Limit cycles from a monodromic infinity in planar piecewise linear systems

LIMIT CYCLES FROM A MONODROMIC INFINITY IN PLANAR PIECEWISE LINEAR SYSTEMS EMILIO FREIRE, ENRIQUE PONCE, JOAN TORREGROSA, AND FRANCISCO TORRES Abstract. Planar piecewise linear systems with two linearity zones separated by a straight line and with a periodic orbit at infinity are considered. By using some changes of variables and parameters, a reduced canonical form with five parameters is obtained. Instead of the usual Bendixson transformation to work near infinity, a more direct approach is introduced by taking suitable coordinates for the crossing points of the possible periodic orbits with the separation straight line. The required computations to characterize the stability and bifurcations of the periodic orbit at infinity are much easier. It is shown that the Hopf bifurcation at infinity can have degeneracies of co-dimension three and, in particular, up to three limit cycles can bifurcate from the periodic orbit at infinity. This provides a new mechanism to explain the claimed maximum number of limit cycles in this family of systems. The centers at infinity classification together with the limit cycles bifurcating from them are also analyzed. 1. Introduction and main results The analysis of piecewise linear systems is nowadays an active field of research since certain modern devices are well-modeled by this class of systems, see [10]. Even for the simplest situation, as is the case of the aggregation of two planar linear systems, there are still unsolved problems; for instance, it is known that such discontinuous piecewise linear systems can have three limit cycles (see, for instance [4, 16, 20, 24, 25]) but we still do not know if three is indeed the maximum number for them. In the analysis of the dynamical richness in a differential system, an interesting source of knowledge comes from the study of all possible bifurcations that the system can undergo. Furthermore, it should be emphasized the relevance of including in such a bifurcation study the possible bifurcations from infinity. Here, we explore the information on the maximum number of limit cycles that can be obtained by studying the periodic orbit at infinity and its possible bifurcations in the mentioned family of planar discontinuous piecewise linear systems with two zones separated by a straight line. Bifurcations from infinity for planar piecewise linear differential systems has been analyzed before in [23], and more recently in [19]. In [23] only continuous cases with two zones and three zones with symmetry were considered. For the two-zones case, only one bifurcating limit cycle was detected, according to the well-known fact that there can be only one limit cycle in the class of continuous planar piecewise linear differential systems with two zones separated by a straight line, see [12]. In [19], the bifurcation from infinity is addressed for the case of discontinuous piecewise linear differential systems, by perturbing in a non-symmetric 2010 Mathematics Subject Classification. Primary: 37G15, 34C07; Secondary: 34C25, 34C23. Key words and phrases. Planar piecewise linear systems; Bifurcation from infinity; Limit cycles; Centers. 1 2 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES way the canonical continuous linear center ( ˙x, ˙y) = (−y, x), allowing for different linear perturbations in the half planes y < 0 and y > 0. Again, only one bifurcating limit cycle was obtained. In both cases, the technical procedure for the analysis takes advantage of the Bendixson transformation. Such a technique is also used in the recent work [6], where a different family of piecewise linear systems with a symmetric 4-star structure is analyzed. There, after a rather involved computational work, the quoted authors show that up to five limit cycles can bifurcate from infinity. In this paper we propose an alternative and more direct way to work near infinity, so that it is possible to build without excessive computational effort a Poincar´elike return map that allows to characterize in a complete way the periodic orbit at infinity, discriminating several cases where such orbits belong to a period annulus and the cases where such a periodic orbit behaves like a weak-focus. It is shown that the maximum order for the infinity being a weak-focus is three, so we also show that up to three limit cycles bifurcate from infinity. This achievement is very relevant because we provide a new mechanism to generate the supposedly maximum number of limit cycles for the family via just a local analysis. In fact, the phenomenon could be termed a degenerate Hopf bifurcation at infinity. Effectively, to explain the existence of three limit cycles on discontinuous piecewise linear differential systems with two zones separated by a straight line, different mechanisms have been proposed. In [3] authors propose a degenerate boundary equilibrium bifurcation of non-smooth Hopf-like type to pass from a configuration without periodic orbits to another with three limit cycles. In [4] they appear perturbing the harmonic oscillator via piecewise averaging technique of high-order. In [7] they are obtained by perturbing