Hopf bifurcation at infinity in 3D Relay systems

Noname manuscript No. (will be inserted by the editor) Hopf bifurcation at infinity analysis for 3D Relay systems Emilio Freire ·Enrique Ponce ·Javier Ros ·El´ısabet Vela Received: date / Accepted: date Abstract A complete analysis of the limit cycle bifurcation from infinity in 3D Relay systems, which belong to the class of three-dimensional symmetric discontinuous piecewise linear systems with two zones, is presented. A criticality parameter is found, whose sign determines the character of the bifurcation. When such non-degeneracy parameter vanishes, a higher co-dimension bifurcation takes place, giving rise to the emergence of a curve of saddle-node bifurcations of periodic orbits. To justify the existence of a large amplitude limit cycle that bifurcates from infinity, the closing equations method is adapted, so that we provide analytical expressions for its amplitude and period. Derivatives of the corresponding transition maps are rigorously studied in order to characterize the stability of the bifurcating limit cycle. The theoretical results are applied to a specific family of 3D relay systems, where several high co-dimension bifurcation points are detected, organizing the bifurcation set of the family. Keywords Piecewise linear systems ·bifurcations at infinity. 1 Introduction and statement of main results As emphasized in [13] and [30], relay systems appear frequently in practical control applications. Its switching nature leads to the consideration of discontinuous differential systems, which is a subject of intense research in our days. Some obligated references regarding relay systems are [37], [3] and [2]. While, in the recent references [4, 9–11, 24, 29, 31] some bifurcations leading Departamento de Matem´atica Aplicada II, Escuela T´ecnica Superior de Ingenier´ıa, Avda. de los Descubrimientos, 41092 Sevilla, Spain E-mail: ef[email protected], [email protected], ja[email protected], eliv[email protected] ORCiD: https://orcid.org/0000-0001-5852-0699, https://orcid.org/0000-0003-0467-5032, https://orcid.org/0000-0002-6396-1461, https://orcid.org/0000-0003-2344-0351. This is an Accepted Manuscript of an article published by Elsevier in Physica D: Nonlinear Phenomena, Volume 444, February 2023, available at: https://doi.org/10.1016/j.physd.2022.133586. Copyright 2022 Elsevier. En idUS Licencia Creative Commons CC BY-NC-ND 2 Emilio Freire et al. to limit cycles in these systems have been analyzed. However, the possibility of the bifurcation of limit cycles of large amplitude that come from the infinity is a problem which has not been thoroughly studied. Thus, only a few cases can be found in the literature, as the study made in [21] for continuous 3D systems with symmetry, previously analyzed in [14, 18], see also [26], [27] and references in these articles. Recently, the dynamical consequences of a specific bifurcation from infinity in continuous 3D systems with only two linear pieces have been reported in [15]. Here, we will present some contributions to the analysis of bifurcations from infinity in symmetric discontinuous 3D systems. More precisely, we consider relay systems ˙x =Ax−bϕc⊤x where ϕis the sign function. Under generic hypothesis to be later specified, we can assume that c=e1(the first canonical vector), so that these systems can be written as ˙x =F(x) = F−(x) = Ax+b,if x≤0, F+(x) = Ax−b,if x≥0,(1) where x= (x, y, z)⊤and the ambiguity in the definition of the vector field will be not relevant as long as we only consider the crossing dynamics. Furthermore, here the dot represents derivative with respect to the time τ, the vector b= (b1, b2, b3)⊤∈R3is constant, while the matrix A=  t−1 0 m0−1 d0 0  (2) rules the dynamics and the coefficients t,mand dare the linear invariants (trace, sum of principal minors and determinant) of such matrix. To start from the above canonical form (1)-(2) does not suppose any loss of generality, if it is assumed that the initial system is observable, see Prop. 10 in [7]. Therefore, system (1) is a symmetric discontinuous piecewise linear system with two linearity zones separated by the plane Σ={x∈R3:x= 0}.(3) This set is the common boundary for the two regions of R3where x≤0 and x≥0, which will be denoted by L(left) and R(right) zones, respectively. As emphasized in [11], system (1) has a sliding set Σs={x∈R3:x= 0,−|b1| ≤ y≤ |b1|}, to be atractive or repulsive depending on the sign of b1. We will not enter into more details, since our goal is to study big crossing periodic orbits that necessarily surround this siliding set without interacting with it. Guided by a similar result obtained in [21], we study the possible bifurcation from infinity, using the closing equations technique, under the eigenvalue Hopf bifurcation at infinity analysis for 3D Relay systems 3 configuration of matrix Aassociated to a center, that is, we assume that the eigenvalues of Aare λ,σ±iω with ω > 0 and analyze the dynamical effects of a sign transition for the real part σ. Thus, we suppose that the linear invariants satisfy t= 2σ+λ, m= 2σλ +σ2+ω2,(4) d=λσ2+ω2. The so-called closing equations technique dates back to Andronov [1] and [32], and has been succesfully applied to a large number of piecewise linear systems. See for example [33], [16], [17], [19], [20], [35] and [5]. When d6= 0 or equivalently λ6= 0, there appears one equilibrium for each vector field in (1), namely xR= (xR, yR, zR) = A−1b=b3 d,t db3−b1,m db3−b2,xL=−xR.