The Focus-Center-Limit Cycle Bifurcation in Discontinuous Planar Piecewise Linear Systems without Sliding

The Focus-Center-Limit Cycle Bifurcation in Discontinuous Planar Piecewise Linear Systems without Sliding Enrique Ponce, Javier Ros and El´ısabet Vela Abstract Planar discontinuous piecewise linear systems with two linearity zones, one of them being of focus type, are considered. By using an adequate canonical form under certain hypotheses,the bifurcation of a limit cycle, when the focus changes its stability after becoming a linear center, is completely characterized. Analytic expressions for the amplitude, period and characteristic multiplier of the bifurcating limit cycle are provided. The studied bifurcation appears in real world applications, as shown with the analysis of an electronic Wien bridge oscillator without symmetry. 1 Introduction and main results Nowadays, the analysis of discontinuous piecewise-linear systems is an active field of research since certain modern devices are well modeled by this class of systems, see [1]. For the simplest situation however, as is the case of the aggregation of two planar linear systems, there are bifurcations that still require a thorough analysis. Recently, in [2] it has been proposed a canonical form for the case of planar discontinuous systems with two zones of linearity, to be denoted D2PWLS2for short. In the quoted paper, there are shown some bifurcation results for the case when both linear dynamics are of focus type without visible tangencies,that is, there are no real equilibrium points in the interior of each half-plane. Here, by resorting to the canonical form given in [2], we consider a different situation when we have an equilibrium point of focus type in the interior of a half plane without specifying the linear dynamics type in the other half plane. Our goal is to describe qualitatively and quantitatively the possible bifurcation of limit cycles through the change of Enrique Ponce, e-mail: eponce[email protected] ·Javier Ros, e-mail: [email protected] ·El´ısabet Vela, email: eliv[email protected] Departamento Matem´atica Aplicada II, E.T.S. Ingenier´ıa, 41092-Sevilla (Spain) 1 2 Enrique Ponce, Javier Ros and El´ısabet Vela stability of such an equilibrium point. Thus, this work is a relevant generalization to discontinuous vector fields of the bifurcation studied in [3] for the continuous case. To begin with, we assume without loss of generality that the linearity regions in the phase plane are the left and right half-planes, S−={(x,y):x<0},S+={(x,y):x>0}, separated by the straight line Σ ={(x,y):x=0}. The systems to be studied become ˙x =(F+ 1(x),F+ 2(x)T=A+x+b+,if x∈S+, F− 1(x),F− 2(x)T=A−x+b−,if x∈S−,(1) where x= (x,y)T∈R2,A+= (a+ ij)and A−= (a− ij)are 2×2 constant matrices, b+= (b+ 1,b+ 2)T,b−= (b− 1,b− 2)Tare constant vectors of R2, and the definition of the vector field in Σ is not relevant for our purposes. We assume the generic condition a+ 12a− 12 >0, which means that orbits can cross the discontinuity line in opposite directions, allowing the existence of period orbits that use the two half planes. By applying Proposition 3.1 of [2], system (1) can be written in the canonical form ˙x =T−−1 D−0x−0 a−if x∈S−, ˙x =T+−1 D+0x−−b a+if x∈S+, (2) where a−=a− 12b− 2−a− 22b− 1,b=a− 12 a+ 12 b+ 1−b− 1,a+=a− 12 a+ 12 (a+ 12b+ 2−a+ 22b+ 1), and T±=tr(A±),D±=det(A±)are the linear invariants of each zone. The canonical form (2) has seven parameters; apart from the mentioned linear invariants,we find the two parametersa+and a−related to the position of equilibria, and a