have a tangency with the y-axis. If at least one vector field vanishes at the origin we say that it is a boundary equilibrium point. If both vector fields are not zero at the origin we still can have a pseudo-equilibrium point when both vector fields are anti-collinear (i.e. g1(0)g2(0) <0). Then it behaves as an equilibrium point that may be reached in finite time. Its stability and local phase portrait will be determined by studying its nearby orbits, see Figure 1. Proposition 1. For system (3) the following statements hold. (a) If g1(0)g2(0) >0then the origin can be thought of a regular point. (b) If g1(0)g2(0) = 0 then the origin is a boundary equilibrium point. (c) If g1(0)g2(0) <0then the origin is a pseudo-equilibrium point, being of saddle type if g1(0) >0and g2(0) <0, and of focus type if g1(0) <0and g2(0) >0. Proposition 1 will be proved in Section 2. From the point of view of practical engineering problems the most interesting case corresponds to the existence of a pseudo-equilibrium point or a proper equilibrium point of focus type at the origin, because then it is possible that the system behaves locally or even globally as an oscillator. Thus we will be mainly interested in possible periodic orbits. In this context, to assure that there is no more singular points the following hypothesis is assumed. (H1) The function gsatisfy xg(x)>0 for x6= 0. We will require that the divergence of the vector field does not change its sign in each side of the discontinuity line, i.e. (H2) The function fsatisfy xf(x)>0 for x6= 0. Under the last hypothesis we have positive divergence for x > 0 and negative divergence for x < 0. Then in order to have some periodic orbit 4 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES surrounding the origin, there must be some balance between the x-positive and x-negative parts of the interior of the bounded region limited by the periodic orbit. This idea will be precisely stated below in Lemma 7 but in the same spirit of comparing the x-positive and x-negative half-planes and following [1], it will be useful to introduce some auxiliary functions as follows. Under Hypothesis H2 and recalling the definition of Fin (3), we define a variable p=p(x) = F(x). As p′(x) = f(x), then p(x)≥0 for all x, and sgn(p′(x)) = sgn(x) for x6= 0. We deduce that the function p(x) has inverse functions both for x≤0 and for x≥0, namely the non-positive decreasing function (5) x1: [0, F(a)] →[a, 0],such that F(x1(p)) = p, and the non-negative increasing function (6) x2: [0, F(b)] →[0, b],such that F(x2(p)) = p. Hence for x6= 0 we have that both systems (3) and (4) are equivalent to the two differential equations (7) dy(xi(p)) dp =g(xi(p)) F(xi(p)) −y 1 f(xi(p)) =1 p−y g(xi(p)) f(xi(p)), where i= 1,2,according to x < 0 or x > 0 respectively, and these new differential equations are both meaningful only for p > 0. Now by considering the functions (8) hi(p) = g(xi(p)) f(xi(p)), equations (7) can be written in the more compact form (9) dy (xi(p)) dp =hi(p) p−y. Note that hi(p)>0 for p > 0 and i= 1,2,and that the effect of considering equations (9) instead of the original systems (3) or (4) can be thought of as if the plane (x, y) had been folded into the half-plane (p, y) with p > 0. When h1(p) = h2(p) for psufficiently small and the origin is a topological focus it is not difficult to show that we have indeed a center, see for instance Theorem 11.3 in [8]. We add a third hypothesis precluding such possibility. It is written in a dual way to facilitate the checking of its validity in the applications. (H3) Assume that there exist the two limits lim x→0− g(x) f(x)= lim p→0+h1(p) = l1,lim x→0+ g(x) f(x)= lim p→0+h2(p) = l2 LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES 5 satisfying 0≤l2≤l1<∞, and if l2=l1then h2(p)< h1(p) for p > 0 and sufficiently small (when l2< l1this last requirement is always fulfilled). It is worth