(5) Furthermore, to each equilibrium point there is associated an invariant plane, usually called focal plane. For instance, if wdenotes the left-hand eigenvector associated to the real eigenvalue λ6= 0 of the matrix A, then the focal plane for xRis w⊤(x−xR) = 0, or equivalently, w⊤x=w⊤xR=w⊤A−1b=1 λw⊤b. Effectively, the invariance of such a plane comes from the equalty w⊤˙ x=w⊤(Ax+b) = λw⊤x−w⊤b= 0, which is true for any point belonging to such a plane. If b3dis positive, then both equilibria are real; otherwise, when b3dis negative the above equilibria are located out of the half space whose dynamics is ruled by them, and so they are called virtual equilibria. In any case, for each vector field F±, there appears an invariant plane passing through the corresponding equilibrium point. From such focal planes, only the half part with x < 0 (x > 0) is invariant for F−(F+), and so we will speak of invariant focal half-planes. Taking advantage of the fact that the left-hand eigenvector associated to the real eigenvalue of Ais w⊤= (λ2,−λ, 1), and assuming λ6= 0 we obtain for the right focal half-plane the following expression λ2x−λy +z=1 λλ2b1−λb2+b3.(6) Obviously for the left focal half-plane, we similarly get λ2x−λy +z=−1 λλ2b1−λb2+b3.(7) 4 Emilio Freire et al. x y z p0 −p0 Fig. 1 Any symmetric periodic orbit must intersects Σat two symmetrical points between the intersections lines of the focal planes. Note that the focal half-planes do not depend on σnor ω. In this work, we will only be concerned about symmetric periodic orbits with just two intersections points with the plane Σ. The symmetry of these periodic orbits and the invariance of the focal planes, impose that the two intersection points ±p0= (0,±y0,±z0) belong to the band defined by the intersecction lines of the two focal half-planes (6)-(7) with Σ, see Figure 1. Therefore, for λ6= 0 |−λy0+z0| ≤  1 λλ2b1−λb2+b3 .(8) Obviously, when a periodic orbit bifurcates from infinity at least one of the coordinates y0and z0becomes unbounded, but their combination in the left hand side in (8) is always bounded. In the non-generic case λ= 0, we have d= 0, and there will be no equilibrium points whenever b36= 0, but nevertheless symmetric periodic orbits are possible, as shown later. If both λ= 0 and b3= 0, then the third equation in system (1), becomes ˙z= 0, so that all the horizontal planes are invariant, and consequently symmetric periodic orbits are only possible on the plane z= 0. Clearly, the dynamics on such a plane corresponds with the dynamics of a planar relay, a subject which has been studied in [25]. In the quoted paper, authors consider different bifurcations in the parameter plane (b1, b2), but without analyzing possible bifurcations from infinity. We remark that when λ=b3= 0 there appear an infinity of non-isolated equilibrium points, which for σ= 0 undergo a Zero-Hopf bifurcation, to be analyzed elsewhere. We will assume that the half of the periodic orbit corresponding to x≤0, starts at the point p0= (0, y0, z0), so that we impose that ˙x|(0,y0,z0)=e⊤ 1A  0 y0 z0 +e⊤ 1b < 0,(9) Hopf bifurcation at infinity analysis for 3D Relay systems 5 that is −y0+b1<0. The arrival point for this half-orbit will be −p0= (0,−y0,−z0) and then the crossing assumption needs y0+b1>0. Therefore, we need y0>|b1| ≥ 0 and work only with positive values of y0. In particular, to deal with possible periodic orbits bifurcating from infinity, we introduce the reciprocal quantity r0= 1/y0, so that the unbounded values of y0are converted to small positive values for r0. In short, under the conditions given in (4) and assuming σto be the principal bifurcation parameter, we will state our main results, to be proved in Section 4. They give sufficient conditions for system (1) to undergo a Hopf bifurcation from infinity, characterizing the existence and stability of the bifurcating limit cycle. Our first result deals with the generic case under a nondegeneracy condition. Theorem 1 Consider system (1) under the assumptions (4) with ω > 0and define the non-degeneracy parameter δ=b3−b2λ−b1ω2.