a global center with a different piecewise linear system in each zone. In [16], starting form a situation possessing one limit cycle coexisting with a boundary focus, two new limit cycles are obtained by taking advantage of the boundary focus unfolding. We emphasize that our alternative formulation of the closing equations, whose local analysis near infinity is the subject of this work, might be useful to get the upper bound for the total number of limit cycles in the family of systems under study. This should be the subject of a future work. The existence of such upper bound has been proved only for some special non-generic classes; see for instance [26], where focus type dynamics is not allowed. We start our analysis by assuming without loss of generality that the two regions in the phase plane are the left and right half-planes, SL={(x, y)∈R2:x < 0}, SR={(x, y)∈R2:x > 0}, separated by the straight line Σ = {(x, y)∈R2:x= 0}. The systems to be studied become ˙x =(ALx+bL,if x∈SL∪Σ, ARx+bR,if x∈SR, where x= (x, y)T∈R2,AL= (aL ij) and AR= (aR ij) are 2 ×2 constant matrices with real coefficients and bL= (bL 1, bL 2)T,bR= (bR 1, bR 2)Tare constant vectors in R2. Over the separation line Σ we define the vector field using the Filippov convention, see [11]. As we have commented before, we are interested only in solutions near the periodic orbit at infinity. Under the generic condition aL 12aR 12 >0, orbits sufficiently LIMIT CYCLES FROM INFINITY IN PLANAR PWLS 3 far from the origin cross the discontinuity line, allowing the existence of periodic orbits living in both half-planes. This kind of orbits are usually called of crossing type. Under such a generic condition, the points in Σ that cannot be part of a crossing orbit, i.e. sliding or escaping ones, where Filippov convention is necessary, form a bounded set. Therefore, by using a similar approach to the one followed in getting Proposition 3.1 of [14] and denoting Λ ∈ {L, R}, we obtain the new canonical form ˙x =TΛ−1 DΛ0x−bΛ aΛif x∈SΛ,(1) with TΛ= tr(AΛ) and DΛ= det(AΛ) are the linear invariants in each zone and bL=band bR=−b. The canonical form (1) has seven parameters; apart from the mentioned linear invariants, we find two parameters aL, aR,related to the position of equilibria and a parameter bwhich is responsible for the existence of a sliding set. In fact, the sliding set is the segment joining the points (0,−b) and (0, b), see [14] for more details. These two endpoints are the tangency points of system (1), so that the sliding segment becomes attractive for b < 0 and repulsive for b > 0, shrinking to the origin when b= 0. By computing the sign of ¨xat the tangency points, we obtain ¨x|(x,y)=(0,−b)=aL,¨x|(x,y)=(0,b)=aR, so that the left (right) tangency is called visible if aL<0 (aR>0), being invisible if aL>0 (aR<0), see again [14]. Thus, the aΛparameters are related with the location of the equilibria and determine the visibility of the tangencies; when some of them vanish then we have a boundary equilibrium point, see [21, 29]. Our main hypothesis will be the monodromy of the point at infinity, that is the existence of a periodic orbit at infinity, which requires to have no equilibrium points there. This is equivalent to ask for having dynamics of focus type in both regions, see [23], namely T2 Λ−4DΛ<0.We note that under the above conditions both determinants are positive. As a preliminary result, necessary to stated our main theorems, we introduce a new (symmetric) canonical form, that it will be used in our approach to study limit cycles bifurcating from infinity. Proposition 1. Under the hypotheses T2 Λ−4DΛ<0(both dynamics are of focus type), system (1) is topologically equivalent to system ˙x= 2γLx−y−b, ˙y= (1 + γ2 L)x−αL,if x≤0,and ˙x= 2γRx−y+b, ˙y= (1 + γ2 R)x−αR,if x > 0,(2) where, for each zone Λ∈ {L, R}, we introduce the new parameters γΛ=σΛ ωΛ , αΛ=aΛ ωΛ ,(3) with 2σΛ=TΛand ωΛ>0is such that 4ω2 Λ= 4DΛ−T2 Λ. We have reduced by two the number of parameters in (1) but, what is more important, we make patent the intrinsic features of the dynamics in each region. Effectively, the eigenvalues for the foci in (2) are now γΛ±i, so that the natural 4 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES frequencies are scaled to 1 for both dynamics and the dynamical expansion or contraction for each focus depends only on the coefficients γΛ,being again Λ ∈ {L, R}. A direct consequence of the above proposition is the characterization of the continuity for system (2), leading to new reduced canonical form with only three free parameters. Corollary 2. System (2) becomes continuous if and only if b= 0 and αL=αR. Hence, it writes as ˙x= 2γLx−y, ˙y= (1 + γ2 L)x−α, if x≤0,and ˙x= 2γRx−y, ˙y= (1 + γ2 R)x−α, if x≥0, being αthe common value for the non-homogeneous terms. Before to state our first result about the characterization of the existence of a period annulus at infinity for system (2), we recall the notion of time-reversibility with respect to straight lines. Whenever a planar system is invariant under the change (x, y, τ)7→ (x, −y, −τ) or (x, y, τ)7→ (−x, y, −τ),we say that is time-reversible with respect to the x-axis or y-axis, respectively. Theorem 