Hopf bifurcation from infinity for planar control systems

Publicacions Matem`atiques, Vol 41 (1997), 181–198. HOPF BIFURCATION FROM INFINITY FOR PLANAR CONTROL SYSTEMS Jaume Llibre and Enrique Ponce Abstract Symmetric piecewise linear bi-dimensional systems are very common in control engineering. They constitute a class of non-differentiable vector fields for which classical Hopf bifurcation theorems are not applicable. For such systems, sufficient and necessary conditions for bifurcation of a limit cycle from the periodic orbit at infinity are given. 1. Introduction and statement of main results In this paper we are concerned with the appearance of one limit cycle from infinity for symmetric piecewise linear bi-dimensional systems. This phenomenon can be considered as a kind of generalized Hopf bifurcation from the infinity. The systems under study are of great importance in direct control theory [1], [2], being very common in control engineering as they include the case where the nonlinearities involved are of saturation type. They constitute a class of non-differentiable vector fields for which classical Hopf bifurcation theorems are not applicable so that specific techniques are needed in their analysis. Thus, we consider differential systems of the form (1) ˙ x=Ax +ψ(cTx)b, where Aisa2×2 real matrix and x,b,cbelong to R2. Here the dot denotes derivatives with respect to the variable s. The nonlinearity of these systems results from the presence of the characteristic function ψ. A common assumption in control theory is to consider odd piecewise linear characteristic functions of the form (2) ψ(σ)=     k2σ−(k1−k2)wif σ≤−w, k1σif −w<σ<w, k2σ+(k1−k2)wif w≤σ. 182 J. Llibre, E. Ponce Note that for k2= 0 the nonlinearity ψcorresponds with a saturation function, one of the most frequent nonlinearities involved in practice. Clearly a system (1)-(2) splits into three linear systems on the regions cTx≤−w,−w≤cTx≤wand cTx≥wand it is invariant under the symmetry x→−x. So, system (1)-(2) is also called an odd threepiecewise linear system. Systems (1)-(2) are linearly dominated at infinity, that is, there exists a constant matrix B=A+k2bcTsuch that lim ||x||→∞ ||Ax +ψ(cTx)b−Bx|| ||x|| =0, and so the results of Glover [3] and He [4], which give sufficient conditions in order that a periodic orbit bifurcates from infinity, apply. However, assuming that this bifurcation occurs for a critical value of a parameter, say µ= 0, they do not provide any information about: (1) whether the bifurcated periodic orbit exists for µ<0orµ>0 with |µ|sufficiently small, (2) the uniqueness of the bifurcated periodic orbit, (3) the stability of the bifurcated periodic orbit, and (4) an asymptotic estimate for the size of the bifurcated periodic orbit. Our results answer the four questions just mentioned. System (1) is called observable if the subspace span{c,ATc}=R2. For systems (1)-(2) we define the following four parameters T= trace(A+k1bcT),t= trace(A+k2bcT), D= det(A+k1bcT),d= det(A+k2bcT). Our first result is the following. Proposition 1. Systems (1)-(2) have a periodic orbit at infinity if and only if they are observable and 4d−t2>0. As it will be shown, by means of a linear change of variables, observable systems (1)-(2) can be written in the form (3) ˙x ˙y=0−d 1tx y+ϕ(y)d−D T−t, Hopf bifurcation from infinity 183 where (4) ϕ(σ)=sign(σ)if|σ|≥ 1, σif |σ|≤ 1. We remark that (3)-(4) correspond to the observable canonical form of systems (1)-(2) where the nonlinearity ϕis further a normalized saturation. Selecting tas the bifurcation parameter, we will use this form to show our main result, which is the following. Theorem 2. For systems (1)-(2) having