(10) If δ6= 0, then, for σ= 0 the system undergoes a Hopf bifurcation from infinity, that is, one symmetric limit cycle of large amplitude appears for δσ < 0and σsufficiently small. Moreover, the period Tof the periodic oscillation is an analytic function at 0, in the variable σ, and the first terms in its series expansion are T=2π ω−2πλδ +λ2+ω2b2 ω3δσ+Oσ2. A measure for the amplitude a, measured as y0= 1/r0>0, has a series expansion in powers of σ, starting with a=2ωδ π(λ2+ω2)σ+O(1) . In particular, when λ6= 0, if δ > 0and λ < 0,then the bifurcating limit cycle for σ < 0is orbitally asymptotically stable. Otherwise, if δ < 0or λ > 0the bifurcating limit cycle is unstable. In the case λ= 0, assuming δ=b3−b1ω2>0a sufficient condition for the stability of the bifurcating limit cycle for σ < 0is b1>0. Remark 1 From conditions (4), and the trivial computation mt −d= (2σλ +σ2+ω2)(2σ+λ)−λσ2+ω2= 2σ((λ−σ)2+ω2), we get that sign(σ) = sign(mt −d), so that σ= 0 implies mt −d= 0. Furthermore, we have then t=λ,d=λω2and m=ω2>0. Reciprocally, if m > 0 and mt −d= 0, then the characteristic polynomial of the matrix A becomes ξ3−tξ2+mξ −d=ξ3−tξ2+mξ −mt = (ξ−t)(ξ2+m) 6 Emilio Freire et al. and then there exists ω > 0 such that ω2=m, and the matrix Ahas the eigenvalues λ=tand ±iω, so that d=λω2. Therefore, the two conditions mt −d= 0 and m > 0 allow to detect the bifurcation from infinity of a periodic orbit with period 2π/ω = 2π/√m. Moreover, under these conditions, the non-degeneracy parameter becomes δ= b3−b2t−b1m, and so the bifurcation can be characterized without computing the eigenvalues. Theorem 1 is proved in Section 4 as a consequence of Proposition 5, included in that section. If the non-degeneracy parameter δvanishes, that is, b3=b2λ+ω2b1, then under certain hypotheses system (1) still undergoes a Hopf bifurcation at infinity now with a degenerate caracter. We assume however b16= 0 and b26= 0 to avoid additional degenerations. Regarding Theorem 1, here the dependence of period and amplitude respect to the parameter σchanges and their expansions have fractional exponents. To alleviate notation, in the sequel we make use of the positive and even function Υ(x) =      tanh(x) x, x 6= 0 1, x = 0. Theorem 2 Consider system (1) under the assumptions (4) with ω > 0, b1b26= 0 and b3=b2λ+ω2b1, so that we are in the degenerate case δ= 0. For σ= 0 the system undergoes a Hopf bifurcation at infinity, that is, one symmetric limit cycle appears for b1b2σ < 0and σsufficiently small. In particular, if b1b2>0and λ < 0then the limit cycle bifurcates for σ < 0 and is orbitally asymptotically stable, being unstable if b1b2<0or λ > 0. For the case λ= 0, if b1>0and b2>0, then the limit cycle bifurcates for σ < 0and is stable. Moreover, the series expansion in the variable σof the period Tof the periodic oscillation starts with T=2π ω−2πb2 ωΥλπ 2ω−1/2−σ b1b21/2 σ+O(σ), while the series expansion of the amplitude a(measured as y0= 1/r0>0) starts with a=Υλπ 2ω1/2−σ b1b2−1/2 +O(1) . The degeneration of Theorem 2 suggests the existence of a co-dimension two bifurcation point, from where a saddle-node bifurcation curve of periodic orbits emerges, analogously to what happens in the focus-center-limit cycle bifurcation for continuous piecewise linear systems, see [19, 20]. In fact, the following theorem guarantees the existence of such bifurcation curve. Hopf bifurcation at infinity analysis for 3D Relay systems 7 01 1 2 0 σ δ σSN 10 0 2 1 σ δ σSN Fig. 2 Bifurcation set for b1b2>0 (left panel) and b1b2<0 (right panel) in a neighborhood of the origin. The digits indicate the number of limit cycles associated with the Hopf bifurcation at infinity in each region. Theorem 3 Consider system (1) under the assumptions (4) with ω > 0, b1b26= 0, and b3in a neighborhood of the value b∗ 3=b2λ+ω2b1, so that the non-degeneracy parameter becomes δ=b3−b∗ 3. Consider a neighborhood of the origin in the parameter plane (σ, δ). As stated in Theorem 1, at the straight line σ= 0 with δ6= 0, the system undergoes a limit cycle bifurcation from infinity, which gives rise to a limit cycle for σδ < 0. Furthermore, the following statements hold. (a) If b1b2<0 (b1b2>0), then from the point (σ, δ) = (0,0) a saddle-node bifurcation curve of periodic orbits emanates in the second quadrant (fourth quadrant) of the parameter plane (σ, δ), see Figure 2. This curve is the graph of a function with the following local expression σSN (δ) = ω2 π2Υλπ 2ω(λ2+ω2)2b1b2 δ2+O(δ)3.(11) (b) For sufficiently small (σ, δ)with b1b2<0,δ > 0and σSN (δ)< σ < 0 (b1b2>0,δ < 0and 0< σ < σSN (δ)) two limit cycles coexist, see Figure 2. (c) For points at the curve σ=σSN there is only one non hyperbolic limit cycle, and no limit cycles out of the regions indicated in item (b) for b1b2<0 (b1b2>0) and with σ < 0 (σ > 0). The proof of Theorem 3 appears in Section 4. The rest of the paper is organized as follows. First, in Section 2, we include some results on the analysis and stability of periodic orbits, which are based upon a study of transition maps and their derivatives, see Appendix A. In Section 3, we illustrate the usefulness of the previous theoretical results by considering a specific family of 3D relay systems, which had already been considered but whose rich dynamics is far of being completely described. The proofs of some preliminary results, along with the ones of main theorems, appear in Section 4. Some concluding remarks appear in Section 5. 