parameter bwhich is responsible for the existence of a sliding set, where both vector fields cannot be concatenated in a natural way. In fact, there exists a sliding set which is a segment joining the origin and the point (0,b), see [2] for more details. These two endpoints are tangency points, the origin for the left region and the point (0,b)for the right one. Furthermore,the sliding set 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)=(0,0)=a−, ¨x|(x,y)=(0,b)=a+,so that the left (right) tangency is called visible for a−<0 (a+>0), being invisible for a−>0 (a+<0), see [2]. Thus the a±parameters are related to the visibility of the tangencies, and when they vanish we have boundary equilibrium points, see [4] and also [5]. The Focus-Center-Limit Cycle Bifurcation in D2PWLS2without Sliding 3 The possible equilibria (real or virtual) are located at (a−/D−,a−T−/D−)and (a+/D+,b+a+T+/D+)where it is assumed D+D−6=0. Without loss of generality, we assume that there exists an equilibrium of focus type in the left zone, that is, T2 −−4D−<0 (what implies D−>0), and a−<0. Our interest is to study what happens when the trace T−passes through the critical value zero, that is when the focus passes from stable to unstable or vice versa. Note that for T−=0 in the left half plane we have a center configuration which terminates in a visible tangency at the origin. To avoid other non-local phenomena it is then natural to impose that in the right zone we have also a tangency at the origin of invisible character, what amounts to require b=0. In fact, if we allow to move this parameter bin a neighborhood of zero, then we should have the possibility of new bifurcations, namely the collision of tangencies, which has been reported in [4]. Thus, our study can be seen as a first step in the analysis of the codimensiontwo bifurcation that appears when the parameter bis allowed to be moved. Such codimension-two bifurcation will be the aim of a future work, since it turns out to be a non-generic case, not included in the extensive analysis done in [6]. Thus, there are no proper sliding set nor jumps in the right dynamics with respect to the critical center. Under these assumptions our first result is the following. Proposition 1. Under the hypotheses T2 −−4D−<0, a−<0(left focus dynamics with visible tangency at the origin) and assuming that in the right zone we have an invisible tangency at the origin, that is, b =0and a+<0, system (2) is topologically equivalent to the system ˙x=Tx−y, ˙y=Dx+a,if x >0,˙x=2 γ x−y, ˙y= (1+ γ 2)(x+1),if x <0,(3) where T =T+, D=D+, γ = α / ω , with 2 α =T−, ω >0issuchthat4D−−T2 −=4 ω 2, and a =D−a+/( ω a−)>0. The proof of Proposition 1 appears in Sect. 3. With this result, we manage to describe the left dynamics with only one parameter, needing other three parameters to deal with the right region. The following remark should be taken into account. Remark 1. System (3) represents in general a discontinuous vector field since for x=0 we will have generically a6=1+ γ 2. The original system (2) is continuousonly in the case a+=a−and b=0, but even in such non-generic case, the new system provided by Proposition 1 will be discontinuous. Thus, the analysis of system (3) is relevantfor some continuous cases that could be also studied by alternativemethods within a continuous vector field context. This fact will be illustrated later in Sect. 2. Regarding equilibrium points, in the zone x<0, there exists a focus at (x,y) = (−1,−2 γ ), to be stable for γ <0 and unstable for γ >0. When γ =0, we have a linear center. In the zone x>0, since a>0, there will be no equilibrium points if D=0; otherwise, thepossible equilibriumpoint will be located at (−a/D,−aT/D). We will take γ in (3) as the bifurcation parameter, having its critical value at γ =0, where the center configuration takes place, see Figure 1. Our first main result is the following. 