mentioning that this hypothesis implies that the origin is topologically an unstable focus when l2>0, see Lemma 11. Next result states a necessary condition for the existence of periodic orbits under the above hypotheses. Theorem 2. Let fand gbe the functions defined in (2) such that fiand giare of class C1in [a, 0] and [0, b]for i= 1,2, respectively, where −∞ < a < 0< b < ∞. Let Fand hibe the functions defined in (3) and (8) and assume that hypotheses H1-H3 are fulfilled. If system (3) has a periodic orbit contained in the band a < x < b, then the system (10) F(x1) = F(x2),g(x1) f(x1)=g(x2) f(x2), has at least one solution (x1, x2) = (s1, s2)with a < s1<0< s2< b, or equivalently there exists at least one solution ˆp∈(0, F(a))∩(0, F (b)) for the equation h1(p) = h2(p). Theorem 2 is proved in Section 2. Now we give a result on uniqueness of limit cycles for Li´enard equations where discontinuities are allowed at x= 0. Theorem 3. Under the same conditions of Theorem 2, assume that system (10) has exactly one solution (x1, x2) = (s1, s2)with a < s1<0< s2< b, or equivalently there exists exactly one solution ˆp∈(0, F(a)) ∩(0, F(b)) for the equation h1(p) = h2(p). The following statement holds. If the positive function (11) α(x) = g(x) f(x)F(x) is increasing for x∈(a, 0),or equivalently the positive function (12) h1(p) p is decreasing for p∈(0, F(a)),then system (3) has at most one periodic orbit contained in the band a < x < b, and if it exists has a negative characteristic exponent. Theorem 3 is proved in Section 2. Although our main motivation is the case of discontinuous systems, it should be noted that the above results can be useful also for continuous differential equations. For instance we can state the following result. 6 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES Proposition 4. The following Li´enard system (13) x′=αx2+βx3+x4−y, y′=x, where β > 0and 9β2−32α < 0has no limit cycles in the plane. Proposition 4 is proved in Section 4. We finish by considering an application of the above results to discontinuous piecewise linear differential systems. This class is increasingly used in engineering and applied sciences to model a large variety of technological devices and physical systems [2, 15]. Similar differential systems had been considered before in [6] but under the assumption of continuity for the corresponding vector field. Theorem 5. Consider the Li´enard piecewise linear differential system (14) ˙x=t1x−y, ˙y=d1x+a1,if x < 0,˙x=t2x−y, ˙y=d2x+a2,if x≥0, where it is assumed t1<0, d1>0, a1<0, t2>0, d2>0, a2>0. Then the following statements hold. (a) If a2/t2< a1/t1then a necessary condition for the existence of periodic orbits is d2/t2 2> d1/t2 1.If the system has periodic orbits, then it has a unique periodic orbit which is a stable limit cycle. (b) If a1/t1< a2/t2then a necessary condition for the existence of periodic orbits is d1/t2 1> d2/t2 2.If the system has periodic orbits, then it has a unique periodic orbit which is an unstable limit cycle. (c) If a2/t2=a1/t1then either the system has no periodic orbits when d1/t2 16=d2/t2 2, or it has a center at the origin when d1/t2 1=d2/t2 2. Theorem 5 is proved in Section 5. Observe that statement (c) of Theorem 5 when 0 < a2/t2=a1/t1and d1/t2 1=d2/t2 2says that the origin is a center even when the dynamics of the linear differential system in each half–plane could be of node type. This situation happens when di t2 i≤1 4 for i= 1,2. When both dynamics are of focus type and we are under the assumptions of statements (a) and (b) of Theorem 5 the necessary condition for the existence of limit cycles is also sufficient, as stated in our last main result. LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES 7 Theorem 6. Under the assumptions of Theorem 5 and if di t2 i >1 4 for i= 1,2,then the following statements hold. (a) If a2/t2< a1/t1then the system has periodic orbits if and only if d2/t2 2> d1/t2 1, and in such case it has a unique periodic orbit which is a stable limit cycle. (b) If a1/t1< a2/t2then the system has periodic orbits if and only if d1/t2 1> d2/t2 2, and in such case it has a unique periodic orbit which is an unstable limit cycle. Theorem 6 is proved in Section 5. 