8 Emilio Freire et al. 2 Some tools for the analysis of symmetric periodic orbits in 3D relay systems Here, we include some results that will allow us to determine the existence and stability of a symmetric periodic orbit Γof system (1) with only two transversal crossing points (0, y0, z0)∈Σand (0,−y0,−z0)∈Σ. We will use some properties of transition maps, also called Poincar´e maps, see [34] and the Appendix. Taking Σas transversal section, let us consider the Poincar´e return map of system (1), defined as the composition of two Poincar´e half-return maps. If a point p0= (0, y0, z0) in the plane x= 0 is transformed by means of the flow in the zone x < 0 into the point p1= (0, y1, z1)∈Σafter a time τL(y0, z0) arriving transversaly at x= 0 for the first time, then we define the left Poincar´e half-return map in terms of local coordinates as ψL(y0, z0) = (y1, z1), see (78) at the Appendix for more details. Analogously, by using the flow in the zone x > 0, a point p1= (0, y1, z1) in the plane x= 0 is transformed by means of the flow in the zone x > 0 into the point p2= (0, y2, z2)∈Σafter a time τR(y1, z1) arriving transversaly at x= 0 for the first time the Poincar´e half-return map ψR(y1, z1) and the right flight time τR(y1, z1) can be defined. The composition of these two half-return maps is the complete Poincar´e map. Clearly, for the points p0,p1and p2belonging to a symmetric periodic orbit Γwith only two transversal crossing points, we have p0=p2= (0, y0, z0)∈ Σand p1=−p0= (0,−y0,−z0)∈Σ. Note that the solution starting from p0can be written as   x(τ) y(τ) z(τ) =eAτ   0 y0 z0 +Zτ 0 eA(τ−s)bds =eAτ   0 y0 z0 +Zτ 0 eAubdu, after the change u=τ−s. Thus, the existence of symmetric periodic orbits can be just associated to the existence of solutions of the closing equations eAτL  0 y0 z0 +ZτL 0 eAubdu =  0 −y0 −z0 ,(12) with y0>|b1|and τL>0. The condition y0>|b1|assures that condition (9) is satisfied, while the assumption of only two crossing points implies that x(τ)<0 for all 0 < τ < τL. Once we have a solution for (12), to assess the stability of the corresponding periodic orbit, we must compute the eigenvalues of the derivative of the composition ψR(ψL(y, z)) at the fixed point (y0, z0), that is, the eigenvalues of the product DψR(−y0,−z0)·DψL(y0, z0). The symmetry of the vector field suggests that there should be some relationship between both matrices and, in fact, Corollary 3 from Appendix allows us to state the following result. Hopf bifurcation at infinity analysis for 3D Relay systems 9 Proposition 1 If Γis a symmetric periodic orbit of system (1) with only two transversal crossing points p0= (0, y0, z0)∈Σand p1=−p0∈Σ, then the matrices QR=1DτR(−y0,−z0) 0DψR(−y0,−z0), QL=1DτL(y0, z0) 0DψL(y0, z0)(13) are similar, satisfying DτR(−y0,−z0) = −DτL(y0, z0),and DψR(−y0,−z0) = DψL(y0, z0).(14) In fact, if τLis the half-period of the orbit, then we have QL=   1 b1+y00 0 b2+z0 b1+y0−1 0 b3 b1+y00−1  eAτL  b1−y00 0 b2−z0−1 0 b30−1 .(15) Proof Equalities (14) are a direct consequence of the symmetry of the vector field, which imposes that τR(−y, −z) = τL(y, z),and ψR(−y, −z) = −ψL(y, z), for every (y, z) where the return map is defined. We will apply Corollary 3 of the Appendix by taking p1=−p0, and Fas the vector field F−from (1). Thus, we obtain that F−(p1)0 −I1DτL(y0, z0) 0DψL(y0, z0)= exp (AτL(y0, z0)) F−(p0)0 −I,(16) where DτLand DψLare the derivatives of the flight time function and Poincar´e halfmap in the left zone and F−(p0) = Ap0+b= (b1−y0, b2−z0, b3), F−(p1) = F−(−p0) = −Ap0+b= (b1+y0, b2+z0, b3). The matrix equality (16), after defining the matrices B− 0=F−(p0)0 −I=  b1−y00 0 b2−z0−1 0 b30−1  and B− 1=F−(p1)0 −I=  b1+y00 0 b2+z0−1 0 b30−1 , can be abridged as B− 1QL= exp(AτL)B− 0, and then a simple computation leads to (15). 16 Emilio Freire et al. Fig. 5 Two limit cycles of system (20) for for ˜κ=−1, ˜ λ= ˜σ= ˜ω= 1, and ˜ρ= 2, near the saddle-node bifurcation that appears at ˜ρ= 2.05. 4 Proof of main results In order to simplify as much as possible the different proofs, we will introduce a new time variable θ=ωτ and some suitable linear change and scaling in phase variables and parameters, looking also to put in horizontal position both focal planes. This is accomplished by means of the following result, whose proof is a direct computation to be omitted. Proposition 4 Under hypotheses (2) and (4), the change of time, variables and parameters θ=ωτ, ˜x=x, y =ω˜y, z =ω2˜z, λ =ωµ, σ=ωγ, b1=ω˜ b1, b2=ω2˜ b2, b3=ω3˜ b3,(24) transforms system (1) into the form d dθ   ˜x ˜y ˜z =  2γ+µ−1 0 2γµ +γ2+ 1 0 −1 µγ2+ 10 0    ˜x ˜y ˜z −sgn(˜x)  ˜ b1 ˜ b2 ˜ b3 .