Bifurcation of relative equilibria generated by a circular vortex path in a circular domain

DISCRETE AND CONTINUOUS doi:10.3934/dcdsb.2019265 DYNAMICAL SYSTEMS SERIES B Volume 25, Number 2, February 2020 pp. 749–760 BIFURCATION OF RELATIVE EQUILIBRIA GENERATED BY A CIRCULAR VORTEX PATH IN A CIRCULAR DOMAIN David Rojas∗ Departament d’Inform`atica, Matem`atica Aplicada i Estad´ıstica Universitat de Girona, 17003 Girona, Spain Pedro J. Torres Departamento de Matem´atica Aplicada Universidad de Granada, 18071 Granada, Spain Abstract. We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number. 1. Introduction. The passive particle transport in a 2D incompressible inviscid flow with prescribed vorticity is a research topic of the highest relevance in Fluid Dynamics [2,8,10]. In the Lagrangian formulation, the advection of single particles is ruled by a Hamiltonian system where the stream function plays the role of the Hamiltonian. In this paper, we consider the dynamics induced in an ideal flow confined in a circular domain of radius Runder the action of a prescribed T-periodic vortex path. Such dynamics model the stirring process of an agitator plunged into a fluid inside a cylindrical tank free surface. The main interest of this model is to investigate the amount of fluid that actually mix and how the path of the vortex affects stirring. We refer to [1,3,4,6] and references therein for more information on the model and its historical overview in the literature. Let BR⊂R2be the open ball of center (0,0) and radius R, and consider a T-periodic vortex path given by z:R→BR. Then, the stream function of the fluid confined in BRand under the action of the vortex is given by Ψ(t, ζ) = Γ 2πln |ζ−z(t)| − ln  ζ−R2 |z(t)|2z(t). Here, Γ is the strength or charge of the vortex, and its sign gives the sense of rotation. In this function, the first term accounts for the vortex action, whereas the 2010 Mathematics Subject Classification. Primary: 25C34, 37N10, 76B47. Key words and phrases. Vortex, passive transport, stirring protocol, stream function, periodic orbit, winding number. All the authors are partially supported by the MINECO/FEDER grant MTM2017-82348-C21-P. The first author is also partially supported by the MINECO/FEDER grant MTM2017-86795C3-1-P. ∗Corresponding author: David Rojas (david.ro[email protected]). 749 750 DAVID ROJAS AND PEDRO J. TORRES second term models the influence of the solid circular boundary. It is useful to see ζas a complex variable, then the corresponding Hamiltonian system is ˙ ζ∗=Γ 2πi   1 ζ−z(t)−1 ζ−R2 |z(t)|2z(t) ,(1) where the asterisk means the complex conjugate. In the related literature, z(t) is called the stirring protocol. When it is constant, then Ψ is a conserved quantity and all the particles rotate around the vortex in circular trajectories. In particular, the model does not mix appropiately. When z(t) is time-dependent, then the Hamiltonian ceases to be a conserved quantity and one must in general expect to observe chaotic particle motion. In this direction, Aref [1] proves the existence of chaotic regions in the case when the protocol z(t) is piece-wise constant (also known as blinking protocol) by showing that the Poincar´e map is semiconjugate to the Bernoulli shift on two symbols. In particular, the blinking protocol produce an efficient mixing. In [3], it is proved that any smooth stirring protocol z(t) induces an infinite number of periodic trajectories rotating around the vortex (non-zero winding number). A natural question is to try to identify the stirring