On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry

http://www.gsd.uab.cat ON THE EXISTENCE AND UNIQUENESS OF LIMIT CYCLES IN PLANAR PIECEWISE LINEAR SYSTEMS WITHOUT SYMMETRY JAUME LLIBRE, MANUEL ORD ´ O˜ NEZ, AND ENRIQUE PONCE Abstract. Some techniques to show the existence and uniqueness of limit cycles, typically stated for smooth vector fields, are extended to continuous piecewise-linear differential systems. New results are obtained for systems with three linearity zones without symmetry and having one equilibrium point in the central region. We also revisit the case of systems with only two linear zones giving shorter proofs of known results. 1. Introduction and statement of main results For planar differential systems, the analysis of the possible existence of limit cycles and their uniqueness is a problem which has attracted the interest of many works in the past. For smooth systems, good classical references in the field are the books [24, 25]. The restriction of this problem to polynomial differential equations is the well-known 16th problem Hilbert’s problem [14]. In this context, a celebrated and rather general canonical form is the Li´enard equation. Since Hilbert’s problem turns out to be a strongly difficult one, Smale [22] has particularized it to Li´enard differential systems in his list of problems for the present century. For just continuous or even smooth Li´enard systems there are many results on the non-existence, existence and uniqueness of limit cycles, see for instance [1, 4, 6, 18, 23, 25]. Going beyond the smooth case, the first natural step is to allow non-smooothness while keeping the continuity, as has been done in some recent works [9, 15, 16]. In a further step, other authors have considered a line of discontinuity in the vector field defining the planar system, see [12, 26]. In this paper, we adapt some techniques from the smooth case to continuous piecewise linear differential systems, obtaining new results for systems without symmetry. We also revisit the case of systems with only two linear zones giving shorter proofs of known results. J. Llibre is partially supported by a MICINN/FEDER grant number MTM2008–03437, by an AGAUR grant number 2009SGR–410 and by ICREA Academia. M. Ord´o˜nez is partially supported by a MICINN/FEDER grant number ECO2010-17766. E. Ponce is partially supported by a MICINN/FEDER grant number MTM2009–07849 and Junta de Andaluc´ıa grant number P08-FQM-03770; he also wants to acknowledge Prof. Marco Sabatini for his helpful suggestions and discussions on Massera’s method. 1 This is a preprint of: “On the existence and uniqueness of limit cycles in planar continuous piecewise linear systems without symmetry”, Jaume Llibre, Manuel Ord´o˜nez, Enrique Ponce, Nonlinear Anal. Real World Appl., vol. 14(5), 2002–2012, 2013. DOI: [10.1016/j.nonrwa.2013.02.004] 2 JAUME LLIBRE, MANUEL ORD ´ O˜ NEZ, AND ENRIQUE PONCE -1 1 x y F(x) g(x) y=δ Figure 1. The graphs of Fand gconsidered in the paper have no symmetries. The point indicates the only equilibrium point, which is initially not at the origin. In most of interesting applications, piecewise linear differential systems have two or three different linearity regions separated by parallel straight lines, see [2]. For such systems we assume without loss of generality that the lines separating these regions are x=−1 and x= 1. Furthermore, it is rather usual for these systems to exhibit only one equilibrium point, which is supposed to be in the central linearity region when the system has three linear zones, to be denoted in the sequel as L (left), C (central), and R (right). If the system has only two zones, we assume that the left and the central zones are in fact only one. Then, under