4 Enrique Ponce, Javier Ros and El´ısabet Vela 01 Y 0011 Y bx0 bx1 Fig. 1 (Left) The critical situation for γ =0. (Right) The bifurcating limit cycle for γ T<0 and | γ | small. Theorem 1. Consider system (3) with a >0and under the assumption T 6=0. The linear center configuration restricted to the zone x ≤0, that exists for γ =0gives place to a unique periodic oscillation for γ T<0and | γ |sufficiently small. More precisely, for T <0the limit cycle bifurcates for γ >0and it is stable, while for T >0the limit cycle bifurcates for γ <0and it is unstable. If we denote with bx0= (0,by0)Tthe lower crossing point of the bifurcating limit cycle, then the peak-topeak amplitude App in x, the period P of the periodic oscillation, the characteristic multiplier ρ of the periodic orbit and the coordinate by0are analytic functions at 0 in the variable γ 1/3. Namely, for γ T<0and | γ |sufficiently small we have App =2+(3 π )2/3 2 1+a (T2a)1/3 γ 2/3+O( γ 4/3), P=2 π +2(3 π )1/3a−1 (a2T)1/3 γ 1/3−2 π 15 (15a2−12D+T2−a(3D+T2)) a2T γ +O( γ 4/3), ρ =1−2(3 π )1/3T a2/3 γ 1/3+2(3 π )2/3T a4/3 γ 2/3+ +2 π 15 15a2+12D−31T2 a2 γ +O( γ 4/3), and by0=3 π a T1/3 γ 1/3+ π 2T 3a1/3 γ 2/3+ π 15a2+3D+T2 15aT γ +O( γ 4/3). It should be noted that the sign of the parameter Dis not relevant for the bifurcation and thus our result covers the cases the focus-antisaddle case (D>0), the case of D=0 (parabolic case) and the saddle case (D<0). Of course, the uniqueness of the bifurcating limit cycle is referred to a neighborhood of the most external periodic orbit of the linear center existing for γ =0, and so it is a local uniqueness. The case T=0 is explicitly excluded from our result because then we should have The Focus-Center-Limit Cycle Bifurcation in D2PWLS2without Sliding 5 a global center for the critical value γ =0; the possible bifurcation of limit cycles for such non-generic case requires different techniques, as considered in [7]. Undoing the changes of variables introduced in the proof of Proposition 1, it is easy now to get a similar result for system (2) with b=0 and adequate hypotheses. Theorem 2. Consider system (2) under the hypotheses T2 −−4D−<0, a−<0, a+<0, T+6=0and assume that in the zone x >0we have an invisible tangency at the origin, that is b =0. The linear center configurationrestricted to the zone x ≤0, that exists for T−=0gives place to a unique periodic oscillation for T−T+<0and |T−|sufficiently small. More precisely, for T+<0the limit cycle bifurcates for T−>0and it is stable, while for T+>0the limit cycle bifurcates for T−<0and it is unstable. If we denote by bx0= (0,by0)Tthe lower crossing point of the bifurcating limit cycle, the peakto-peak amplitude App in x, the period P, the characteristic multiplier ρ and by0 are analytic functions at 0in the variable T1/3 −. Namely, for T−T+<0and |T−| sufficiently small we have App =−2a− D−−1 23 π 22/3a++a− D−a− a+1/3T− T+2/3 +O(T−)4/3,(4) P=2 π √D−−(12 π )1/3a−−a+ a1/3 −a2/3 +√D−T− T+1/3 + + π 15 15a2 +D−−3D+(4a2 −+a−a+)+ T2 +a−(a−−a+) a2 +D3/2 − T− T+ +O(T−)4/3,(5) ρ =1+(12 π )1/3 √D−T+a− a+2/3−T− T+1/3 +(18 π 2)1/3T2 + D−a− a+4/3T− T+2/3 + + π 15 45a2 ++a2 −(12D+−31T2 +) a2 +D3/2 − T−+O(T−)4/3, and by0=3 π 21/3a1/3 +a2/3 − √D−−T− T+1/3 − π 2 121/3a4/3 −T+ a1/3 +D−T− T+2/3 − − π 30 15a2 +D−+a2 −(3D++T2 +) a+D3/2 − T− T+ +O(T−)4/3. (6) This theorem is an extension of the results obtained in [3], where only the continuous case was analyzed. Consequently,if we put in the aboveexpressions a+=a− the continuous case is recovered. The expressions (4)-(7) have been computed with the help of symbolic computation systems (both Mathematica [8] and Maple [9]) and only the first coefficients of the series are explicitly shown. More terms can be computed with the same techniques, if required. 