protocols that generate periodic trajectories with zero winding number, that is, particles moving periodically but not rotating around the vortex. This question was posed explicitly as an open problem in [9, Section 7.3]. The identification of periodic solutions of zero winding number is relevant in the interpretation of the model as a stirring problem. Indeed, in the case of the existence of stable solutions, the stability region conforms a confined portion of fluid that will not mix with the fluid around the vortex. Our intention is to advance on the comprehension of this problem by analyzing the family of circular protocols z(t) = r0exp(iθ0t). In this case, the change to a corotating frame ζ(t) = η(t) exp(iθ0t) transforms equation (1) into the autonomous system ˙η∗=iθ0η∗+Γ 2πi 1 η−r0 −1 η−R2 r0!.(2) The Hamiltonian structure is preserved, so the streamlines are just the level curves of the corresponding Hamiltonian, which is indeed a conserved quantity. In [6], the authors study the case when the stirring protocol moves according with the passive motion of the fluid (free vortex). That is, the trajectories of the vortex are circles with radii r0and angular speed θ0=Γ 2π 1 R2−r2 0 . In this case, the phase portrait of the disk consists on a heteroclinic connection between two points on the boundary of BR separating the vortex from a center generated by the passive fluid (see Figure 1b.) The authors evidence regions of chaotic motion via Melnikov method due to the transverse crossing of the stable and unstable manifolds of the heteroclinic orbit once the radius of the stirring protocol is perturbed by a 2π-periodic function. Our first contribution in this direction is to classify all possible phase portraits for all circular stirring protocols. That is, we will consider θ0∈Ras a free parameter. By a complete bifurcation analysis of the relative equilibria, performed in Section 2, setting φ0:= 2πR2θ0 Γand ρ0:= r0 R, we obtain five regions in the parameter space (φ0, ρ0)∈R×(0,1) with phase portraits as shown in Figures 1a to 1e divided by four codimension one curves with phase portraits as shown in Figures 1f to 1h (the curve φ0= 0 correspond to the constant stirring protocol.) In particular, the center inside CIRCULAR VORTEX PATH IN CIRCULAR DOMAIN 751 (a) (b) (c) (d) (e) (f) (g) (h) Figure 1. Phase portrait of system (3) depending on the parameters according to Theorem 2.1. the ball BRis not always present and there are no periodic orbits with zero winding number at least in the integrable scenario. In Section 3we consider the case when the radius of the stirring protocol is perturbed by a T-periodic function. As in [6] the natural scenario is that, in the cases that homoclinic or heteroclinic orbits appear separating the vortex and the center, such stable and unstable manifolds generically intersect transversally giving rise to chaotic regions. Our second contribution is an analytic proof showing that, under small periodic perturbations of the radius of z(t), there are still periodic trajectories surrounding a certain perturbed critical point and giving rise to periodic orbits with zero winding number. Consequently, in the terminology of Fluid Dynamics the model turns to be a not efficient stirring process. 