a periodic orbit at infinity, the following statements hold. (a) If T=0then for t=0a unique limit cycle bifurcates from the periodic orbit at infinity. (b) If T<0then, for ε>0sufficiently small, the bifurcated limit cycle exists for t∈(0,ε)and is unstable, and it does not exist for t∈(−ε, 0). (c) If T>0then, for ε>0sufficiently small, the bifurcated limit cycle exists for t∈(−ε, 0) and is stable, and it does not exist for t∈(0,ε). (d) For their observable canonical form (3)-(4), the bifurcated limit cycle is near the ellipse of x-semiaxis a=Td−tD d· 1 + exp −πt √4d−t2 1−exp πt √4d−t2, and y-semiaxis b=a/√d. (e) If T=0then for t=0no limit cycles bifurcate from the periodic orbit at infinity. (f) For t=0no limit cycles bifurcate from the periodic orbit at infinity. Particular cases which can be studied by Theorem 2 appeared in Lum and Chua [5], Llibre and Sotomayor [6] and Llibre and Ponce [7]. Our main tool for proving Theorem 2 is the study of the first derivatives of the Poincar´e map in a neighborhood of infinity. To compute these derivatives, we use a extension to piecewise smooth systems of some results of Lloyd [8]. Bifurcation of a periodic orbit from infinity has been also studied for polynomial planar vector fields, see for instance Sotomayor and Paterlini [9], Blows and Rousseau [10], and Gu´ı˜nez, S´aez and Szant´o[11]. 184 J. Llibre, E. Ponce Other papers about bifurcation of periodic orbits from infinity are due to Keith and Rand [12], Malaguti [13] and Sabatini [14], where they study the Rayleigh, Van der Pol and Li´enard systems. The paper is organized as follows. In Section 2, we summarize the results in a previous work [15] about formulas for the first derivatives of the Poincar´e map for piecewise smooth systems. The proof of Proposition 1 and the use of these formulas to prove Theorem 2 are included in Section 3. 2. First derivatives of the Poincar´e map at infinity for piecewise smooth systems The study of the Poincar´e map in a neighborhood of infinity for planar vector fields, when it is well defined, can be conveniently made by using the Bendixson transformation. This reduces the problem to a similar study in a neighborhood of the origin for the transformed system, see for instance Andronov and others [16]. To fix ideas, we consider the planar system, (5) ˙x=f(x, y), ˙y=g(x, y), where fand gare Lipschitz functions. Through the inversion given by the Bendixson change of variables u v=1 x2+y2x y, we can formulate an equivalent system which behaves in a neighborhood of the origin like system (5) near infinity. Using now the polar coordinates u=rcos θ,v=rsin θ, what of course corresponds to do from the beginning the change of variables x=cos θ r,y=sin θ r, the system becomes (6) ˙r=−r2fcos θ r,sin θ rcos θ+gcos θ r,sin θ rsin θ, ˙ θ=−rfcos θ r,sin θ rsin θ−gcos θ r,sin θ rcos θ, Hopf bifurcation from infinity 185 and we will be interested in the flow defined in the half-cylinder R+× S1={(r, θ):r≥0,θ∈[0,2π)}. System (6) has in most cases no sense for r= 0, but this difficulty can be normally overcomed by a time reparametrization, and typically it suffices to multiply its vector field by an adequate power of r. Also note that, after extending continuously the flow to r= 0 if needed, the existence of a periodic orbit at infinity for system (5) is equivalent to have r= 0 as a periodic orbit on the cylinder for system (6). To be more precise, we assume in the sequel that system (6) can be extended to the system, (7) ˙r=R(r, θ), ˙ θ=Θ(r, θ), where the functions Rand Θ verify the following assumptions: (A1) Rand Θ both are Lipschitz functions and they have period 2π in θ. (A2) R(0,θ) = 0 for all θ, and