(25) Furthermore, the additional linear change of variables X= ˜x, Y = ˜y−µ˜x, Z = ˜z−µ˜y+µ2˜x(26) Hopf bifurcation at infinity analysis for 3D Relay systems 17 transforms system (25) in the system d dθ   X Y Z =  2γ−1 0 γ2+ 1 0 −1 0 0 µ   X Y Z −sgn(X)  β1 β2 β3 ,(27) where β1=˜ b1,β2=˜ b2−µ˜ b1,β3=˜ b3−µ˜ b2+µ2˜ b1. Remark 3 Note that in the first part of Proposition 4, leading to system (25), the main goal is to reduce the number of parameters by one, normalizing the frequency; that is, the final effect is equivalent to take ω= 1 in (4). Next, in the second part of Proposition 4, which allows to get system (27), we look for a twofold aim, namely to decouple the dynamics of the third variable putting the two focal planes in a horizontal position, and to write the linear part describing the focus dynamics into the canonical form introduced in [22]. After applying Proposition 4, the equilibria become (XL, YL, ZL) = −µβ2+β3 µ(γ2+ 1),2γXL+β1,−β3 µ and (XR, YR, ZR) = −(XL, YL, ZL), while the new focal planes are Z=β3 µ(for X > 0) and Z=−β3 µ(for X < 0). Remark 4 Regarding system (27), if β3= 0 then the plane Z= 0 is invariant (being attractive or repulsing acording to the sign of µ) and the long-term dynamics of the system becomes esentially two-dimensional. This case corresponds to the condition λ2b1−λb2+b3= 0 in system (1), when the two half focal planes (6) and (7) are contained in the plane λ2x−λy +z= 0, which in the new setting after applying Proposition 4 becomes the plane Z= 0. Thus, when β3= 0 the dynamics of the 3D system is determined by the dynamics in the plane Z= 0, and in such a plane we have the 2D system d dθ X Y=2γ−1 γ2+ 1 0 X Y−sgn(X)β1 β2,(28) so that our results also cover the bifurcation from infinity for γ= 0 of the above planar system, a bifurcation not studied in [25]. Note that the planar system (28) for γ= 0 is a piecewise Hamiltonian system. For instance, a piecewise Hamiltonian function is X2∓2β2X+Y2±2β1Y for ±X > 0. If furthermore β1= 0, then the plane is foliated by crossing periodic orbits, leading to an unbounded periodic annulus. However, for γ6= 0 and β1= 0 the system cannot have periodic orbits because the divergence of the vector field is constant and there is no sliding set, see Remark 3.8 in [22]. In short, for β1=β3= 0, the bifurcation of periodic orbits when γ= 0 is not 18 Emilio Freire et al. possible. These conditions translate to ˜ b1= 0 and ˜ b3=µ˜ b2, and so b1= 0 and b3=b2λ, which from (10) leads to δ= 0. Reciprocally, if δ= 0 and b1= 0, then b3=b2λ, which leads to β1=β3= 0, and so the bifurcation is not possible at all. In this work, we will be concerned only about symmetric periodic orbits, which can be associated with solutions of the so-called closing equations. Thus, if we assume that a symmetric periodic orbit intersects transversally the plane Σin the two points (0,±Y0,±Z0), we only need to impose for the semi-orbit in the left zone (X < 0) that after a time θLthe closing equations φ(θL; (0, Y0, Z0)) = (0,−Y0,−Z0),(29) are satisfied, where φstands for the flow of the system for X < 0. It is assumed that e⊤ 1 dφ dθ (0; (0, Y0, Z0)) <0,e⊤ 1 dφ dθ (0; (0,−Y0,−Z0)) >0,(30) which from (27), become ˙ X(0,Y0,Z0)=−Y0+β1<0,˙ X(0,−Y0,−Z0)=Y0+β1>0, and so we get the transversality condition Y0>|β1| ≥ 0,(31) which is true whenever Y0is positive and sufficiently big. Since the focal planes are invariant in their corresponding half-spaces, it is clear that the semiorbit cannot cross such planes, and so the coordinate Z0of the initial point must satisfy − β3 µ≤Z0≤ β3 µ ,(32) and so any periodic orbit, including the ones coming from or going to infinity are bounded in the third coordinate. Thus, in order to work with orbits of large amplitude for system (27), the trick to facilitate the analysis is to do a change of variables of the form r0=1 Y0 ,(33) where r0>0 belongs to a neighbourhood of 0. Thanks to this strategy and taking advantage of the block structure of system (27), it is possible to get a complete characterization of the bifurcation from infinity for such a system. The general expression of the flow of system (27) to define the closing equations given in (29) is φ(θL;X) = eAθX+V(θ)β, (34) Hopf bifurcation at infinity analysis for 3D Relay systems 19 where X= (X, Y, Z)⊤, A=  B0 −1 0 0 µ , B =2γ−1 γ2+ 1 0 , β =  β1 β2 β3 , and V(θ) = Rθ 0eAudu satisfies V′(θ) = eAθwith V(0) = 0. In what follows, the symbol Iwill stand for the identity matrix with the suitable order in each case. Using (29) and (34), we obtain for the closing equations the expression eAθL  0 Y0 Z0 +V(θL)β=  0 −Y0 −Z0 , that is, eAθL+I  0 Y0 Z0 +V(θL)β=0.(35) Remark 5 Corollary 2 can be applied to study the stability of periodic orbits of system (27), which are associated to solutions (0, Y0, Z0)⊤of equation (35). It suffices to take into account that the derivative of the Poincar´e map can be obtained from the matrix QL=   1 β1+Y00 0 β2+Z0 β1+Y0−1 0 β3−µZ0 β1+Y00−1  eAθL  β1−Y00 0 β2−Z0−1 0 β3+µZ00−1 ,(36) so that trace QLbecomes eµθ + 2β1cos θ−γY0sin θ β1+Y0 eγθ −2µeµθ −eγθ(µcos θ−(1 −µγ +γ2) sin θ) (β1+Y0)(1 + (µ−γ)2)Z0 (37) while det QL=β1−Y0 β1+Y0 e(µ+2γ)θ.(38) If we assume µ6= 0 and study the structure of matrices eAθand V(θ), we easily conclude that eAθ=eBθ W(θ) 0eµθ , V (θ) = B−1eBθ −IRθ 0W(u)du 0eµθ −1/µ, where W(θ) is the unique solution of the initial value problem W′(θ) = BW (θ) + eµθ 0 −1, W(0) = 0. 20 Emilio Freire et al. Computations give W(θ) = eBθ −eµθIC, with C=1 1 + (γ−µ)2−1 µ−2γ.