2. On the origin and the periodic orbits In this section we prove Proposition 1 and we give some preliminary results necessary for the proof of Theorem 2. Proof of Proposition 1. The vector fields at the origin are X1(0,0) = (0, g1(0)) and X2(0,0) = (0, g2(0)), see (4). Then from (1) we have x′′(0) = −gi(0) for i= 1,2. Therefore if g1(0) and g2(0) are both positive or both negative then X1(0,0) and X2(0,0) are collinear and the orbits of both vector fields in a neighborhood of the origin have the same convexity. Consequently we can define the vector field at the origin in such a way the orbit through the origin has a quadratic tangency with the y-axis. This completes the proof of statement (a). Statement (b) follows directly from the definitions. If g1(0)g2(0) <0 then the vector fields at the origin are anti-collinear and so the origin is a pseudo-equilibrium point. Assume g1(0) >0 and g2(0) <0. Then the vector field X1has a visible quadratic tangency, that is, the orbit of x′=X1(x) through the origin is locally contained in x≤0 for backward and forward times. Similarly, the vector field X2has also a visible quadratic tangency in x≥0, see Figure 1. Hence the origin is a topological saddle. When g1(0) <0 and g2(0) >0 the vector fields X1and X2have invisible quadratic tangencies. That is, the unique point of the orbit of x′=X1(x) through the origin locally contained in x≤0 for backward and forward times is the origin itself, and similarly for X2; see Figure 1. Now the origin is a topological focus. This ends the proof of statement (c).  Now we extend a necessary condition for the existence of periodic orbits fulfilled by smooth vector fields to the case of our discontinuous differential systems. 8 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES Lemma 7. Consider the functions fand gdefined as in (2). If system (3) has a periodic orbit Γand the interior of the bounded region limited by Γ includes the origin and it is denoted by ∆, then Γcrosses the y-axis in two points different from the origin, and the function fsatisfies the condition ZZ∆ f(x)dxdy = 0. Proof. Since x′=−yon x= 0 and the origin is in ∆, it follows that Γ intersects the y-axis in two points M= (0, yM) and N= (0, yN) with yM<0< yN. We define ∆1,Γ1,and ∆2, Γ2to be the parts of ∆ and Γ contained in x < 0 and x > 0 respectively. We denote by Λ the oriented segment on the y-axis from the point Mto the point Nwhile the same segment with the opposite orientation is denoted by −Λ. Then by applying the Green’s Theorem we have ZZ∆ f(x)dxdy =ZZ∆1 f(x)dxdy +ZZ∆2 f(x)dxdy =ZΓ1 [F(x)−y]dy−g(x)dx+ZΛ [F(x)−y]dy−g(x)dx+ ZΓ2 [F(x)−y]dy−g(x)dx+Z−Λ [F(x)−y]dy−g(x)dx = 0 + ZyN yM (−y)dy + 0 + ZyM yN (−y)dy = 0, and the conclusion follows.  3. Proof of Theorems 2 and 3 First we prove Theorem 2. Proof of Theorem 2. We start by noticing that if system (3) has singular points they must be on the y-axis because xg(x)>0 if x6= 0. Also we have g1(0) ≤0 and g2(0) ≥0. Since x′=−ywhen x= 0 the unique possible singular point is the origin, and from Proposition 1 it is a boundary equilibrium point or a pseudo-focus because g1(0)g2(0) ≤0. Assume that system (3) has a periodic orbit Γ contained in the band a < x < b. As a consequence of the Poincar´e-Bendixson Theorem for phase portraits in the plane (see for instance [4]) the interior of the bounded region limited by Γ must contain a singular point. Such point in our system must be the origin. Next some geometrical properties of the periodic orbit Γ will be established. First since x′=−yon the y-axis, the orbit Γ is run in counterclockwise sense. Let