3. System (2) has a center (period annulus) at infinity if and only if it is time-reversible with respect to y= 0 or x= 0. The centers have a time-reversibility with respect to y= 0 if and only if b= 0 and γL=γR= 0.The centers have a time-reversibility with respect to x= 0 if and only if b= 0, γL=−γR6= 0 and either αL=αR= 0 or αL=−αR6= 0. The proof of this result is a direct consequence of a more complete one, where we detail also the global qualitative behavior, see Theorem 5. Its proof is based upon the derivation of an adequate Poincar´e return map that allows to study a neighborhood of infinity as if it were a standard monodromic point. This idea has been used many times, see [6, 8, 19, 22, 23], by resorting to the Bendixson transformation; the computations become rather involved since, as shown later, to ‘determine’ the stability of the equilibrium point one needs to compute derivatives of high-order of such a Poincar´e map. Here, we exploit an alternative and more convenient approach, by introducing a new suitable coordinate u0associated to one of the two intersection points of the periodic orbit with the separation straight line, the value u0= 0 corresponding to the periodic orbit at infinity. Thus, we are able to compute much more easily a displacement map in the form ∆(u0)=∆1u0+ ∆2u2 0+ ∆3u3 0+ ∆4u4 0+· · · ,(4) for u0>0 and small, such that its positive zeros have a one-to-one correspondence with periodic orbits near infinity. Furthermore, the coefficients ∆idetermine the stability and the weak-focus or center character of the periodic orbit at infinity. In particular, when there exists a period annulus at infinity then we can say that the infinity behaves like a center and all the above coefficients vanish. The reciprocal statement is also true. When ∆1= 0 the periodic orbit at infinity is non-hyperbolic and then, provided that the first non-vanishing coefficient in the above expansion is ∆i, we say that the periodic orbit at infinity behaves like a weak-focus of order i−1. Thus, our second main result assures that the maximum order of the periodic orbit at infinity when it behaves like a weak-focus is three, see Section 5 for a proof. LIMIT CYCLES FROM INFINITY IN PLANAR PWLS 5 Theorem 4. For system (2), the periodic orbit at infinity is hyperbolic and stable (unstable) whenever γL+γR>0 (γL+γR<0). When γL+γR= 0 the periodic orbit at infinity is non-hyperbolic so that it behaves like a weak-focus or a center. The possible weak-focus orders are only 1,2,and 3and there exist perturbations such that the system exhibits 1,2,and 3limit cycles of big amplitude, respectively. We notice that it is the first time that in this family of systems the associated Hopf bifurcation is shown to be up to of co-dimension three; furthermore, it is proved that up to three limit cycles can bifurcate from infinity. This paper is structured as follows. Section 2 presents a part from some properties satisfied by system (2), other canonical forms associated to system (1). Proposition 1 is proved also here. How are the half-return maps near infinity and the computation of the coefficients of the displacement function (4) are done in Section 3. The center characterization result, Theorem 3, is shown with more details through Theorem 5 in Section 4. In Section 5 we get the different possible weak-focus orders and the corresponding limit cycles bifurcation that the periodic orbit at infinity can have, see Theorems 7 and 8, jointly leading to Theorem 4. The limit cycles near infinity that bifurcate from the centers are studied in Section 6, see Propositions 10, 11, and 12. Finally, Section 7 deals with an explicit example where the three limit cycles that bifurcate from infinity are numerically shown. 2. About the canonical forms In this paper we basically work with the canonical form (2) but some other equivalent forms are also interesting. First we introduce some notation and properties on the equilibrium points of (2) which are of focus type: (xL, yL) = (xL,2γLxL−b) = αL 1 + γ2 L ,2αLγL 1 + γ2 L −b(5) and (xR, yR) = (xR,2γRxR+b) = αR 1 + γ2 R ,2αRγR 1 + γ2 R +b.(6) As the vector fields in (2) are linear, it is clear that the equilibrium points are stable (unstable) for γΛ<0 (γΛ>0).When γΛ= 0, we have linear centers. Such equilibria will be real when αL<0 or αR>0, boundary equilibria for αΛ= 0, and virtual ones when αL>0 or αR<0. In terms of the equilibrium coordinates (5) and (6), we can rewrite system (2) as follows, ˙x= 2γL(x−xL)−(y−yL), ˙y= (1 + γ2 L)(x−xL),˙x= 2γR(x−xR)−(y−yR), ˙y= (1 + γ2 R)(x−xR),(7) for x≤0 and x > 0, respectively, and note that now the family is described with 6 parameters, one more than in (2). The parameter bhas been rewritten after introducing the equilibrium ordinates yΛand the parameters αΛhave been substituted by the corresponding equilibrium abscissas xΛ, for Λ ∈ {L, R}.We have that αΛ= (1 + γ2 Λ)xΛ, b =−(yL−2γLxL) = yR−2γRxR,(8) and the last equality gives the condition to be fulfilled by the six parameters in a system (7) to be equivalent to a system in the form (2). However, every system (7) 6 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES not fulfilling the last equality in (8) can be rewritten