(39) Clearly, the closing equations (35) can be decoupled by taking apart the third component, namely eµθL+ 1Z0+1 µeµθL−1β3= 0,(40) and writing the first two components as follows eBθL+I0 Y0+W(θL)Z0+B−1eBθL−Iβ1 β2+β3ZθL 0 W(u)du =0. (41) Multiplying (41) by the non-singular matrix Band taking into account that Bcommutes with eBθ, and the equalities B0 Y0=−Y0 0, and Bβ3ZθL 0 W(u)du =β3ZθL 0 BW(u)du =β3eBθL−I−eµθL−1 µBC, whose right hand side using (40) reduces to β3eBθL−IC+eµθL+ 1Z0BC, we finally see that (41) becomes eBθL+I−Y0 0+Z0BC+eBθL−Iβ1 β2+β3C=0.(42) Note that from (40) we get an explicit value for Z0in terms of parameters and flight time θL, namely Z0=−β3 µ eµθL−1 eµθL+ 1 =−β3 µtanh µθL 2.(43) We remark that when µ= 0 the continuous extension of the expression in (43) gives Z0=−β3θL 2,(44) and it is indeed the value we should obtain by considering this particular case from the beginning. In short, thanks to Proposition 4 we can reduce in one dimension the closing equations, which can be rewritten as in (42) with Z0as in (43) or in (44). Hopf bifurcation at infinity analysis for 3D Relay systems 21 In order to determine orbits of large amplitude for system (1), or equivalently for system (27) we will do the change of variables (33) in the closing equations (42-43). It is clear that a periodic orbit of large amplitude has a big positive value Y0verifying (31), so that we can choose Y0as the reference coordinate for the intersection of the orbit with the plane Σ. To obtain a new set of equations that allows to work easily near the point at infinity, instead of Y0, we take r0= 1/Y0as new variable. Consequently, after such substitution and multiplying (42) by r0, we obtain the reduced closing equations eBθL+I−1 0+r0Z0BC+r0eBθL−Iβ1 β2+β3C=0.(45) If we detect a parameterized branch of solutions of equation (45) and we can follow such a branch making r0tend to 0, then the associated periodic orbit will approach certain periodic orbit at infinity. Thus if we analyze the set of closing equations (45) for small r0>0, we could obtain limit cycles with a very big value for Y0. Therefore, these equations turn out to be useful to detect the bifurcation of a limit cycle from the periodic orbit at infinity of system (1) that corresponds to the solution of (45) with r0= 0, which is analyzed in the following Lemma. Lemma 2 The reduced closing equations (45) with r0= 0 and 0< θL<2π are only satisfied for θL=πand γ= 0. Proof Putting r0= 0 in (45) equation reduces to eBθL+I−1 0=0, so that, (−1,0)⊤is an eigenvector associated to an eigenvalue −1 for the matrix exp (BθL). Since Bhas eigenvalues γ±i, we obtain the condition exp (γθL±iθL) = −1, that is, γ= 0, θL=π. The conclusion follows. Remark 6 For γ= 0 and θL=π, we have indeed eBθL+I=0 so that, equation (45) reduces to −2r0β1 β2+β3 1 + µ2−1 µ=−2r0 1 + µ2(1 + µ2)β1−β3 (1 + µ2)β2+µβ3=0.(46) Therefore, if the non degeneracy parameter ˜ δ=β3−β1(µ2+ 1) (47) and the auxiliary parameter ǫ=µβ3+β2(µ2+ 1) (48) 22 Emilio Freire et al. Fig. 6 The unbounded center for system (27) in the critical situation that corresponds to the case δ=b2= 0 in the original system. vanish, then the reduced closed equations are satisfied not only for r0= 0 but also for any value r0>0. This fact indicates that there is an unbounded isochronous center in such a situation. Effectively, as shown in Figure 6, for system (27) with γ= 0, µ= 1, β1= 1, β2=−1, and β3= 2, we obtain an unbounded manifold with a hole completely foliated by periodic orbits. Note that from Proposition 4, the configuration ˜ δ=ǫ= 0 leads to ˜ b1=˜ b3, and ˜ b2= 0, that is, δ=b2= 0. This is the reason for which we require the condition b26= 0 in theorems 2 and 3. Remark 7 Note that in the original system (1), the existence of the solution of Lemma 2 implies σ=ωγ = 0, and so the orbit at infinity from which a limit cycle bifurcates has a semi-period τ=π/ω. Taking advantage of the reduced closing equations (45), we will derive the proofs of the Theorems 1, 2 and 3 by considering system (27), studying now the possible existence of a branch of solutions emerging from the solution predicted in Lemma 2. If we wish to apply the implicit function theorem to equations (45), to be written in the form G(z) = 0,(49) with z= (r0, θL, γ), near the point ¯z = (0, π, 0),(50) Hopf bifurcation at infinity analysis for 3D Relay systems 23 then we must study the Jacobian matrix at such point, namely DzG(¯ z) = −2β2−2µβ3 µ2+1 1 0 2β1−2β3 µ2+1 0−π!= −2ǫ µ2+1 1 0 −2˜ δ µ2+1 0−π!,(51) where we have introduced the parameter ˜ δand ǫ, defined in Remark 6. From (51), it is evident that the point ¯zis always a regular point of system (49) and so, we can obtain their solutions for γand θLin terms of r0, see [8]. Computations lead to the following result, where we use (47) and (48) to avoid the appearance of parameters β1and β2. Lemma 3 The point ¯z in (50) is a regular point of the reduced closing equations (45). Consequently, in a neighborhood of ¯z, there exists only one branch of solutions (r0, θL(r0), γ(r0)), with the analytic expansions θL=π+2ǫ µ2+ 1r0+8β3˜ δ π(µ2+ 1)3µr02− πβ3Υπµ 2˜ δ−ǫµ (µ2+ 1)2r02+Or03. (52) γ=−2˜ δ π(µ2+ 1)r0+8˜ δǫµ2+ 1+β3µ π2(µ2+ 1)3r02− β3Υπµ 2ǫ+˜ δµ (µ2+ 1)2r02+Or03. From (43), using (52), we get the expansion Z0=−π 2β3Υπµ 2−4eπµβ3ǫ (µ2+ 1)2(eπµ + 1)2r0−16eπµβ2 3˜ δ π(µ2+ 1)3(eπµ + 1)2r2 0 + 2πeπµβ3Υπµ 2ǫ2µ2+β3(δ−ǫµ) (µ2+ 1)2(eπµ + 1)2r2 0+O(r3 0).