2. Phase portrait and bifurcation analysis. This section is devoted to the bifurcation analysis of the phase portrait of system (2). Working on Cartesian coordinates, the streamlines are level curves of the Hamiltonian function Ψ(x, y) = −θ0 2(x2+y2) + Γ 2πln s(x−r0)2+y2 (x−R2 r0)2+y2. Here (R, Γ, θ0)∈(0,+∞)×(R\ {0})2and r0∈(0, R). From now on, for the sake of further simplicity, we denote a(x):= a(x, r0) = x−r0, b(x):= b(x, R, r0) = x−R2 r0 and c:= Γ 2πθ0 . Thus, system (2) can be written in the (x, y)-variables as        ˙x=∂Ψ ∂y =−θ0y+cθ0y1 a(x)2+y2−1 b(x)2+y2, ˙y=−∂Ψ ∂x =θ0x−cθ0a(x) a(x)2+y2−b(x) b(x)2+y2. (3) 752 DAVID ROJAS AND PEDRO J. TORRES Let DR⊂R2be the closed ball of center (0,0) and radius R. It is an immediate calculation to show that DRis invariant by the flow of system (3). Next result deals with the phase portrait of the system on DR. It will be shown that the position r0 and angular velocity θ0of the vortex path are the main parameters on the system, whereas the remaining ones can be normalized. To this end, and for the sake of simplicity on the statement, we set ρ0:= r0 Rand φ0:= R2 c=2πR2θ0 Γ. Thus, the parameter space of system (3) turns Λ:= {(ρ0, φ0)∈R2: 0 < ρ0<1 and φ06= 0}. Moreover, let us define f(ρ0, φ0):= 27ρ2 0(ρ2 0−1) + φ02−3ρ2 0−3ρ4 0+ 2ρ6 0−2(1 −ρ2 0+ρ4 0)3 2, and B:= (ρ0, φ0)∈Λ : φ0f(ρ0, φ0)ρ0−1−φ0 1 + φ0ρ0−φ0−1 1 + φ0= 0. We claim that there is no ρ0∈(0,1) such that 2−3ρ2 0−3ρ4 0+ 2ρ6 0−2(1 −ρ2 0+ρ4 0)3 2= 0. Indeed, the previous equality is equivalent to −27ρ4 0(ρ0−1)2(ρ0+ 1)2= 0 by means of elementary algebraic manipulations. Therefore, the curve Bis the union of three curves, namely C1:= {(ρ0, φ0)∈Λ : ρ0=1−φ0 1+φ0}, C2:= {(ρ0, φ0)∈Λ : ρ0=φ0−1 1+φ0}, C3:= {(ρ0, φ0)∈Λ : f(ρ0, φ0)=0}, and splits the parameter space Λ into five connected components, namely Ri,i= 1,...,5, according with Figure 2. Theorem 2.1. Let (ρ0, φ0)∈Λ. The set Λ\ B corresponds to regular parameters of system (3). On each connected component, the phase portrait is the following: (a)If (ρ0, φ0)∈ R1then the dynamics on DRis a global vortex at (r0,0) (see Figure 1a). (b)If (ρ0, φ0)∈ R2then the system has a vortex at (r0,0), a center at (x∗ c,0) with x∗ c∈(−R, 0) and two hyperbolic saddles (x∗ s,±y∗ s)at ∂DRwith a saddle connection inside DR(see Figure 1b). (c)If (ρ0, φ0)∈ R3then the system has a vortex at (r0,0), a center at (x∗ c,0) with x∗ c∈(−R, 0) and a hyperbolic saddle at (x∗ s,0) with x∗ s∈(r0, R)(see Figure 1c). (d)If (ρ0, φ0)∈ R4then the system has a vortex at (r0,0), a center (x∗ c,0) and a hyperbolic saddle (x∗ s,0) satisfying 0< x∗ c< x∗ s< r0(see Figure 1d). (e)If (ρ0, φ0)∈ R5then the dynamics on DRis a global vortex at (r0,0) (see Figure 1e). Moreover, the set Bcorresponds to bifurcation parameters of system (3). On each curve the phase portrait is the following: (f)If (ρ0, φ0)∈C1then the system has a vortex at (r0,0) and a degenerated saddle at (−R, 0) (see Figure 1f). (g)If (ρ0, φ0)∈C2then the system has a vortex at (r0,0), a center at (x∗ c,0) with x∗ c∈(−R, 0) and a degenerated saddle at (R, 0) (see Figure 1g). CIRCULAR VORTEX PATH IN CIRCULAR DOMAIN 753 Figure 2. Bifurcation diagram of the phase-portrait of system (3) on DR. The bold curve corresponds to bifurcation parameters B, whereas the remaining ones correspond to regular parameters. In Theorem 2.1 the phase portrait at each region is given. (h)If (ρ0, φ0)∈C3then the system has a vortex at (r0,0) and a cusp at (x∗ p,0) (see Figure 1h), where x∗ p:= x∗ p(R, r0) = R2+r2 0−pR4−R2r2 0+r4 0 3r0 . Proof. For the sake of simplicity we first begin the proof by showing that the only critical points in DRthat do not lie on the line {y= 0}are the hyperbolic saddles (x∗ s,±y∗ s) at ∂DRof case (b) on the statement. To this end, assuming y6= 0, from equations in (3) we have that ˙x= 0 if and only if −1 + c1 a(x)2+y2−1 