A= (xA, yA) and B= (xB, yB) be the points on Γ LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES 9 A B C L M N O K Λ Γ ˜ Γ Ω Figure 2. Notable points associated to a periodic orbit (thick line). For x < 0, it is sketched the orbit passing through (xC, yC), where yB=yC=F(xC). The line AOB is the graph of the curve y=F(x). for which the variable xassumes its minimum and maximum values, then xA<0< xB.Since for x6= 0 we have (15) dx dy =F(x)−y g(x), and this derivative vanishes for x=xAand x=xB,one obtains yA=F(xA) and yB=F(xB).Moreover this derivative only vanishes at the points A and B. Indeed when dx/dy = 0 the second derivative is given by d2x dy2=dF dx dx dy −1g(x)−[F(x)−y]dg dx dx dy g(x)2dx dy =0 =−1 g(x), which has a definite sign, in fact the opposite sign to x. Then the derivative (15) vanishes only once for x > 0 and only once for x < 0,and so the points Aand Bare the unique points where the orbit Γ intersects the curve defined by the equation y=F(x) denoted by Ω. It follows that Γ intersects any straight line Ldefined by x=qwith xA< q < xBin exactly two points (q, yα) and (q, yβ) with yα< F(q)< yβ. 16 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES and from (23) we conclude that I3<0.We compute I4=Z Γ:N2N f(x(t))dt +Z ˜ Γ:LL1 f(˜x(t))dt = 0 Z s2 f(x) F(x)−yu(x)dx + s1 Z 0 f(x) F(x)−˜yu(x)dx = ˆp Z 0 dp yu(x2(p)) −p− ˆp Z 0 dp ˜yu(x1(p)) −p= ˆp Z 0 [˜yu(x1(p)) −yu(x2(p))] dp [yu(x2(p)) −p] [˜yu(x1(p)) −p], and from (23) we conclude that I4<0.Hence I=I1+I2+I3+I4<0 and ρ=I+ZΓ:NAM f(x(t))dt −Z˜ Γ:LCK f(˜x(t))dt. We define J1=ZΓ:NA f(x(t))dt −Z˜ Γ:LC f(˜x(t))dt, J2=ZΓ:AM f(x(t))dt −Z˜ Γ:CK f(˜x(t))dt, so that ρ=I+J1+J2. Now we will show that J1<0 and J2<0. We compute the integral J1= yA Z 0 dp p−yu(x1(p)) − yB Z 0 dp p−˜yu(x1(p)) = yA Z 0 dp p−yu(x1(p)) − yA Z 0 dp p−ˆyu(p) = yA Z 0 yu(x1(p)) −ˆyu(p) [p−yu(x1(p))] [p−ˆyu(p)]dp where the function ˆyu(p) is given by ˆyu(p) = µ˜yux1µ−1p and µ=yA yB >1. Clearly the function ˆyu(p) is a solution of the differential equation dy dp =ˆ h1(p) p−y, LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES 17 where ˆ h1(p) = µh1µ−1p. The function αdefined in (11), which can be written for x < 0 as α(x1(p)) = g(x1(p)) f(x1(p)) p=h1(p) p, is an increasing function of x1, and so a decreasing function of p. Then h1µ−1p> µ−1h1(p),so h1(p)< µh1µ−1p=ˆ h1(p) for p≥0. We recall that for y=yu(x1(p)) we knew that dy dp =h1(p) p−y. Now from the equality ˆyu(yA) = yA yB ˜yux1yB yA yA=yA=yu(x1(yA)) , and using the statement (b) corresponding to Remark 9 for h1and ˆ h1in the interval [0, yA] we get yu(x1(p)) <ˆyu(p) for 0 < p < yA,and consequently J1<0. Similarly we can show that J2<0,and the proof is complete.  4. Proof of Proposition 4 Hypotheses H1 clearly holds for system (13). We see that f(x) = x(2α+ 3βx + 4x2). Obviously α > 0 and since the discriminant of the quadratic factor is negative Hypotheses H2 holds. Regarding Hypothesis H3 we see that g(x) f(x)=1 2α+ 3βx + 4x2, so that l1=l2= 1/(2α) but since β > 0 it is clear that h2(p)< h1(p) for p > 0 and sufficiently small because x1(p)<0 but x2(p)>0. Now we look for solutions (x1, x2) = (s1, s2) with s1<0< s2of system αx2 1+βx3 1+x4 1=αx2 2+βx3 2+x4 2,1 2α+ 3βx1+ 4x2 1 =1 2α+ 3βx2+ 4x2 2 , or equivalently of system α(x2 2−x2 1) + β(x3 2−x3 1) + x4 2−x4 1= 0,3β(x2−x1) + 4(x2 2−x2 1) = 0. After removing the obvious factor x2−x1>0 we get the system α(x1+x2)+β(x2 1+x1x2+x2 2)+x3 1+x2 1x2+x1x2 2+x3 2= 0,3β+4(x1+x2) = 0. Substituting the second equation in the first one multiplied by 4, we get −3αβ+4β(x2 1+x1x2+x2 2)−3β(x2 1−x1x2+x2 2)−3βx1x2= 0,3β+4(x1+x2) = 0, 18 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES and new simplifications lead to the quadratic system −3α+x2 1+ 4x1x2+x2 2= 0,3β+ 4(x1+x2) = 0. A new substitution in the first equation multiplied by 16 gives the system 9β2−48α+ 32x1x2= 0,3β+ 4(x1+x2) = 0. So the solutions x1and x2must be roots of the quadratic x2+3β 4x+48α−9β2 32 = 0, which has real different solutions only if 9β2−32α > 0, and this is contrary to the initial assumption. Then from Theorem 2 no limit cycles are possible. 