these generic assumptions, continuous piecewise linear differential systems (CPWL, for short) can be written in the Li´enard form ˙x=F(x)−y, ˙y=g(x)−δ, (1) where F(x) =    tR(x−1) + tC,if x≥1, tCx, if |x| ≤ 1, tL(x+ 1) −tC,if x≤ −1, (2) and g(x) =    dR(x−1) + dC,if x≥1, dCx, if |x| ≤ 1, dL(x+ 1) −dC,if x≤ −1. (3) Our assumption on the equilibrium point requires that the determinant in central region be positive, that is, dC>0, and also −dC< δ < dC, along with dL, dR≥0. This means that the only equilibrium point is located at the line x= ¯x=δ/dC∈(−1,1). The corresponding traces tL, tC, tRcould be arbitrary, but we know from Bendixson theory that they cannot have the same sign for the existence of limit cycles. EXISTENCE AND UNIQUENESS OF LIMIT CYCLES IN CPWL SYSTEMS 3 Remark 1. The above formulation includes as particular cases the following ones. If tC=tLand dC=dLthen we have a system with only two different linearity zones, thoroughly analyzed in [9]. If tR=tL,dR=dLand δ= 0, then we have a symmetric system with three different linearity zones, thoroughly analyzed in [10]. The case tR=tL,dR=dLand δ6= 0 was considered in [19]. Examples of this last situation appeared in [3]. Under suitable hypotheses that include systems without any symmetries, see Fig. 1, our main results are the following. Theorem 1. Consider the differential system (1)–(3) with only one equilibrium point in the central zone, i.e. dC>0,−dC< δ < dC, and dL, dR≥0. If the external traces satisfy tL, tR<0, while the central trace is positive, that is tC>0, then the equilibrium point is surrounded by a limit cycle which is unique and stable. Theorem 1 is shown in Section 5, and of course, by using the opposite sign distribution for the traces, we could state a similar theorem on existence and uniqueness of an unstable limit cycle. For the case of only two linearity zones the corresponding result is a bit more involved. Theorem 2. Consider the differential system (1)–(3) with only two linearity zones, more specifically, under the assumptions tC=tLand dC=dL>0, and one equilibrium point in the left zone, i.e. δ < dC, and dR≥0. Assume also that the left trace satisfy tL>0, while the right trace is negative, that is tR<0. The following statements hold. (a) A necessary condition for the existence of periodic orbits is that the equilibrium point be a topological focus, that is t2 L−4dL<0. (b) If t2 L−4dL<0then the system has periodic orbits if and only if the following condition holds tL √dL +tR √dR <0. In such case, the equilibrium point is surrounded by a limit cycle which is unique and stable. Theorem 2 is not new. It was stated in [8] in a equivalent way without an explicit proof and indicating how several existing results could be concatenated to get such theorem. It can also be considered as a byproduct of the case-by-case study made in [9]. We include here to emphasize how the useful techniques introduced in proving Theorem 1 allow us to obtain a much shorter proof of Theorem 2. The rest of the paper is organized s follows. First, in Section 2 we illustrate through some relevant examples the usefulness of the achieved results. Next section is devoted to review the ideas underlying the Massera’s approach for uniqueness of limit cycles. Finally, our main results are shown in Section 5. 