with a simple translation in the variable yin another equivalent system, already satisfying the mentioned condition. In fact, such a condition amounts to have the origin in the middle of the sliding set, which is a segment in the y-axis. Now we can prove our first main result. Proof of Proposition 1. Under the hypotheses, if we define ωR>0 such that ω2 R= DR−T2 R/4 and σR=TR/2, the eigenvalues of the matrix ruling the dynamics on the half-plane SRin (1) are σR±iωR. Note that DR=σ2 R+ω2 R.We make first the change X=ωRx,Y=y,τ=ωRtfor the variables in SR, without altering variables and time on the half-plane SL. Note that we do not change the coordinate y, so that periodic orbits using both half-planes are preserved. Then, for X > 0 we have dX dτ =1 ωR dX dt =dx dt =TRX ωR −Y+b, dY dτ =1 ωR dY dt =1 ωR dy dt =1 ωRDR X ωR −aR=DR ω2 R X−aR ωR . Introducing the parameter γRas in (3), we see that TR ωR = 2γR,DR ω2 R =γ2 R+ 1. Thus, the new vector field for the right half-plane is as given in the statement with αRas in (3). Doing the analogue transformation for the left half-plane we get a similar result, and the proposition is proved.  As an intermediate option between the two forms (2) and (7), we can also use the 5-parameter formulation ˙x= 2γLx−y−b, ˙y= (1 + γ2 L)(x−xL),˙x= 2γRx−y+b, ˙y= (1 + γ2 R)(x−xR),(9) for x≤0 and x > 0, respectively. Regarding the form (9), the system becomes invariant under the transformations (x, y, τ, γL, xL, b, γR, xR)7→ (−x, y, −τ, −γR,−xR,−b, −γL,−xL),(10) (x, y, τ, γL, xL, b, γR, xR)7→ (x, −y, −τ, −γL, xL,−b, −γR, xR), and its composition (x, y, τ, γL, xL, b, γR, xR)7→ (−x, −y, τ, γR, xR, b, γL, xL). The new time τhas been introduced in the proof of Proposition 1. These properties are useful to reduce the number of configurations to be considered for the analysis of the family. In fact, the parameter bis modal in the sense that by means of a homogeneous scaling in the variables (x, y), which also implies to scale accordingly the parameters (xL, xR) in (9), only the three cases b= 1 (repulsive sliding segment), b= 0 (sewing case), and b=−1 (attractive sliding segment) should be considered. We will not take advantage of this last observation as we are interested in a bifurcation approach to our problem, which requires as much as possible to modify the parameters in a continuous way. LIMIT CYCLES FROM INFINITY IN PLANAR PWLS 7 3. Half-return maps near infinity The periodic orbits of system (2) near infinity can be determined from the halfreturn maps, Land R, near infinity on each side. We take a point (0, y0) with y0>0 as initial point of an orbit for the left system, and integrate the solution forward in time up to arrive again, after approximately a half tour around the focus at (xL, yL), to the y-axis. The existence of an arrival point of the form (0, y1) with y1<0 is guaranteed as long as y0is chosen sufficiently big. Similarly for the right side, we just integrate the right system backward in time, also starting at the point (0, y0) and arriving to a point (0, y2) with y2<0.These intersection points in the negative vertical axis define the half-return maps L(u0)=1/y1and R(u0) = u2= 1/y2being u0= 1/y0,for u0>0 and small. Then, we can define the displacement map ∆(u0) = L(u0)−R(u0).(11) We will see in the following that L(u0) = L1u0+L2u2 0+L3u3 0+L4u4 0+· · · , R(u0) = R1u0+R2u2 0+R3u3 0+R4u4 0+· · · . It should be clear that the positive zeros of the difference function (11) correspond with periodic orbits near the periodic orbit at infinity. Thus the periodic orbit at infinity will be stable (unstable) when for u0>0 and small we have y1−y2<0 (y1−y2>0). Furthermore, we will see that when L1−R1=L0(0) −R0(0) is nonvanishing, the periodic orbit at infinity will be hyperbolic and its sign determines its stability. More concretely, when L1−R1>0 (L1−R1<0) the infinity of (2) is stable (unstable). Alternatively, the orbit passing through the point (0, y0) with y0= 1/u0>0 can be thought as the orbit that terminates at the point (0, y1) with y1= 1/u1<0 after a complete turn starting at the point (0, y2) with y2= 1/u2<0, defining a pseudo-Poincar´e return map u1= Π(u2) = L(R−1(u2)). The first derivative is Π0(u2) = L0(R−1(u2)) 1 R0(R−1(u2)), so that for u2= 0 we have R−1(0) = 0 and then Π0(0) reduces to L1/R1, being this quotient the unity when γL+γR= 0. Thus, these half-return maps will allow to determine the stability of the periodic orbit at infinity and the birth of other periodic orbits from infinity in a degenerate Hopf type bifurcation. The Taylor series of the displacement map (11) has all monomials, contrary to what happens in the analytical case in which it is shown that the first non-vanishing coefficient always corresponds to an odd exponent, see [1]. Moreover, in piecewise differential systems, the return map near a monodromic equilibrium point has a constant term due to the existence of a sliding segment, see [15]. However, here ∆(0) = 0 since the infinity remains invariant under any perturbation. Let us start by considering the left side. Thanks to Proposition 1, we already can assume that AL=2γL−1 1 + γ2 L0, 8 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES and, instead of writing the solution of the differential system starting at the point (0, y0), we can take advantage of the fact that the exponential matrix exp(ALτL) is a fundamental matrix for the corresponding variational system, where τLis the time elapsed between two points of a given orbit. Thus, we have the following relation between the vector field at the arrival point and the vector field at the starting point, −y1−b −αL= eALτL−y0−b −αL, or equivalently, y1+b αL−eALτLy0+b αL=0 0.