5. Proof of Theorems 5 and 6 First we show Theorem 5. Proof of Theorem 5. We first check the hypotheses H1–H3 in order to see that both Theorem 2 and Theorem 3 can be applied. Hypotheses H1 and H2 are immediate. We will use the functions hiin checking Hypothesis H3, and noting that xi(p) = p/tifor i= 1,2,we have hi(p) = di t2 i p+ai ti , for i= 1,2. Now Hypothesis H3 is fulfilled whenever l2=a2/t2< l1= a1/t1, or if a2/t2=a1/t1when d2/t2 2< d1/t2 1. The equation h1(p) = h2(p) becomes equivalent to (24) d1 t2 1−d2 t2 2p=a2 t2−a1 t1 , which has a unique positive solution only if d1 t2 1−d2 t2 2a2 t2−a1 t1>0. Now statement (a) of Theorem 5 is a direct consequence of Theorems 2 and 3. Statement (b) can be shown by using statement (a) applied to the system ˙x= (−t2)x−y, ˙y=d2x+ (−a2),if x≤0,˙x= (−t1)x−y, ˙y=d1x+ (−a1),if x > 0, which corresponds to systems (14) after doing the change of variable (x, τ)→ (−x, −τ). Regarding statement (c), it is obvious that equation (24) has no solutions different from zero when d1/t2 1−d2/t2 26= 0 and the first assertion then comes LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES 19 −y −y −P(y) y=p y=p y =−p p p P2(y) −P(y) = −P−1 1(P2(y)) ε1 ε2 Figure 4. Building the Poincar´e map near the origin by using the half–return maps P1and P2. from Theorem 2. In the remaining case we have h1(p) = h2(p) for all pand the conclusion on having a center comes from the application of Theorem 11.3 in [8] to system (19). This completes the proof of Theorem 5.  Starting from system (19) corresponding to system (14), when both dynamics are of focus type (stable for x < 0, unstable for x > 0) it is possible to define globally a Poincar´e return map by introducing a transversal section to the flow. We select for that the negative y–axis and define P: (0,∞)→(0,∞) which maps the coordinate y > 0 of the point (0,−y) into the vertical coordinate P(y) of the point (0,−P(y)), where both points are the initial and the final point, respectively, of one orbit that gives counterclockwise a complete turn around the origin. For more details on the definition of this Poincar´e map see the proof of Lemma 10. The explicit computation of this map Pshould solve the problem of determining the exact number of periodic orbits; however this is not possible in general. The following result will be needed later. Lemma 10. Under the assumptions of Theorem 6 the derivative of the Poincar´e map Psatisfy lim y→∞ dP dy =eπ(κ2−κ1), 20 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES where for i= 1,2, κi=1 q4di t2 i−1 . Proof. To work in a more compact way we choose the folded plane, that is systems (16), namely dp dτ =p−y, dy dτ =di t2 i p+ai ti , for p≥0, and we define ωi=sdi t2 i−1 4, pe i=−aiti di for i= 1,2. Integrating both linear systems taking as initial point (0,−y), we have pi(τ)−pe i yi(τ)−ye i= exp τ 2Ci(τ)0−pe i −y−ye i where ye i=pe i, and Ci(τ) =      cos(ωiτ) + sin(ωiτ) 2ωi−sin(ωiτ) ωi disin(ωiτ) t2 iωi cos(ωiτ)−sin(ωiτ) 2ωi      . After one half–turn around the origin following these solutions (pi(τ), yi(τ)), we will arrive up to the positive part of the y–axis for certain values τisuch that pi(τi) = 0 with 0 < ωiτi< π, see Figure 4. The corresponding values of yi(τi) allows to define the half–return maps Pi: (0,∞)→(0,∞) with Pi(y) = yi(τi) and pi(τi) = 0 with 0 < ωiτi< π, for i= 1,2. Now the return map P(y) of system (19) corresponding to system (14) can be recovered by taking P(y) = P−1 1(P2(y)). As it is shown in [6] for the continuous case, the study of such half–return Poincar´e maps is not possible explicitly and must be done in a parametric way. Thus introducing the notation θi=ωiτiand κi= 1/(2ωi) for i= 1,2, the map Piis determined