4 JAUME LLIBRE, MANUEL ORD ´ O˜ NEZ, AND ENRIQUE PONCE 2. Examples We analyze here a celebrated piecewise linear model in mathematical biology. Following [3], the equations for the two-dimensional McKean model of a single neuron take the form C˙v=f(v)−w+I, ˙w=v−γw, (4) where vstands for the voltage, wis the gating variable and f(v) =    −v, if v < a/2, v−a, if a/2≤v≤(1 + a)/2, 1−v, if v > (1 + a)/2. (5) Here, C > 0, γ > 0, Iis a constant drive, and f(v) is a PWL caricature of the cubic FitzHugh-Nagumo nonlinearity f(v) = v(1 −v)(v−a), provided that 0 < a < 1, see [3] for more details. First, we put system (4) in the form given in (1). We start by using instead of wa new variable usuch that w=C(u+γv), so that we get a new system where the dynamics of the second variable depends only on the first one, namely C˙v=f(v)−Cγ −Cu +I, ˙u=v−γf(v)−γI. (6) An appropriate translation and scaling of variables suffices now to arrive at the form (1). We take x= 4v−2a−1, Cy = 4Cu −Cγ(2a+ 1) −4I+ 2a−1,(7) and the computations give tL=tR=−1 C−γ, tC=1 C−γ, dL=dR=1 + γ C, dC=1−γ C,(8) and δ=γ(1 −2a+ 4I)−2a−1 C.(9) Since dL=dR>0, it is now direct to conclude that to have always only one equilibrium point we need γ < 1, for then dC>0. Furthermore, to be able of having oscillations in the model, we also need that γ < 1/C, since then tC>0. Finally, in order to apply Theorem 1, we translate the condition −dC< δ < dC, getting the equivalent inequalities 1 + γ 2γa < I < 1 + γ 2γa+1−γ 2γ, so that, under these conditions we can guarantee a unique stable limit cycle in the model. The uniqueness of the limit cycle and this quantitative information on the admissible range for the drive I, deduced from Theorem 1, is an interesting information which complements the study made in [3]. Note that our results also apply to more general versions of the PWL function f(v), when the slopes of external pieces are different. This is the case EXISTENCE AND UNIQUENESS OF LIMIT CYCLES IN CPWL SYSTEMS 5 for instance of the PWL models considered in [7] to study the oscillations in an electronic circuit whose nonlinearity comes from an Esaki diode. 3. The Massera’s method for uniqueness of limit cycles We review in this section a geometrical argument which is usually known as the Massera’s method; it will allow us, after adequate adaptations, to show the uniqueness of limit cycles in the CPWL differential systems considered in this paper, when they satisfy certain hypotheses. Uniqueness results for limit cycles are typically rather involved; see [24, 25], for a review in the subject. Here we reformulate in a specific way the simple and elegant idea proposed by J.L. Massera in his brief note extending a previous result of G. Sansone, see [20] and the recent study on the legacy of the latter author in [21] . First, we recall some notions and introduce some definitions. A period annulus is a region in the plane completely filled by non-isolated periodic orbits. We say that a vector field has the non-negative rotation property whenever along any half-ray starting from the origin the angle of the vector field measured with respect the positive direction of the x-axis does not decrease as long as one moves far from the origin. For a closed orbit surrounding the origin, we say that it is star-like with respect to the origin when any segment joining the origin and a point of the closed orbit has no other points in common with the closed orbit, and consequently such segments are in the interior of the closed orbit. The following result can be stated. Lemma 1. (Massera’s method) Consider a Li´enard system with a continuous vector field given by ˙x=F(x)−y, ˙y=g(x), and assume that xg(x)>0for x6= 0, and that F(0) = 0, so that the only equilibrium point is at the origin. Assume that the system has the non-negative rotation property and that period annuli are not possible. If the system has a closed orbit then it is star-like with respect to the origin and it is a limit cycle which is unique and stable. Proof. First, we will show that if the system has a closed orbit then it is star-like with respect to the origin. Obviously, since the vector field is continuous the periodic