(12) In order to work near infinity, we introduce new suitable variables that allow us to work as if we were working near an ordinary equilibrium point, without needing to transform the differential equation (as it happens with the Bendixson transformation, see [23, 19]). The key point is to introduce a suitable change of variables once written the closing equations that determine the periodic orbits of the system; recently, the same idea has been successfully extended to 3D systems in [13]. First, as the time τLshould be near πwhen y0is very big, it seems natural to take a new time variable sL=τL−πbut, what is more relevant, we also introduce in equation (12) the new variables u0=y−1 0, u1=y−1 1 so that we get, after some standard manipulations, the equation u0+bu0u1 αLu0u1−eAL(π+sL)u1+bu0u1 αLu0u1=0 0, where u0>0, u1<0,and sLare assumed to be small enough. Thus, we want to solve the above equation in a neighborhood of the point (u0, u1, sL) = (0,0,0), which turns out to be an equilibrium point. It is convenient to split the exponential matrix into the product of two matrices, by noting that exp(ALπ) = −exp(γLπ)I. After multiplying the last equation by the scalar e− L= e−γLπ, we get e− Lu0+bu0u1 αLu0u1+ eALsLu1+bu0u1 αLu0u1=0 0.(13) Now, to desingularize equation (13), it is enough to remove from the second component the trivial factor u1, and write the equation e− Lu0+bu0u1 αLu0+u10 0 1 eALsL1 + bu0 αLu0=0 0,(14) whose Jacobian with respect to (u0, u1, sL) at (0,0,0) is the full-rank matrix e− L1 0 (1 + e− L)αL0 1 + γ2 L. It is possible now to apply the Implicit Function Theorem at the point (u0, u1, sL) = (0,0,0), to assure the existence of unique expansions for u1=L(u0) and sL=β(u0) LIMIT CYCLES FROM INFINITY IN PLANAR PWLS 9 in terms of u0, namely u1=L(u0) = L1u0+L2u2 0+L3u3 0+L4u4 0+· · · , sL=β(u0) = β1u0+β2u2 0+β3u3 0+β4u4 0+· · · . Computations can be done in a degree by degree manner regarding the powers of u0, taking into account that eALsL=I+β1ALu0+β2AL+β2 1 2A2 Lu2 0+β3AL+β1β2A2 L+β3 1 6A3 Lu3 0 +β4AL+β2 2+ 2β1β3 2A2 L+β2 1β2 2A3 L+β4 1 24A4 Lu4 0+· · · , and separating the left hand side terms of (14) in the form e− Lu01 αL+u0u1b 0+u10 0 1 eALsL1 0+u0b αL. For instance, the vanishing of the first degree terms in (14) gives e− L1 αL+L1 αL+ (1 + γ2 L)β1=0 0, so that L1=−e− L, β1=−1+e− L 1 + γ2 L αL=−(1 + e− L)xL.(15) Regarding second order terms, we have e− LbL1 0+L2+ 2γLβ1L1+bL1 (1 + γ2 L)(β2+bβ1+γLβ2 1)=0 0, so that L2= e− L(1 + e− L)(b−2γLxL) = −e− L(1 + e− L)yL, β2=−bβ1−γLβ2 1,(16) and so on. We have also obtained L3,β3,L4,and β4. Here, we write the final expressions for L3and L4, which will be needed later for the analysis, namely L3=−e− L(1 + e− L)(1 + γ2 L)e− L−1 2x2 L+ (1 + e− L)y2 L, L4=−e− L(1 + e− L)QL, (17) where QL= (1 + γ2 L)2γL(1 −e− L+(e− L)2) 3x3 L+(e− L−1)(2 e− L+3) 2x2 LyL+ (1 + e− L)2y3 L. The procedure can be repeated step by step for the right half-plane, starting from equation −y0+b −αR= eARτR−y1+b −αR, just, by the symmetry of our model, changing (y0, y1, b, L) by (y1, y0,−b, R),respectively. Because now the point (0, y1) is the initial point and (0, y0) is the final point, the parameter bhas now a plus sign, and all the subscripts are Rinstead of L. We introduce the equivalent values sR=τR−πand e+ R= eγRπ. 16 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES So that there exists, in the working neighborhood, a one-to-one correspondence between the ∆i-values and the three free parameter values. The proof finishes using the Implicit Function Theorem and the Weierstrass Preparation Theorem, that allow us to take new local coordinates δ= (δ1, δ2, δ3) around zero so that, because ∆46= 0 in (23), we have ∆(u, δ) = q(u, δ)e ∆(u, δ) = (δ1u+δ2u2+δ3u3+u4)e ∆(u, δ),(24) where e ∆ is an analytical non-vanishing function at (0,0).It is also clear that, under these conditions, there will be no more than three positive zeros. The bifurcation δ1 δ2 01 2 δ1 δ2 01 2 Figure 3. The number of positive solutions for q(u, δ) in the plane (δ1, δ2) for δ3<0 (left) and δ3≥0 (right). The region in red corresponds to the existence of 3 positive solutions. curves shown in Figure 3 follow directly studying when the discriminant of qwith respect to uvanishes, namely on the varieties δ1= 0 and 4δ1δ3 3−δ2 2δ2 3−18δ1δ2δ3+ 4δ3 2+ 2δ2 1= 0.The cusp point in Figure 3 (left) is located at (δ1, δ2) = (δ3 3/27, δ2 3/3). This explains the small size of the parameters region where 3 positive zeros exist for δ3<0. Clearly, by the Descartes’ rule there cannot be three positive zeros when δ3>0. We remark that, as an alternative approach to assure that the bifurcation in the above proof is determined with only the first four terms of the Taylor series of ∆ whenever ∆46= 0, we can take advantage of the Z2-Classification theorem in [18]. Effectively, we can write the function qin (24), naming u=v2and removing a factor v, as ˆq(v) = δ1v+δ2v3+δ3v5+v7, which corresponds with case 5 of Figure 6.2 in [18, page 269]. As we have mentioned before, this bifurcation is by no means different from a degenerated Hopf bifurcation for a monodromic non-degenerated equilibrium point, i.e. those for which its Jacobian matrix have zero trace and positive determinant. 