by the equation eκiθi  cos θi+κisin θi−2κisin θi 1+κ2 i 2κisin θicos θi−κisin θi −pe i −y−pe i=−pe i Pi(y)−pe i LI´ ENARD DIFFERENTIAL EQUATIONS ALLOWING DISCONTINUITIES 21 and it is parametrically described for each value of θi∈(0, π) as follows, y=−pe i e−κiθi−cos θi+κisin θi 2κisin θi , Pi(y) = −pe i eκiθi−cos θi−κisin θi 2κisin θi , for i= 1,2. Hence a straightforward computation now shows that for the derivatives we also have the parametric representation dPi dy (θi) = 1−eκiθi(cos θi−κisin θi) 1−e−κiθi(cos θi+κisin θi)=e2κiθiy Pi(y), so that lim y→∞ dPi(y) dy = lim θi→π− 1−eκiθi(cos θi−κisin θi) 1−e−κiθi(cos θi+κisin θi)=1 + eκiπ 1 + e−κiπ=eκiπ. We can conclude by the chain rule and the inverse function theorem that lim y→∞ dP(y) dy = lim y→∞ dP−1 1(P2(y)) dy =1 eκ1πeκ2π=eπ(κ2−κ1), and the lemma follows.  This last result can be also obtained by resorting to the techniques followed in [11]. We finish by giving the proof of Theorem 6. For that we show first another technical result Lemma 11. Hypothesis H3 implies that the origin is an unstable topological focus if l2>0. Proof. We will show that when y > 0 is sufficiently small the Poincar´e map introduced in this section satisfies P(y)> y. Taking a point (εi, εi) on the line y=psufficiently near the origin, we know that for the orbit passing through this point dpi dy = 0,d2pi dy2=−1 hi(εi), and then the corresponding orbits can be approximated by pi(y) = εi−1 2hi(εi)(y−εi)2+O(y−εi)3, which cuts the y–axis in the points y± i≈ ±p2hi(εi)εi+εi≈ ±p2liεi+εi. Note that for εisufficiently small we have that y− i<0. 22 JAUME LLIBRE, ENRIQUE PONCE AND FRANCISCO TORRES We choose ε1and ε2for the systems with i= 1,2,in such a way that the two quadratic approximations for the orbits coincide in the positive y–axis, namely (25) ε1+p2h1(ε1)ε1=ε2+p2h2(ε2)ε2. From Remark 9 and considering only the part of the orbits contained in the region y > p we can assure that ε1< ε2, see Figure 4. Now taking y=−y− 2 we can make the approximation P(y)≈ −y− 1so that P(y)−y≈ −(ε1−p2h1(ε1)ε1) + ε2−p2h2(ε2)ε2= 2(ε2−ε1)>0, where we have taken into account the equality (25). This implies that the origin is an unstable topological focus, see Figure 4.  Proof of Theorem 6. Reasoning like in the proof of Theorem 5 we must only show statement (a). From Lemma 11 we know that the origin is unstable and in particular that for the Poincar´e map Pintroduced in this section we have P(y)> y for y > 0 and sufficiently small. The assumptions assure that d2/t2 2> d1/t2 1, and using that the function 1/√4x−1 is decreasing for x > 1/4 we see that κ2−κ1<0. Therefore from Lemma 10 we have L= lim y→∞ dP dy =eπ(κ2−κ1)<1. We will now claim that there exists y∗>0 with P(y∗)< y∗so that from the intermediate value theorem we deduce the existence of a periodic orbit. Then the conclusion of the theorem follows from Theorem 3. Effectively we can assure that there exists a certain value ¯ysuch that for y≥¯ywe have dP dy <1 + L 2=¯ L < 1. If P(¯y)<¯ywe are done. 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Smale,Mathematical Problems for the Next Century, Mathematical Intelligencer 20 (1998), 7–15. [13] D. Xiao and Z. Zhang,On the uniqueness and nonexistence of limit cycles for predator–prey systems, Nonlinearity 16 (2003), 1185–1201. [14] Zhang Zhifen, Ding Tongren, Huang Wenzao and Dong Zhenxi,Qualitative Theory of Differential Equations, Translations of Math. Monographs, Vol. 101, Amer. Math. Soc, Providence, 1992. [15] Z.T. Zhusubaliyev and E. Mosekilde,Bifurcations and Chaos in Piecewise– Smooth Dynamical Systems, World Scientific, Singapoore, 2003. Jaume LLibre, Departament de Matem` atiques, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain E-mail:jllib[email protected]ab.cat Enrique Ponce and Francisco Torres, E.T.S. Ingenieros, Camino de los Descubrimientos, 41092 Sevilla, Spain, E-mail:eponc[email protected], [email protected]