orbit must surround the origin, see Theorem 3.1 in [13]. Suppose that such an orbit is not star-like with respect to the origin. Then there must be a half-ray that starting from the origin intersects the closed orbit in more than one point; in fact such half-ray must have at least three points in common with the closed orbit, see Fig. 2. It is easy to conclude that the angle of the vector field measured with respect the positive direction of the x-axis cannot be monotone, that is, first decreases to increase later or vice versa. This is not compatible with the non-negative rotation property, getting the desired contradiction. We now assume that there exists a closed orbit Γ that surrounds the origin, which must be star-like with respect to it by the above argument, see Fig. 2 (right). Then, using Γ as starting point, one can build a geodesic 6 JAUME LLIBRE, MANUEL ORD ´ O˜ NEZ, AND ENRIQUE PONCE x y x y Γ Figure 2. (Left) Closed orbits that are not star-like with respect to the origin cannot appear in systems with the nonnegative rotation property. (Right) If Γ is a star-like closed orbit, one can build a geodesic system of closed curves by homotetical transformations. Then, the non-negative rotation property assures the stability of the closed orbit Γ and hence its uniqueness. system of closed curves by homotetical transformations, foliating the entire plane by the curves kΓ for all k > 0. Consider now a half-ray starting from the origin and take into account the non-negative rotation property. Of course the vector field is tangent to Γ at the point where the half-ray intersects Γ, see Figure 2 (right). Now the non-negative rotation property assures that in the points where the half-ray intersects the closed curves of the geodesic system near Γ the vector field points in such a direction that it is guaranteed the stability of the periodic orbit, even in the case the periodic orbit considered is not isolated. Since we exclude the possibility of any period annulus and there cannot be consecutive nested stable periodic orbits, if there exists such an orbit then it must be isolated and stable, that is, it should be the unique stable limit cycle. The conclusion follows.  4. Preliminary results and uniqueness of limit cycles In this section we give some preliminary results and also include uniqueness results for possible periodic orbits. Note that if we make in system (1) the change X=−x, Y =−ywe get the system ˙ X=e F(X)−Y, ˙ Y=eg(X) + δ, (10) where the new functions e Fand egare obtained from the given in (2) by interchanging the subscripts L and R. Thus, there is no loss of generality in assuming δ≥0 hereafter. EXISTENCE AND UNIQUENESS OF LIMIT CYCLES IN CPWL SYSTEMS 7 Our first result is just a preparation lemma, reducing by one the number of parameters and looking for a more compact equivalent expression for system (1)–(3). Lemma 2. System (1)–(3) with 0≤δ < dCis topologically equivalent to the system ˙x=Fn(x)−y, ˙y=gn(x),(11) where Fn(x) =    aR(x−xR) + aCxR,if x≥xR, aCx, if xL≤x≤xR, aL(x−xL) + aCxL,if x≤xL, (12) and gn(x) =    bR(x−xR) + xR,if x≥xR, x, if xL≤x≤xR, bL(x−xL) + xL,if x≤xL, (13) with xL=−1−¯x,xR= 1−¯x, for 0≤¯x=δ/dC<1, and the new piecewise slopes satisfy aZ√dC=tZ, bZdC=dZ,for each Z∈ {L, C, R}. Proof. First, we put the equilibrium point at the origin by the translation ex=x−¯x,ey=y−tC¯x. This makes that the new vertical lines separating the zones be ex=xLand ex=xRand the δ-term in the second equation disappears. Next, we make the change of variables and time defined by X=ex,ωY =ey, and τ=ωt, with ω2=dC. We obtain dX dτ =F(X+ ¯x) ω−Y, dY dτ =g(X+ ¯x) ω2, so that the conclusion follows from the two obvious equalities F(X+ ¯x) = ωFn(X) and g(X+ ¯x) = ω2gn(X).  