6. Limit cycles bifurcating from the centers In a fixed class of systems having centers, the maximum number of limit cycles that can bifurcate from a center is known as the local cyclicity of that center. This is a very difficult problem and it is solved for a very few classes of differential systems. The analytic quadratic class is one of them and it was proved 70 years ago by Bautin LIMIT CYCLES FROM INFINITY IN PLANAR PWLS 17 that at most three limit cycles of small amplitude can bifurcate, see [2]. Another instance is the class of cubic systems without quadratic nonlinearities, that was studied by Sibirski˘ı in [31], but the complete proof that only 5 limit cycles of small amplitude bifurcate at the origin was done 30 years later by ˙ Zo l¸adek in [32]. Up to our knowledge, this question has not solved for other general classes, either for the complete cubic polynomial class. In the non-smooth differential systems world, the existence of sliding segments, as is the case when bis not zero in our main system (2), makes this problem very intricate because the return map is not analytic. This is not the case however when the return map is studied near the periodic orbit of infinity for system (2), as we have seen in Section 3. Typically, the main technique used to bound the local cyclicity is the study of the Bautin ideal formed by the coefficients of the difference map ∆ as we have defined in (4) for our study. These coefficients are known as the Liapunov quantities associated to the center-focus problem of a planar differential system, see more details in [1]. We recall that in this context the Liapunov quantities are polynomials in the perturbation parameters, see [9]. The finiteness property is proved usually using that the Bautin ideal is defined in a Noetherian ring, because the number of parameters is finite, and that such ideal is radical. In our context, the coefficients ∆iare not polynomials in the perturbation parameters, as we have already seen in Section 4. Due to the difficulties to study upper bounds, in this section we deal with the study of lower bounds for the maximum number of limit cycles of big amplitude bifurcating from the centers presented in the classification Theorem 3. We will see that, for some of them, see Propositions 11 and 12, a transversal weak-foci curve of order 3 emerges at the critical point in the parameters space corresponding to a center configuration. The transversality ensures again the existence of three limit cycles in a neighborhood of such a curve. A higher-order analysis is required and we prove that the local cyclicity changes when moving the parameters in the selected center family. That is, although generically the cyclicity of a family of centers remains constant, over some singular locus it can increase. We will closely follow the scheme of [17]. We start perturbing center family (c) in Theorem 5 because of its simplicity. The necessary computations for the other two are more involved. For simplicity, we will use the equivalent canonical form (7) instead of (2). Proposition 10. Let η, ξ be non-zero real numbers. The local cyclicity of the periodic orbit at infinity of center type defined by γL=−γR=−η, xL=−xR=−ξ, and b= 0,when perturbed in the class of systems (7), is at least 2. Proof. We start taking the perturbed system (7) being (γL, γR, b, xL, xR)=(−η+ε1, η +ε2, ε3,−ξ+ε4, ξ +ε5) and computing the coefficients of ∆i(η, ξ;ε),for i= 1,...,4,defined in Section 4, being ε= (ε1, . . . , ε5).The second step is the computation of their first-order expansions ∆1 i(η, ξ;ε) for the Taylor series of ∆iwith respect to ε= (ε1, . . . , ε5).We can easily check that the matrix (of size 3×5) defined by the coefficients of (∆1 1,∆1 2,∆1 3) with respect to εhas rank 3 whenever the parameters η, ξ are non-vanishing. In fact, 18 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES the Jacobian matrix of (∆1 1,∆1 2,∆1 3) with respect to (ε1, ε3, ε4) has a determinant 2πξ(eηπ)3(eηπ −1)(eηπ +1)2(η2+ 1) 6= 0. We notice that the rank of the matrix (of size 4 ×5) defined by the coefficients of (∆1 1,∆1 2,∆1 3,∆1 4) with respect to εremains unchanged, being also 3. Hence, using the Implicit Function Theorem there exist new local coordinates εδ= (δ1, ε2, δ2, δ3, ε5) in a neighborhood of the origin in the parameters space, such that ∆i(η, ξ;εδ) = δi for i= 1,2,3.The proof finishes using the same argument as in Section 5 because we have a transversal curve of weak-foci of order 2 bifurcating from each center value (η, ξ), in the 2-dimensional manifold in the parameters space. Moreover, the transversality assures the bifurcation of up to 2 limit cycles of big amplitude.  