b(x)2+y2= 0. Since a(x)2< b(x)2for all x < R, if φ0<0 then c < 0 and so the left-hand side of the previous equality is negative. Then assume φ0>0. In this case, ˙x= 0 if and only if y2=−1 2(a(x)2+b(x)2) + 1 2p(b(x)2−a(x)2)(4c+b(x)2−a(x)2). Substituting the previous equality on the expression of ˙yin (3) and equaling to zero one gets the equation (2x−a(x)−b(x))(a(x) + b(x)) + p(b(x)2−a(x)2)(4c+b(x)2−a(x)2) 2(a(x) + b(x)) = 0. Thus, using that a(x) = x−r0and b(x) = x−R2 r0, the previous equation has the unique solution x∗ s=R2+r2 0 2r0 −c 2r01−r2 0 R2 and so (y∗ s)2=1 42(c2+R4) R2−(c−R2)2 r2 0 −(c+R2)2r2 0 R4. 754 DAVID ROJAS AND PEDRO J. TORRES It is a computation to show that x∗ s∈(−R, R) if and only if ρ0>max{1−φ0 1+φ0,φ0−1 1+φ0} (that is, (ρ0, φ0)∈ R2) and (x∗ s)2+ (y∗ s)2=R2. It is only remaining to prove that (x∗ s,±y∗ s) are hyperbolic saddles. This can be done evaluating the previous expression of the points (x∗ s,±y∗ s) on the Jacobian matrix of system (3). In the case of (x∗ s, y∗ s) the determinant of the Jacobian matrix is det(DX(x∗ s, y∗ s)) = (cR −R3−(c+R2)r0)(cR −R3+ (c+R2)r0)θ2 0 c2(R2−r2 0) which is negative if and only if ρ0>max{1−φ0 1+φ0,φ0−1 1+φ0}. Then, (x∗ s, y∗ s) is a hyperbolic saddle. The same argument is valid for (x∗ s,−y∗ s). Moreover, since ∂DRis an invariant curve of system (3) and (x∗ s, y∗ s)∈∂DR, then ∂DRis the stable manifold of one saddle (and unstable of the other). The corresponding unstable (stable) manifold cuts transversally the disk of radius Rdue to the hyperbolicity of the saddles and so the connection between the saddles follows by Poincar´e-Bendixon’s theorem. The previous argument shows that out of case (b) on the statement, all the critical points of system (3) lie on {y= 0}. Let us prove now the remaining cases of the result. Let us first consider φ0>0. This corresponds to the statements (a)−(c) and (f)−(g). We can assume with no loss of generality that θ0>0 and Γ >0. The case with θ0<0 and Γ <0 follows by reversion of time. Notice that the hypothesis φ0>0 implies c > 0. System (3) has a critical point at (x∗,0) inside the disk of radius Rif and only if the function F(x):= θ0x−c1 a(x)−1 b(x) satisfies F(x∗) = 0 for some x∗∈(−R, R). Multiplying by a(x)b(x) the previous condition turns into F(x∗)a(x∗)b(x∗) = 0. We point out that, on account of the expressions of a(x) and b(x), the previous two conditions are equivalent if x∗/∈ {r0,R2 r0}(those correspond to singularities on the Hamiltonian function and so no critical points). Thus, system (3) has a critical point at (x∗,0) in DRif and only if x∗a(x∗)b(x∗) = cr0−R2 r0=:λ=λ(r0, R, c).(4) Notice that, since φ0>0 then λ < 0. The cubic polynomial P(x):= xa(x)b(x) has zeros at x= 0, x=r0and x=R2 r0.P(x) is negative if x∈(−∞,0) ∪(r0, R2/r0) and it is positive if x∈(0, r0)∪(R2/r0,+∞), and the local maximum and minimum are, respectively, xM=R2+r2 0−pR4−R2r2 0+r4 0 3r0 and xm=R2+r2 0+pR4−R2r2 0+r4 0 3r0 . Moreover, since φ0>0 then λ=λ(r0, R, c) varies from zero to −∞. Thus P(x)− λ= 0 has always a solution x∗∈(−∞,0), it has a double solution x∗=xmif P(xm) = λand two solutions in (r0, R2/r0) if P(xm)< λ. Let us study when these solutions correspond to critical points in DR. To this end, we define Q(u):= CIRCULAR VORTEX PATH IN CIRCULAR DOMAIN 755 1 R3(P(Ru)−λ). Elementary algebraic manipulations show that 0<xM R<1 + ρ2 0−p(1 −ρ2 0)2+ρ2 0 3ρ0 < ρ0<1<xm R, Q(1) = (1 −ρ0)(φ0+ 1) φ0ρ0ρ0−φ0−1 1 + φ0, Q(−1) = −(1 + ρ0)(φ0+ 1) φ0ρ0ρ0−1−φ0 1 + φ0, Q(0) = Q(ρ0) = 1−ρ2 0 φ0ρ0 >0. We point out that since xm> R then at most