(53) If in expressions (37)-(38) we do the change of variables (33) and substitute the expansions (52) and (53), then we obtain trace(QL) = eπµ + 2 ˜ δ+eπµ(ǫµ −β3) µ2+ 1 r0−8˜ δǫ(µ2+ 1) + β3µ(1 −2eπµ) π(µ2+ 1)3r2 0 + 2eπµ(β2 3+ǫ2µ2)−˜ δ2 (µ2+ 1)2r2 0−πβ3Υπµ 2˜ δµ(e2πµ −1) −2ǫ (µ2+ 1)2r2 0 −2β3ǫµ eπµ(eπµ + 3) (µ2+ 1)2(eπµ + 1)r2 0+O(r3 0),(54) and det(QL) = −eπµ + 2eπµ β3+˜ δ−ǫµ µ2+ 1 r0−8˜ δeπµ ǫ(µ2+ 1) + 3β3µ π(µ2+ 1)3r2 0 −2eπµ (β3+˜ δ−ǫµ)2 (µ2+ 1)2r2 0−πβ3eπµΥπµ 2ǫ(µ2−2) −3˜ δµ (µ2+ 1)2r2 0+O(r3 0). (55) 24 Emilio Freire et al. When ˜ δ6= 0, it is possible to invert the series for γof (52) and determine the unique branch of solutions in terms of parameter γ. In fact, the determinant of the submatrix obtained by removing the last column in (51), which corresponds to γ, is equal to 2˜ δ µ2+ 1 6= 0, so that the rank of such submatrix is 2 and we could also apply at ¯ zthe implicit function theorem for analytic functions to solve equation (49) for (r0, θL). In any case, we obtain r0=−µ2+ 1 2˜ δπγ +O(γ2),(56) θL=π−ǫ ˜ δπγ +O(γ2).(57) Now, we can give easily the proof of Theorem 1. 4.1 Proof of Theorem 1 From the expansion for r0in (56) it is clear that a change of sign in the parameter γdetermines a passing through zero of r0, that is a bifurcation of a periodic orbit from infinity, which appears when r0becomes positive. From the condition r0>0, we obtain that the product ˜ δγ must be negative for the bifurcation to take place, so that from the branch of solutions obtained in Lemma 3, only the solutions verifying ˜ δγ < 0 correspond with periodic orbits of system (27). Assuming this inequality, we can apply Corollary 2 to determine the stability of the bifurcating limit cycle. From (55), the first condition of (19), namely |det(QL)|<1, is fulfilled for µ < 0 and r0sufficiently small. Under such hypotheses, from (54) and (55) we get the expression |trace(QL)−1|−1−det(QL) = −trace(QL)−det(QL) = −2˜ δeπµ + 1 µ2+ 1 r0+O(r2 0), so that the second condition for stability in (19) leads to ˜ δ > 0, and so we need γ < 0. Therefore, the bifurcating limit cycle is stable for µ < 0 and γ < 0. Using now that the bifurcating limit cycle exists for ˜ δγ < 0, we can state the following result, which is a direct consequence of the above reasoning. Proposition 5 Consider system (27) and define the non-degeneracy parameter ˜ δas given in (47). If ˜ δ6= 0, then, for γ= 0 the system undergoes a Hopf bifurcation from infinity, that is, one symmetric limit cycle of large amplitude appears for ˜ δγ < 0and γsufficiently small. Hopf bifurcation at infinity analysis for 3D Relay systems 25 Moreover, the period Tof the periodic oscillation is an analytic function at 0, in the variable γ, and the first terms in its series expansion are T= 2π−2ǫ ˜ δπγ +O(γ2). Taking Y0= 1/r0>0as a measure for the amplitude of the periodic orbit, we have the asymptotic estimate Y0≈ − 2˜ δ π(µ2+ 1)γ. In particular, when µ6= 0, if ˜ δ > 0and µ < 0,then the bifurcating limit cycle for γ < 0is orbitally asymptotically stable. Otherwise, if ˜ δ < 0or µ > 0, then the bifurcating limit cycle is unstable. Remark 8 In the case µ= 0, we have ˜ δ=β3−β1, so that to study the stability of the bifurcating limit cycle, we will use that trace(QL) = 1 + 2β1r0+O(r2 0), and det(QL) = −1 + 2(2β3−β1)r0+O(r2 0). If we assume γ < 0 and ˜ δ=β3−β1>0, then a sufficient condition for the stability of the bifurcating limit cycle is β1>0. Effectively, then we have 0< β1< β3<2β3and the first condition in (19) is fulfilled for r0>0 and sufficiently small. The second condition for stability in (19) now gives |trace(QL)−1|−1−det(QL) = −4β3r0+O(r2 0), which becomes negative for r0>0 and sufficiently small. Proposition 5 and Remark 8 lead to Theorem 1 after applying Proposition 4, that is, taking into account that ωβ1=b1, ω2β2=b2−λb1, ω3β3=b3−λb2+λ2b1, and ω3˜ δ=ω3β3−ω3(µ2+ 1)β1=b3−λb2−ω2b1=δ. 4.2 Proof of Theorem 2 For the proof of Theorem 2, which considers the situation δ= 0 leading to ˜ δ= 0, we can rewrite Lemma 3 under such additional hypothesis, as follows. Note that for ˜ δ= 0 we have β3= (µ2+ 1)β1and ǫ= (µβ1+β2)(µ2+ 1). 