In what follows we extend the notation ∆j i(ε) for the j-th order truncation of the Taylor series of ∆iwith respect to ε. We notice that in the above proposition we have not get more limit cycles using ∆4,even arriving up to fourth-order. Indeed, ∆4 4vanishes when (δ1, ε2, δ2, δ3, ε5) = (0, ε2,0,0, ε5). Next result provides the number of limit cycles that can bifurcate from center family (a) in Theorem 5. It gives the bifurcation diagram of the number of limit cycles in a 2-dimensional manifold. Proposition 11. Let η, ξ be non-zero real numbers. The local cyclicity of the periodic orbit at infinity of center type defined by γL=γR= 0, xL=η,xR=ξ, and b= 0, when perturbed in the class of systems (7), is at least 1when η=−ξ, at least 2when η6=±ξ, and at least 3when η=ξ. Proof. We consider a perturbation in system (7) with (γL, γR, b, xL, xR)=(ε1, ε2, ε3, η +ε4, ξ +ε5). The Taylor series of the coefficients of ∆i(η, ξ;ε),defined in Section 4, with respect to ε= (ε1, . . . , ε5) write as ∆1(η, ξ;ε) = πε1+πε2+O2(ε), ∆2(η, ξ;ε) = −4ηε1+ 4ξε2+ 4ε3+O2(ε), ∆3(η, ξ;ε) = πη2ε1+πξ2ε2+O2(ε), ∆4(η, ξ;ε) = −4 3η3ε1+4 3ξ3ε2+O2(ε). (25) The matrix of the first three linear terms with respect to (ε1, ε2, ε3) has a determinant 4π2(η2−ξ2).As it is non-zero when η6=±ξ, reasoning as in the proof of Proposition 10, it is clear that there exists a change of variables in the parameters space such that ∆i(η, ξ;ε) = δi,for i= 1,2,3,and that the local cyclicity is at least 2.Straightforward computations show that ∆2 4= 0 when δ1=δ2=δ3= 0 and we can not get more limit cycles up to a second order analysis. When η=−ξ6= 0,the first two linear Taylor series in (25) are linearly independent and the rank of the corresponding matrix adding the next two rows does not increase. Using again the Implicit Function Theorem, we can use new local coordinates (δ1, ε2, δ2, ε4, ε5) in a neighborhood of the origin so that ∆i=δi,for i= 1,2. Straightforward computations show that ∆2 3= ∆2 4= 0 when δ1=δ2= 0 and we can not get more limit cycles up to a second order analysis. LIMIT CYCLES FROM INFINITY IN PLANAR PWLS 19 Finally, when η=ξ6= 0 we need to work with Taylor series of second order. Doing as above and using again the Implicit Function Theorem, we can take new local coordinates (δ1, δ2, ε3, ε4, ε5) such that ∆i=δi,for i= 1,2.Taking δ1=δ2= 0, we have that (25) reduces to ∆3(ξ; ˆε) = πε3(ε4−ε5) + O3(ˆε), ∆4(ξ; ˆε) = −4 3ξ2ε3+O2(ˆε),(26) with ˆε= (ε3, ε4, ε5).From now on, we can simplify the computations taking ε5= 0. Then, doing a blow-up change of coordinates (ε3, ε4)=(ε3,˜ε4ε3),equation (26) writes as ∆3(ξ;ε3,˜ε4) = ε2 3(π˜ε4+ε3O0(ε3,˜ε4)), ∆4(ξ;ε3,˜ε4) = ε3(−4 3ξ2+ε3O0(ε3,˜ε4)). The Implicit Function Theorem allows us to define a new local coordinate δ3so that ∆3(ξ;ε3,˜ε4) = ε2 3δ3.The proof finishes by imposing that the new coordinate δ3to be zero and checking that when ε3is small but not zero, the fourth coefficient ∆4is non-vanishing because ξ6= 0. Consequently, we have a third-order weak-focus curve that is born from the critical parameter values corresponding to the center, from which the 3 limit cycles can bifurcate.  The strategy used at the end of the last proof is an interesting non-standard use of the Implicit Function Theorem in this field, even it frequently employed in desingularization procedures in singularity theory. Up to the best of our knowledge, this procedure goes back to Loud in [27]. We finish this section perturbing the remaining family (b) in Theorem 5. Proposition 12. Let ηbe a non-zero real number. The local cyclicity of the periodic orbit at infinity of center type defined by γL=−γR=−η, xL=xR= 0,and b= 0, when perturbed in the class of systems (7), is at least 3. Proof. As in the previous two proofs, we consider a general perturbation in system (7) with (γL, γR, b, xL, xR) = (−η+ε1, η +ε2, ε3, ε4, ε5). The Taylor series of the coefficients of ∆i(η, ξ;ε),defined in Section 4, with respect to ε= (ε1, . . . , ε5) write as ∆1(η;ε)=eηπ π(ε1+ε2) + O2(ε), ∆2(η;ε) = 2 eηπ(eηπ +1)(ε3+η(ε4+ε5)) + O2(ε), ∆3(η;ε) = O2(ε), ∆4(η;ε) = O2(ε). As before we can use Taylor series of order 1 and the Implicit Function Theorem to define new local coordinates (δ1, ε2, δ2, ε4, ε5) so that ∆1=δ1and ∆2=δ2. Restricting our attention to the manifold δ1=δ2= 0,the next two coefficients write, up to order 3 terms, as ∆3(η; ˜ε) = −2˜ε4˜ε5(η2+ 1)(e2ηπ −1) eηπ +O4(˜ε), ∆4(η; ˜ε) = 4 3η(η2+ 1) eηπ ˜ε4(e3ηπ +1)˜ε2 4+ (eηπ +1)(9 e2ηπ −6)˜ε2 5+O4(˜ε), 20 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES where ˜ε= (˜ε4,˜ε5) and we have taken ε2= 0, ε4= ˜ε4+ ˜ε5,and ε5= ˜ε4−˜ε5to simplify the above expressions. In the plane (˜ε4,˜ε5) the curve ∆3= 0 has, near the origin, two branches, one tangent to ˜ε4= 0 and another to ˜ε5= 0.As the above Taylor series vanish over the first one, we should work with the second one, where ∆3vanishes but not ∆4,being ˜ε4small but not zero. The proof finishes doing a new local change of coordinates of blow-up type (˜ε4,˜ε5) = (˜ε4,˜ε4δ3).Clearly, we have again for every nonzero ηa transversal curve of third-order weak-foci on the parameters space that is born at the critical value where system has a center and from which 3 limit cycles can bifurcate.  