Θ(0,θ)= 0 for all θ. Note that the last assumption implies that r= 0 is a periodic orbit of system (7) and that it has no equilibrium points in [0,ρ)×S1for some ρsufficiently small. This represents a sufficient and necessary condition for system (5) to have a periodic orbit at infinity. Since Θ = 0 in a neighborhood of r= 0, we can regard (7) as the first order equation (8) dr dθ =S(r, θ)=R(r, θ) Θ(r, θ), where r∈[0,ρ),θ ∈S1.Forξ∈[0,ρ), we denote with r(θ,ξ) the solution of (8) satisfying r(0,ξ)=ξ. We consider the Poincar´e map ξ→ h(ξ)=r(2π,ξ) and assume that h(ξ) is defined in [0,ρ). Of course, h(0) = 0. As it is well-known, his monotonically increasing on its domain of definition and every solution of h(ξ)−ξ= 0 corresponds with a periodic orbit of (8) and consequently of (7), (6) and (5). By studying the behaviour of the first derivatives of h, we can deduce whether additional solutions of the equation h(ξ)−ξ= 0 bifurcate from the solution ξ= 0. New solutions, if any, will correspond with periodic orbits of system (5) bifurcating from the periodic orbit at infinity. When Rand Θ, and so S, are sufficiently smooth, these first derivatives can be computed following Lloyd [8]. However, the lack of smoothness of 186 J. Llibre, E. Ponce differential systems (1)-(2) requires additional caveats. In what follows, we consider the case when the corresponding vector field S(r, θ) is only piecewise differentiable and we are interested in computing the first and second derivatives of a generic Poincar´e map. We assume a domain D={(r, θ)∈[0,ρ)×[θ1,θ 2]}which is halved by the graph of a C1-function ¯ θ:[0,ρ)→ [θ1,θ 2] in the two pieces D1={(r, θ):r∈[0,ρ),θ ∈[θ1,¯ θ(r)]}and D2={(r, θ):r∈[0,ρ),θ ∈ [¯ θ(r),θ 2]}, so that (9) S(r, θ)=S1(r, θ) for (r, θ)∈D 1, S2(r, θ) for (r, θ)∈D 2\D1, where both S1:D1→ Rand S2:D2→ Rare smooth, and S(0,θ)=0, for all θ∈[θ1,θ 2]. r ρ h1(ξ) ξ θ1¯ θ(ξ)θ∗(ξ)θ2 θ h(ξ) D1 D2 Figure 1. Scheme of the situation considered in this section. We suppose that the corresponding equation (8) with Sgiven in (9) has for all ξ∈[0,ρ) a continuous solution r(θ,ξ) defined in θ1≤θ≤θ2which verifies r(θ1,ξ)=ξ. We also assume for all ξ∈[0,ρ) a transversality condition (10) S1(ξ, ¯ θ(ξ)) ·d¯ θ dr (ξ)=1 on the curve ¯ θ([0,ρ)), and the existence of a C1crossing phase function, θ∗:[0,ρ)→ [θ1,θ 2] with θ∗(ξ)=¯ θ(r(θ∗(ξ),ξ)), Hopf bifurcation from infinity 187 which permits to define the intermediate Poincar´e map ξ→ h1(ξ)=r(θ∗(ξ),ξ). See Figure 1. The following result has been proved in a previous work [15]. Proposition 3. Under the previous assumptions, for ξ∈[0,ρ)and θ∈[θ1,θ 2]we define the functions: E(ξ,θ1,θ) = exp θ θ1 ∂S ∂r (r(φ, ξ),φ)dφ, D(ξ,θ)=E(ξ,θ1,θ)·∂2S ∂r2(r(θ,ξ),θ). Then the Poincar´e map ξ→ h(ξ)=r(θ2,ξ)verifies: h(ξ)=1−S2(h1(ξ),¯ θ(h1(ξ))) ·¯ θ(h1(ξ)) 1−S1(h1(ξ),¯ θ(h1(ξ))) ·¯ θ(h1(ξ))E(ξ,θ1,θ 2),(11) h(0) = E(0,θ 1,θ 2).(12) If furthermore the vector field Sgiven in (9) is continuous, we have (13) h(ξ)=E(ξ,θ1,θ 2)·θ2 θ1 D(ξ,θ)dθ +E(ξ,θ1,θ 2)E(ξ,θ1,¯ θ(h1(ξ))) ·¯ θ(h1(ξ)) 1−S1(h1(ξ),¯ θ(h1(ξ))) ·¯ θ(h1(ξ)) ·∂S1 ∂r (h1(ξ),¯ θ(h1(ξ))) −∂S2 ∂r (h1(ξ),¯ θ(h1(ξ))), and (14) h(0) = E(0,θ 1,θ 2)·θ2 θ1 D(0,θ)dθ +E(0,θ 1,¯ θ(0)) ·¯ θ(0) ∂S1 ∂r (0,¯ θ(0)) −∂S2 ∂r (0,¯ θ(0)). We remark that these formulae are in general different of those obtained by Lloyd [8], and of course both coincide if the vector field S turns out to be sufficiently differentiable. Proposition 3 will be used in the next section to prove Theorem 2. 