The following remark will be useful to split the analysis of system with three zones into two different subcases with only two zones. Remark 2. System (11)-(13) is invariant under the following symmetry (x, y, t, aC, aL, aR, bL,bR, xL, xR)→(−x, −y, t, aC, aR, aL, bR, bL,−xR,−xL). Now, we consider a system with only two linearity zones which can be obtained from system (11)-(13) by suppressing the left zone and extending the central zone to the left, which is equivalent to assume aL=aCand bL= 1. Proposition 1. Consider the two-zones piecewise linear differential system ˙x=F(x)−y, ˙y=g(x), where g(x) = x, if x < xR, bR(x−xR) + xR,if x≥xR, and F(x) = aCx, if x < xR, aR(x−xR) + aCxR,if x≥xR, 8 JAUME LLIBRE, MANUEL ORD ´ O˜ NEZ, AND ENRIQUE PONCE with aC>0,aR<0,bR≥0and xR>0. The following statements hold. (a) If bR= 1 then the system has the non-negative rotation property. (b) If 0≤bR6= 1 then the system can be transformed in an equivalent system with the non-negative rotation property. Proof. To show the non-negative rotation property we will compute the slope of the vector field along half-rays of the form y=λx. In the following computations there naturally appears the expression F(x)−λx in some denominators; obviously, we can disregard the points of vertical slope in which F(x)−λx = 0. If bR= 1 then g(x) = xfor all x∈R. In this case, the slope of the vector field along the half-rays y=λx is given by mλ(x) = dy dxy=λx =x F(x)−λx, which is constant for x≤xR. For x > xR, it has the derivative dmλ(x) dx =F(x)−λx −x(aR−λ) [F(x)−λx]2=xR(aC−aR) [F(x)−λx]2, which is always positive. The non-negative rotation property is concluded for this simple case. If bR6= 1, the numerator in the computation of the derivative of mλ(x) turns out to be dependent on λand the sign of numerator could change. However, we can transform the system by introducing a new first variable u=u(x) so that the new second equation become ˙y=ufor all u. For that, it suffices to write u= sgn(x)p2G(x),where G(x) = Rx 0g(s)ds. Note that u=xif x≤xRand then the slope of the vector field in this case is not altered. Now, we study its slope for u > xR. Clearly, from u2(x) = 2G(x) we have u(x)u′(x) = g(x) for all x, and so du dx =g(x) u.(14) Therefore, du dt =du dx dx dt =g(x) u[F(x)−y], and in the new variables the system is equivalent to the equation dy du = dy dt du dt =u F(x(u)) −y, that is, to the system ˙u=F(x(u)) −y, ˙y=u. As in the previous case g(x) = x, we can write mλ(u) = dy duy=λu =u F(x(u)) −λu, EXISTENCE AND UNIQUENESS OF LIMIT CYCLES IN CPWL SYSTEMS 9 and also dmλ(u) du =F(x(u)) −λu −u[F′(x(u))x′(u)−λ] [F(x(u)) −λu]2=F(x(u)) −aRu x′(u) [F(x(u)) −λu]2, finally arriving at dmλ(u) du =xR(aC−aR) + aR[x(u)−u x′(u)] [F(x(u)) −λu]2.(15) We will study the sign of x(u)−u x′(u) for x > xR. From (14) and the equality u2= 2G(x) = bR(x−xR)2+ 2xRx−x2 Rfor x > xR, we have x(u)−u x′(u) = xg(x)−u2 g(x)= =x[bR(x−xR) + xR]−[bR(x−xR)2+ 2xRx−x2 R] g(x), and after obvious simplifications, we get x(u)−u x′(u) = (bR−1)xR(x−xR) g(x). Now, if bR≤1 the above expression is non-positive for x > xR, and then the expression in (15) is obviously positive. The non-negative rotation property follows. The remaining case bR>1 can be also managed by noticing that, when x > xR, we have 0<(bR−1)xR(x−xR) g(x)=(bR−1)xR(x−xR) bR(x−xR) + xR <(bR−1)xR bR , so that xR(aC−aR) + aR[x(u)−u x′(u)] > > xR(aC−aR) + aR (bR−1)xR bR = =xRaC−aR bR>0, and since (15) is again positive, the conclusion follows.  