6 Enrique Ponce, Javier Ros and El´ısabet Vela The rest of the paper is organized as follows. An application of Theorem 2 to the study of the dynamics of an electronic circuit is given in Sect. 2. The proofs of main results along with other technical lemmas appear finally in Sect. 3. 2 Application to the dynamics of a Wien bridge oscillator C2 R1C1 Vo R2 + + - - + - RSRF OA Fig. 2 Scheme of the Wien bridge oscillator. The power supply needed to bias the operational amplifier is not shown. In [10] a realistic model for an operational amplifier is proposed to illustrate the jump transition to oscillation in the Wien bridge oscillator. The problem leads to a idealized symmetric model that can be settled down in a continuous vector field context, see [3]. Here, we will assume a more general situation in the sense that the nonlinear characteristic of the electronic device is allowed to have some lack of symmetry. This assumption implies that in the birth of oscillations only two linear zones are involved, and we can discard one of the most external regions. Thus we assume that the circuit model is given by ¨v+2 Γ ˙v+v=0,v>1, ¨v−2 Γε ˙v+v=0,v≤1, where vis the dimensionless input voltage in the operational amplifier, Γ is a positive constant parameter which depends on the passive elements of the circuit, and ε is taken as the bifurcation parameter, see [10] for more details. The above equations, after the translation v=x+1, can be written in the form ˙x=y,˙y=−x−2 Γ y−1,if x>0, −x+2 εΓ y−1,if x≤0.(7) System (8) is in the formulation (1.1)given in [2] and we can apply the Proposition 3.1 of that paper in order to put system (8) in Li´enard form. The homeomorphism The Focus-Center-Limit Cycle Bifurcation in D2PWLS2without Sliding 7 ex=1 0 −2 Γ −1xif x>0,ex=1 0 2 εΓ −1xif x≤0, transforms system (8) in ˙x=−2 Γ x−y,if x>0, 2 εΓ x−y,if x<0,˙y=x+1,(8) where the tildes have been dropped for the sake of simplicity. Now, we can apply Theorem 2 to system (9) by taking T+=−2 Γ ,T−=2 εΓ , D+=D−=1, a+=a−=−1 and b=0, obtaining that the bifurcation takes place for ε =0, the limit cycle exists for sufficiently small ε >0 and it is stable. The peak-to-peak amplitude, the period and the characteristic multiplier of the limit cycle are App =2+3 π 22/3 ε 2/3− π 4/3(99+52 Γ 2) 40·181/3 ε 4/3+O( ε 5/3), P=2 π +4(18 π 5)1/3 5 ε 5/3+O( ε 2), ρ =1−2(12 π )1/3 Γε 1/3+4(18 π 2)1/3 Γ 2 ε 2/3+2 π 15 Γ (27−124 Γ 2) ε + +4 54 π 91/3 Γ (34 πΓ 3−27 πΓ −5) ε 4/3+O( ε 5/3). We note that the order ε 5/3term in the last expressionof the period Pis notexplicitly shown in Theorem 1. It has been computed with the same procedure of the proof of Theorem 1, increasing the number of terms in (18). We emphasize that the above series are very useful to describe accurately all the features of the nonlinear oscillation for parameters near the critical value corresponding to the bifurcation.Of course, we could have obtained the same results applying Proposition 1 and Theorem 1. 