188 J. Llibre, E. Ponce 3. Proof of the main results We start by showing two intermediate results which will be used in the proof of Proposition 1. Lemma 4. If a system (1)-(2) has a periodic orbit at infinity then it is observable. Proof: Suppose that a system (1)-(2) is not observable. Then, there exists a vector v=0such that c,ATcTv=0, and so both cTv=0 and cTAv = 0. Then vis an eigenvector of the matrix A, because if Av =λvfor all λ∈Rwe get the contradiction 0 = cTAv =λcTv=0. Since A+k1bcTv=Av, the vector vis also an eigenvector of the matrix A+k1bcT. We note that system (1)-(2) is ˙ x=A+k1bcTx, for |cTx|≤u. So that the straightline cTx= 0 is invariant under its flow. Consequently, we have a symmetric pair of critical points in the equator of the Poincar´e sphere and the conclusion follows. Lemma 5. Observable systems (1)-(2) can be written by means of a linear change of variables in the form given in (3)-(4). Proof: If we make in (1)-(2) the changes wy=x,¯ A=A+k2bcTand ¯ b=(k1−k2)b, we obtain the system (15) ˙ y=¯ Ay +ϕ(cTy)¯ b, where ϕis given in (4). This transformation preserves the hypothesis of observability because this hypothesis is equivalent to det(c,ATc)=0, and det(c,¯ ATc) = det c,ATc+(k2bTc)c= det(c,ATc). Now, from (15) and by means of another linear change of variables, we can pass to the observable canonical form (see Theorem 7-2 of Chen [17]): (16) ˙x ˙y=0−d 1tx y+ϕ(y)b1 b2. We remark that the quoted theorem guarantees the existence of an invertible matrix Tsuch that cTT=(0,1),0−d 1t=T−1¯ AT,and b1 b2=T−1¯ b. Hopf bifurcation from infinity 189 Comparing now the vector field near the origin of system (16) with that of (1)-(2), we have b1−d=−Dand b2+t=T, and the lemma follows. In the rest of the paper, we use for system (3)-(4) the notation of (16), where (17) b1=d−D, and b2=T−t, and we remark that it is formed by the following linear systems ˙x ˙y=0−d 1tx y+b1 b2for y≥1,(18) ˙x ˙y=0b1−d 1b2+tx yfor |y|≤1,and(19) ˙x ˙y=0−d 1tx y−b1 b2for y≤−1.(20) Proof of Proposition 1: Assume first that system (1)-(2) is observable. From Lemma 5, we can pass to the equivalent formulation given in (17)- (20). We define three regions on the cylinder R+×S1={(r, θ):r≥0 and θ∈[0,2π)}as follows DI={(r, θ):0≤θ≤πand r≤sin θ}, DII ={(r, θ):0≤θ≤2πand r≥|sin θ|}, DIII ={(r, θ):π≤θ≤2πand r≤−sin θ}. See Figure 2. If we make the Bendixson transformation, defined in Section 2, we obtain from (6) and (18)-(20) the following three systems: (21) ˙r=−r(1 −d) sin θcos θ+tsin2θ+r(b1cos θ+b2sin θ), ˙ θ=dsin2θ+ cos2θ+tsin θcos θ−r(b1sin θ−b2cos θ), where (r, θ)∈D I, (22) ˙r=−r(1 + b1−d) sin θcos θ+(b2+t) sin2θ, ˙ θ=−(b1−d) sin2θ+ cos2θ−(b2+t) sin θcos θ, where (r, θ)∈D II, and (23) ˙r=−r(1 −d) sin θcos θ+tsin2θ−r(b1cos θ+b2sin θ), ˙ θ=dsin2θ+ cos2θ+tsin θcos θ+r(b1sin θ−b2cos θ), 196 J. Llibre, E. Ponce From (40) and Lemma 6, we have ξ(t)≈− 1−exp −πt √4−t2 (b1t+b2) exp −πt √4−t21 + exp −πt √4−t2, and so the bifurcated limit cycle for system (27) is near the circle of radius r=1/ξ(t). Going backwards through (26), the bifurcated limit cycle for system (25) is near the circle of radius 1 r=− 1−exp −πt √4d−t2 (b1t+b2d) exp −πt √4d−t21 + exp −πt √4d−t2√d. Statement (d) follows at once by using (17) and recalling the transformation (24), made to get (25) from (16). 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Chen,“Linear System Theory and Design,” Holt, Rinehart and Winston, Inc., 1984. 198 J. Llibre, E. Ponce 18. S. Wiggins,“Introduction to Applied Nonlinear Dynamical Systems and Chaos,” Text Applied Math. 2, Springer-Verlag, 1990. Jaume Llibre: Departament de Matem`atiques Universitat Aut`onoma de Barcelona 08193 Bellaterra (Barcelona) SPAIN e-mail: [email protected] Enrique Ponce: Departamento de Matem´atica Aplicada II Escuela Superior de Ingenieros Avda. Reina Mercedes 41012 Sevilla SPAIN e-mail: [email protected] Rebut el 30 de Novembre de 1996