two zero of P(x)−λlie in (−R, R). By the second equality above we have that P(R)−λ > 0 if ρ0>φ0−1 1+φ0, that P(R)−λ= 0 if ρ0=φ0−1 1+φ0and P(R)−λ < 0 if ρ0<φ0−1 1+φ0. By the third equality above we have that P(−R)−λ > 0 if ρ0<1−φ0 1+φ0, that P(−R)−λ= 0 if ρ0=1−φ0 1+φ0 and that P(−R)−λ < 0 if ρ0>1−φ0 1+φ0. Thus, if (ρ0, φ0)∈ R1no roots of P(x)−λ are inside [−R, R] and so the result in (a) holds. If (ρ0, φ0)∈C1the unique zero of P(x)−λin [−R, R] is x=−R. This correspond to a critical point of system (3) at (−R, 0). Moreover, it is a degenerated saddle since ∂DRis an invariant curve of system (3) so (f) is proved. If (ρ0, φ0)∈ R2then P(x)−λhas only one zero x∗ C∈(−R, 0). Then, on account of the previous discussion about the hyperbolic saddles (x∗ s,±ys∗) on ∂DR, result in (b) is proved. If (ρ0, φ0)∈C2then P(x)−λ has two zeros: x=x∗ c∈(−R, 0) and x=R. The critical point (R, 0) corresponds to a degenerated saddle since ∂DRis an invariant curve of system (3). Then (g) holds. Finally, if (ρ0, φ0)∈ R3then P(x)−λhas two zeros: x=x∗ c∈(−R, 0) and x=x∗ s∈(r0, R). In order to end with the case φ0>0 it only remains to prove that x∗ cand x∗ sare a center and a hyperbolic saddle, respectively. The Jacobian matrix associated to system (3) with y= 0 is given by DX(x, 0) =     0θ0−1 + c1 a(x)2−1 b(x)2 θ01 + c1 a(x)2−1 b(x)2 0     . (5) Notice that θ01 + c1 a(x)2−1 b(x)2>0 for all x∈(−R, R). Furthermore, setting x=x∗a critical point of (3), we have c1 a(x∗)2−1 b(x∗)2−1 = x∗1 a(x∗)+1 b(x∗)−1 =(x∗)2−R2 (x∗−r0)(x∗−R2 r0), where we used F(x∗) = 0 on the first equality and the expressions of a(x) and b(x) on the second. Thus DX(x∗,0) =    0θ0(x∗)2−R2 (x∗−r0)(x∗−R2 r0) θ02 + (x∗)2−R2 (x∗−r0)(x∗−R2 r0)0  .(6) 756 DAVID ROJAS AND PEDRO J. TORRES Consequently, taking x∗=x∗ c∈(−R, 0) we have (x∗ c)2−R2<0 and (x∗ c−r0)(x∗ c− R2/r0)>0. Therefore (x∗ c)2−R2 (x∗ c−r0)(x∗ c−R2/r0)<0 and so det(DX(x∗ c,0)) >0. This implies that (x∗ c,0) is a center. On the other hand, taking x∗=x∗ s∈(r0, R), (x∗ s)2−R2 (x∗ s−r0)(x∗ s−R2/r0)>0 and so det(DX(x∗ s,0)) <0. This implies that (x∗ s,0) is a hyperbolic saddle. This ends with the proof of statements (a), (b), (c), (f) and (g). Let us now consider the case φ0<0. This corresponds to statements (d), (e) and (h). In this situation we can assume with no loss of generality that θ0>0 and Γ<0. The opposite case follows by reversion of time. Notice that the hypothesis φ0<0 implies c < 0. Consequently, on account of the equation (3) critical points can only belong to {(x, y)∈R2:y= 0}. Similarly as before, system (3) has a critical point at (x∗,0) inside the disk of radius Rif and only if (4) is satisfied. Notice that, since φ0<0, in this case λ=λ(r0, R, c) varies from zero to +∞. Thus, on account of R < R2/r0, if λstays above of the maxima of P(x) inside (0, r0) then P(x)−λ= 0 has a unique zero which is larger than R. This happens when f(ρ0, φ0)<0. If f(ρ0, φ0) = 0 then the maximum of P(x) inside (0, r0) contact λand gives the cusp (x∗ p,0) with x∗ p=xM. Finally, if f(ρ0, φ0)>0 then the maximum of P(x) is greater than λand so P(x)−λhas two real roots inside (0, r0): namely x∗ cand x∗ s, satisfying 0 < x∗ c< xM< x∗ s< r0. It only remains to prove the stability of such critical points. This follows from the expression in (5) of the Jacobian matrix associated to system (3) with y= 0. We point out that, since a(x)2< b(x)2for all x∈(0, r0) and c < 0, we have θ0−1 + c1 a(x)2−1 b(x)2<0. Moreover, setting x=x∗a critical point of system (3), on account of F(x∗) = 0 we have 1 + c1 a(x∗)2−1 b(x∗)2= 2 + (x∗)2−R2 (x∗−r0)(x∗−R2 r0) =3r0(x∗)2−2(R2+r2 0)x∗+R2r0 (r0−x∗)(R2−r0x∗). The previous expression is positive if x∗∈(0, xM) and it is negative if x∗∈(xM, r0). This proves that x∗ cis a center and x∗ sis a hyperbolic saddle and ends with the proof of (d), (e) and (h). 