3 Proofs of main results We give first the proof of Proposition 1. Proof (Proof of Proposition 1). Clearly, the assumption on right tangency at the origin, that is ˙x|(x,y)=(0,0)=0, reduces to b=0, and this tangency will be invisible for a+<0. Under the hypotheses, if we define ω >0 such that ω 2=D−−T2 −/4 and α =T−/2, the eigenvalues of the linear part at S−in (2) are α ±i ω . We make first the change X= ω x,Y=y, τ = ω tfor the variables in the half plane S−, without altering variables and time in S+. Note that we do not change the coordinate y, so that periodic orbits using both half planes are preserved. Then, we get 8 Enrique Ponce, Javier Ros and El´ısabet Vela dX d τ =1 ω dX dt=dx dt=X ω T−−Y, dY d τ =1 ω dY dt=1 ω dy dt=1 ω D−X ω −a−=D− ω 2X−a− ω . Introducing the parameter γ = α / ω , we see that T−=2 γω and D−= ( γ 2+1) ω 2. Making in the whole plane a homothetyof factor ktobe defined below and removing the factor kin the equations, we have for x<0, dx d τ =kdX d τ =k2 γ x k−y k=2 γ x−y, dy d τ =kdY d τ =kh( γ 2+1)x k−a− ω i= ( γ 2+1)x−ka− ω , while for x>0 we have ˙x=Tx −y, and ˙y=Dx−ka+. By imposing now that k=−( γ 2+1) ω /a−>0, we get the expressions given in the statement. Before giving the proof of Theorem 1, we need first some technical results. Let us consider system (1). We assume that the orbit of vector field F+starting at a point bx0= (0,by0)with F+ 1(bx0)>0, lies in S+and eventually comes back to the discontinuity line arriving transversally at the point bx1= (0,by1), where F+ 1(bx1)<0. Lemma 1. If we denote as b δ +(t) = (x(t,bx0),y(t,bx0)) the solution of ˙ x=F+(x) satisfying b δ +(0) =bx0, then a value τ +>0existssuchthatx(t,bx0)>0for0<t< τ + and x( τ +,bx0) = 0, y( τ +,bx0) = by1. Then we can define a right Poincar´ e map PRin a neighborhood of the point bx0such that PR(by0) = by1and the first derivative of map PRat by0is given by P′ R(by0) = F+ 1(bx0) F+ 1(bx1)expZ τ + 0divF+. Proof. Let us consider the differential system ˙ x=F+(x)defined in R2, and let us denote as δ +(t) = (x(t,x0),y(t,x0)), the solution satisfying δ +(0) = x0where x0= (x0,y0). Let us introduce the equation Z(t,x0,y1) = 0, where Z(t,x0,y1) = x(t,x0) y(t,x0)−y1, and from our hypotheses we have Z( τ +,bx0,by1) = 0and F+ 1(0,by1)6=0. Then the Jacobian matrix ∂ (Z1,Z2) ∂ (t,y1)( τ +,bx0,by1) =    ∂ x ∂ t ∂ x ∂ y1 ∂ y ∂ t ∂ y ∂ y1   ( τ +,bx0,by1) =F+ 1(0,by1)0 F+ 2(0,by1)−1 is nonsingular. The Focus-Center-Limit Cycle Bifurcation in D2PWLS2without Sliding 9 Then, we conclude from the implicit function theorem the existence of two functions ϕ (x0), ψ (x0)defined in a neighborhood of x0, such that x( ϕ (x0),x0) = 0,y( ϕ (x0),x0)− ψ (x0) = 0,(9) with ϕ (bx0) = τ +, ψ (bx0) = by1. Thus, we can define the right Poincar´e map PRin a neighborhoodof the point x0 as y1=PR(y0) = ψ (0,y0), which satisfies by1=PR(by0) = ψ (bx0). To compute the first derivative of Poincar´e map, we take derivatives with respect to (x0,y0)in equations (10) to get ∂ Z ∂ t( ϕ (x0),x0, ψ (x0))·Dx ϕ (x0)+ B(x0)−0 Dx ψ (x0)=0,(10) where ∂ Z ∂ t( ϕ (x0),x0, ψ (x0)) =    ∂ x ∂ t( ϕ (x0),x0) ∂ y ∂ t( ϕ (x0),x0)  =F+(x1) and B(x0) = DxZ( ϕ (x0),x0, ψ (x0)) =    ∂ x ∂ x0( ϕ (x0),x0) ∂ x ∂ y0( ϕ (x0),x0) ∂ y ∂ x0( ϕ (x0),x0) ∂ y ∂ y0( ϕ (x0),x0)   satisfies the equality B(x0)F+(x0) = F+(x1),(11) due to the elementary properties of variational equations. By right-multiplying (11) by the vector (0,1)T, we get ϕ y(x0)F+(x1)+B(x0)0 1−0 ψ y(x0)=0, and so B(x0)0 −1=F+ 1(x1) ϕ y(x0) F+ 2(x1) ϕ y(x0)− ψ y(x0).(12) Taking into account (12) and (13) we can write F+ 1(x1)0 F+ 2(x1)−11 ϕ y(x0) 0 ψ y(x0)=B(x0)F+ 1(x0)0 F+ 2(x0)−1 and taking determinants we arrive at the relation F+ 1(x1) ψ y(x0) = F+ 1(x0)expZ ϕ (x0) 0div(F+)dt,(13)