32 Emilio Freire et al. manifolds Σ0and Σ1there are known regular parameterizations r0and r1 and two open sets N0⊂R2and N1⊂R2such that r0(u, v)∈Σ0,r1(u, v)∈Σ1, for all (u, v)∈N0and all (u, v)∈N1, respectively. We also assume that there exist (u0, v0)∈N0with p0=r0(u0, v0), and (u1, v1)∈N1with p1= r1(u1, v1). The transversality hypothesis at the arrival point p1amounts that the vector field F(p1) does not belong to the tangent space to Σ1, which is spanned by the two three-dimensional vectors ∂r1 ∂u (u1, v1) and ∂r1 ∂v (u1, v1); in other words, we can assure that the 3 ×3 matrix   F(p1)∂r1 ∂u (u1, v1)∂r1 ∂v (u1, v1)   is regular. The following result allows to define the transition map that applies points near p0(not necessarily in Σ0) to points in Σ1near p1, by using the orbits of system (71) and resorting in a first step to the local parameterization of the arrival manifold. Proposition 6 Assume for system (71) that the orbit γ(τ)with γ(0) = p0∈ Σ0arrives transversally at p1∈Σ1after a time τ0. If for an open set N1 with (u1, v1)∈N1⊂R2, we denote by r1(u, v) : N1→Σ1a regular, local parameterization of Σ1with p1=r1(u1, v1), then there exist an open set U, where p0∈U⊂R3, and two differentiable functions T:U→Rand R:U→N1satisfying T(p0) = τ0,R(p0) = (u1, v1), and such that φ(T(x),x) = r1(R(x)) ∈Σ1,for all x∈U. Proof We start by considering and open interval for the time with τ0∈(τa, τb) such that the function G: (τa, τb)×U×N1→R3 (τ, x, u, v)7→ φ(τ, x)−r1(u, v)(76) is well defined. By definition, G(τ0,p0, u1, v1) = φ(τ0,p0)−r1(u1, v1) = p1−p1= 0. We also note that ∂G ∂τ (τ0,p0, u1, v1) = ∂φ ∂τ (τ0,p0) = F(φ(τ0,p0)) = F(p1), and so the matrix B1=∂G ∂(τ, u, v)(τ0,p0, u1, v1) =   F(p1)−∂r1 ∂u (u1, v1)−∂r1 ∂v (u1, v1)   Hopf bifurcation at infinity analysis for 3D Relay systems 33 is non-singular by the transversality hypothesis. Thus, a standard application of the implicit function theorem near the point (τ0,p0, u1, v1) allows to solve for τand (u, v), and to conclude the existence of smooth functions τ=T(x) and (u, v) = R(x) in some open neighborhood Uof p0, such that G(T(x),x, R(x)) = φ(T(x),x)−r1(R(x)) = 0. The proof is complete. From Proposition 6, we can define the transition map Pthat applies the nearby points of p0into the arrival points at Σ1when one follows the orbits passing through them; it suffices to take into account the functions Tand R above, which give the ’flight’ time spent by each orbit and the local coordinates of the arrival point, respectively. Thus, we can assure that for the points of U in Σ0, the function P:U∩Σ0→Σ1 x7→ φ(T(x),x) = r1(R(x)) (77) is smooth and well defined. We are interested in the translation of this map Pto its version in terms of local coordinates both for the starting and arrival points, namely ψ=r−1 1◦P◦r0=R◦r0. To do that, we must consider the open sets N0and N1and define the function ψ:N0→N1 (u, v)7→ R(r0(u, v)) (78) along with the corresponding function for the flight time τf:N0→R, such that τf=T◦r0. In the following result, we give an expression that allows us to compute under certain hypotheses the derivatives of the two functions ψ and τf, just introduced. Proposition 7 As in Proposition 6, assume for system (71) that the orbit γ(τ)with initial point γ(0) = p0=r0(u0, v0)∈Σ0arrives transversally at p1=r1(u1, v1)∈Σ1after a time τ0. If for i= 0,1, we define the 3×3 matrices Bi=  F(pi)−∂ri ∂u (ui, vi)−∂ri ∂v (ui, vi)  = F(pi)−Dri(ui, vi)  and the matrix Q=1Dτf(u0, v0) 0Dψ(u0, v0) then we have B1·Q=Dxφ(τ0,p0)·B0. 34 Emilio Freire et al. Proof Starting from the equality φ(T(x),x) = r1(R(x)) for all x∈U, see Proposition 6, if we take derivatives with respect to x, then we get ∂φ ∂τ (T(x),x)·DT (x) + Dxφ(T(x),x) = Dr1(R(x)) ·DR(x). Using the flow property (72) and evaluating the above equality in x=p0, where T(p0) = τ0and R(p0) = (u1, v1), we see that F(p1)·DT (p0) + Dxφ(τ0,p0) = Dr1(u1, v1)·DR(p0). Since for the composed functions τand ψwe have Dτf(u0, v0) = DT (p0)· Dr0(u0, v0) and Dψ(u0, v0) = DR(p0)·Dr0(u0, v0), respectively, if we multiply the above equality by Dr0(u0, v0), we get F(p1)·Dτf(u0, v0) + Dxφ(τ0,p0)·Dr0(u0, v0) = Dr1(u1, v1)·Dψ(u0, v0), so that F(p1)·Dτf(u0, v0)−Dr1(u1, v1)·Dψ(u0, v0) = −Dxφ(τ0,p0)·Dr0(u0, v0). Now, it is evident that from B1·Q= F(p1)F(p1)·Dτf(u0, v0)−Dr1(u1, v1)·Dψ(u0, v0)  we can pass, taking into account that F(p1) = Dxφ(τ0,p0)F(p0), to  F(p1)−Dxφ(τ0,p0)·Dr0(u0, v0) =Dxφ(τ0,p0) F(p0)−Dr0(u0, v0) , and since the last matrix is equal to B0, the proposition follows. In the particular case when Σ0=Σ1=Σas given in (3), the parameterizations riare trivial, since we can take (ui, vi) = (yi, zi) for i= 0,1, and the following corollary is immediate. Corollary 3 Assume for system (71) that the orbit γ(τ)with initial point γ(0) = p0= (0, y0, z0)∈Σarrives transversally at p1= (0, y1, z1)∈Σafter a time τ0. If for i= 0,1, we define the 3×3matrices Bi=  0 0 F(pi)−1 0 0−1  and the matrix Q=1Dτf(y0, z0) 0Dψ(y0, z0) then we have B1·Q=Dxφ(τ0,p0)·B0. Hopf bifurcation at infinity analysis for 3D Relay systems 35 Acknowledgements Authors are partially supported by the Ministerio de Econom´ıa y Competitividad, in the frame of project PGC2018-096265-B-I00, and by the Consejer´ıa de Transformaci´on Econ´omica, Industria, Conocimiento y Universidades de la Junta de Andaluc´ıa, under grants PAIDI P20 01160 and FEDER US-1380740. The authors declare that there is no conflict of interest regarding the publication of this article. Conflict of interest The authors declare that they have no conflict of interest. References 1. Andronov, A.A., Vitt, A., Khaikin, S.E.: Theory of oscillators. 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