From the above result it should be noticed that some systems originally not having the non-negative rotation property can be transformed in equivalent systems satisfying such property. Possible periodic orbits can be deformed in shape by the transformation given in the above proof, but stability and uniqueness results for periodic orbits can be translated between such equivalent systems. More precisely, the following remarks can be stated. Remark 3. From Lemma 1, we can conclude for the systems with two linearity zones given in Proposition 1 that if there is a closed orbit, then it surrounds the origin and it is a limit cycle which is unique and stable. 16 JAUME LLIBRE, MANUEL ORD ´ O˜ NEZ, AND ENRIQUE PONCE [6] F. Dumortier and C. Rousseau,Cubic Li´enard equations with linear damping, Nonlinearity 3(1990), 1015–1039. [7] L. Flatto,Limit cycle studies for circuits containing one Esaki diode, J. Math. Anal, Appl. 9(1964), 360–383. [8] E. Freire, E. Ponce, and F. Torres,Transformaciones de Filippov en sistemas lineales a trozos, Actas V Congreso de Matem´atica Aplicada, Servicio de Publicaci´ons Universidade de Vigo, I(1997), 273–278 (in Spanish). [9] E. Freire, E. Ponce, F. Rodrigo and F. Torres,Bifurcation sets of continuous piecewise linear systems with two zones, Int. J. Bifurcation and Chaos 8(1998), 2073– 2097. [10] E. Freire, E. Ponce, F. Rodrigo and F. Torres,Bifurcation sets of continuous piecewise linear systems with three zones, Int. J. Bifurcation and Chaos 12 (2002), 1675–1702. [11] E. Freire, E. Ponce, and F. Torres,Canonical discontinuous planar piecewise linear systems, SIAM J. Appl. Dyn. Sys., to appear. [12] F. Giannakopoulos and K. Pliete,Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity 14 (2001), 1611–1632. [13] P. Hartman,Ordinary Differential Equations, Second Edition, Birkh¨auser, Boston, 1982. [14] D. Hilbert,Mathematische Probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. G”ttingen Math. Phys. KL. (1900), 253–297; English transl., Bull. Amer. Math. Soc. 8(1902), 437–479. [15] S. J. Hogan,Relaxation oscillations in a system with a piecewise smooth drag coefficiente, J. Sound and Vibration 263 (2003), 467–471. [16] W. T. van Horssen,On oscillations in a system with a piecewise smooth coefficient, J. Sound and Vibration 283 (2005), 1229–1234. [17] D. W. Jordan and P. Smith,Nonlinear Ordinary Differential Equations, 4rd edition, Oxford University Press, Oxford UK, 2007. [18] A.I. Khibnik, B. Krauskopf and C. Rousseau,Global study of a family of cubic Li´enard equations, Nonlinearity 11 (1998), 1505–1519. [19] J. Llibre, E. Nu˜ nez and A.E. Teruel,Limit Cycles for Planar Piecewise Linear Differential Systems via First Integrals, Qualitative Theory of Dynamical Systems 3 (2002), 29–50. [20] J. L. Massera,Sur un th´eor`eme de G. Sansone sur l’equation de Li´enard, Boll. Un. Mat. Ital. 9(1954), 367–369. [21] M. Sabatini and G. Villari,On the uniqueness of limit cycles for Li´enard equations: the legacy of G. Sansone, Le Matematiche, Italy, 65, Dec. (2010). Available at: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/832. [22] S. Smale,Mathematical Problems for the Next Century, Mathematical Intelligencer 20 (1998), 7–15. [23] D. Xiao and Z. Zhang,On the uniqueness and nonexistence of limit cycles for predator–prey systems, Nonlinearity 16 (2003), 1185–1201. [24] Ye Yan-Qian and Others,Theory of Limit Cycles, Translations of Math. Monographs, Vol. 66, Amer. Math. Soc, Providence, 1986. [25] 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. [26] Y. Zou, T. K¨ upper and W.-J. Beyn,Generalized Hopf bifurcations for planar Filippov systems continuous at the origin, J. Nonlinear Science 16 (2006), 159–177. EXISTENCE AND UNIQUENESS OF LIMIT CYCLES IN CPWL SYSTEMS 17 Departament de Matem` atiques, Facultat de Ciencies, Universitat Aut` onoma de Barcelona, Bellatera, 08193 Barcelona, Spain Departamento de Matem´ atica Aplicada, Escuela T´ ecnica Superior de Ingenier´ ıa, Avda. de los Descubrimientos, 41092 Sevilla, Spain