3. Periodic perturbations and local continuation of periodic orbits. Given an autonomous planar Hamiltonian system ˙η=J∇H(η),(7) it is interesting to look for the existence of periodic solutions of the non-autonomous planar Hamiltonian system ˙η=J∇H(t, η;),(8) which are small T-periodic perturbations of (7) meaning that H(t, η; 0) ≡ H(η). A. Fonda, M. Sabatini and F. Zanolin prove in [5] that under the hypothesis of the existence of a non-isochronous period annulus for the autonomous Hamiltonian system and some regularity conditions on H(t, η;) such periodic orbits exist. More precisely, consider H:A → Rtwice continuously differentiable and A ⊆ R2 a period annulus such that the inner and outer components of its boundary are Jordan curves. Assume that Ais not isochronous, that is the period of the periodic CIRCULAR VORTEX PATH IN CIRCULAR DOMAIN 757 orbits in Acovers an interval [Tmin,Tmax], with Tmin <Tmax. Then consider H: R×A×(0, 0)→R, whose gradient with respect to the second variable, denoted by ∇H(t, η;), is continuous in (t, η;), locally Lipschitz continuous in ηand Tperiodic in tfor some T > 0. Under these assumptions, the authors in [5] prove the following result: Theorem 3.1 (Fonda, Sabatini and Zanolin).Given two positive integers mand nsatisfying Tmin <mT n<Tmax,(9) there is an ¯ > 0such that, if || ≤ ¯, then system (8)has at least two mT-periodic solutions, whose orbits are contained in A, which make exactly nrotations around the origin in the period time mT. The authors also emphasize the following immediate consequence: Corollary 1. For any positive integer Nthere is a ¯N>0such that, if ||<¯N, then system (8)has at least Nperiodic solutions, whose orbits are contained in A. Our purpose in this section is to illustrate this situation in the case when system (3) has a non-degenerated center inside DR. This occurs for parameters (ρ0, φ0)∈ R2∪ R3∪ R4, corresponding to the phase portraits (b), (c) and (d) in Figure 1 and Theorem 2.1. Let us denote by Pthe period annulus of the center. The inner boundary of Pis the center itself, namely p, whereas the outer boundary of Pis formed by saddle connections. In both cases the outer boundary has critical points so it is clear that the period function tends to infinity as the orbits approach the outer boundary. Particularly, the center is not isochronous. Next result states the period of the linearized center. Lemma 3.2. Let (ρ0, φ0)∈ R2∪ R3∪ R4and let p= (x, 0) be the non-degenerated center of system (3). Then the period of the associated linearized system at pis T0(x) = 2π qθ2 0ν(x R)(2 + ν(x R)) where ν(x):= x2−1 (x−ρ0)(x−1 ρ0). Proof. From the expression of the Jacobian matrix of the system in (6) we have that the eigenvalues associated to the center are λ±=±ωi =±v u u tθ2 0 x2−R2 (x−r0)(x−R2 r0) 2 + (x2−R2) (x−r0)(x−R2 r0)!i where ωdenotes the frequency of the linearized center. Thus, setting ρ0=r0 R, and using that the period of the linearized center is T0=2π ωthe result holds. Let us now consider the periodically perturbed stirring protocol z(t) = r(t) exp(iθ0t) on system (1) where r(t) is a smooth T-periodic perturbation of r0. More concretely, r(t) = r0+f(t) + g(t;) with fand g(·;)T-periodic analytic functions and g(t;) tending to zero uniformly on t∈Ras tends to zero. The same change