Note that in the proofs of this section, we could have computed even higher-order Taylor series in looking for higher cyclicity but our goal was to get good lower bounds without big computational effort. 7. An explicit example In the previous two sections, we have seen that the maximum number of limit cycles found for system (7) can bifurcate both from the weak-foci of maximal order and also from some of center families. We finish the work with an explicit numerical example showing the existence of 3 limit cycles of big amplitude. As the bifurcation near the centers is more degenerate, we will deal around a weak-focus of maximal order. Let us take for system (7) the parameter values γL=−γR=−1 8, b =−1 4, xL=xR= 1,(27) so that from (19)-(22) we have ∆1= ∆2= ∆3= 0, and ∆4=65 384 eπ/81+e3π/8≈1.06495899308488, where from (8) we have yL=yR= 0. The phase portrait is depicted in Figure 4. -1 1 2 3 -1 1 2 Figure 4. Phase portrait of system (7) with the parameters values given in (27), having a periodic orbit at infinity that behaves like a weak-focus of order three. We draw in red the vertical nullclines and the sliding segment. LIMIT CYCLES FROM INFINITY IN PLANAR PWLS 21 Clearly, the chosen unperturbed system has both the virtual focus and the real one located at the same point (1,0). Obviously, the perturbations providing the three limit cycles will separate them. Curiously, the first found example with three limit cycles in the family of piecewise linear differential systems separated by a straight line, which was numerically presented in [20] and later justified in [24], exhibited a configuration rather far from the weak-focus of order three but with the two foci located at the same point. However, such a pioneering example was not written in any canonical form; if one writes it in our Li´enard canonical form (7) then the two foci become not at the same point and appreciably distant one another. Coming back to our example, we consider the parametric family of perturbed systems (7) with γL=−1 8,xL= 1, and γR=1 8+e−π/8 πε1, b=−1 4+1024 −65π−(1024 −195π) eπ/4 1040π(eπ/4−1) eπ/8ε1+e−π/8 2(eπ/8+1)ε2−8 e−π/8 65(eπ/4−1)ε3, xR= 1 + 16 + 65π−(16 + 195π) eπ/4 130π(eπ/4−1) eπ/8ε1+64 e−π/8 65(eπ/4−1)ε3, so that we have ∆i(γR, b, xR) = εi+O(ε2) for i= 1,2,3, where O(ε2) represents higher-order terms in ε= (ε1, ε2, ε3), obtaining a non-vanishing perturbed value for ∆4. Thus, we have a complete unfolding in the 3-parameter space (γR, b, xR) for a neighborhood of the critical point (γR, b, xR) = (1/8,−1/4,1), which represents a weak-focus of order 3 for the periodic orbit at infinity. In particular, for the concrete perturbed system with γL=−1 8, xL= 1, γR=1638355 13106841, b =−260534 1045519, xR=552751 556327, we get yL=−3383 4182076 ≈ −0.000808928, yR=−6084083513535748 7623599948859945633 ≈ −0.000798059. After using the relations (5) and (6) to get αRand αL, the fourth-degree truncation of function ∆(u0) becomes −4.43719886 ·10−8u0+ 3.993655760 ·10−5u2 0−1.15001344 ·10−2u3 0+ 1.054869499u4 0, which has three simple positive zeros at {0.002467460261,0.003358360933,0.005076128658} with reciprocal values (in reverse order) {197.00052293,297.76430224,405.27501730}. Accordingly, system (7) with such perturbed parameter values has three limit cycles, whose intersection points (0, yi) with the positive y-axis have the yiordinates {196.89979358,297.91820638,405.21567427}, which are very close to the reciprocals of zeros for the fourth-order truncated function for ∆, whose graph is drawn in Figure 5. As a final remark, it should be emphasized that only thanks to the theoretical analysis developed in this work it has been possible to detect the above example. Of course, many other analogous examples could now be built without extra effort. 22 E. FREIRE, E. PONCE, J. TORREGROSA, AND F. TORRES Figure 5. The graph of the fourth-degree truncation of function ∆(u0) for the numerical example obtained after perturbing the weakfocus of order 3 given in (27). 8. Acknowledgements This work has been realized thanks to Consejer´ıa de Econom´ıa y Conocimiento de la Junta de Andaluc´ıa (P12-FQM-1658 grant), Ag`encia de Gesti´o d’Ajuts Universitaris i de Recerca de Catalunya (2017 SGR 1617 grant), Spanish Ministerio de Ci´encia, Innovaci´on y Universidades - Agencia Estatal de Investigaci´on (MTM2016- 77278-P (FEDER), MTM2017-87915-D2-1-P, and PGC2018-096265-B-I00 grants), and European Community (H2020-MSCA-RISE-2017-777911 grant). References [1] A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Ma˘ıer. Theory of bifurcations of dynamic systems on a plane. Halsted Press [A division of John Wiley & Sons], New York- Toronto, Ont.; Israel Program for Scientific Translations, Jerusalem-London, 1973. [2] N. N. Bautin. 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Ingenier´ ıa, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla (Spain) Email address:[email protected] Departament de Matem` atiques, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Barcelona (Spain) Email address:[email protected] Departamento Matem´ atica Aplicada II, E.T.S. Ingenier´ ıa, Universidad de Sevilla, Camino de los Descubrimientos, 41092 Sevilla (Spain) Email address:[email protected]