Bifurcation of relative equilibria generated by a circular vortex path in a circular domain
David Rojas Pedro J. Torres


Impurity scattering and size quantization effects in a single graphene nanoflake
    Mikhail Fonin^1[Email address: mikhail.fonin@uni-konstanz.de]
    December 30, 2023
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 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.




§ INTRODUCTION


The passive particle transport in a 2D incompressible flow with prescribed vorticity is a research topic of the highest relevance in Fluid Dynamics <cit.>. 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 R under the action of a prescribed T-periodic vortex path. This problem is classical in the literature (see for instance <cit.>) and can be seen as a 2D idealization of the mixing of a fluid in a cylindrical tank.

Let B_R⊂^2 be the open ball of center (0,0) and radius R, and consider a T-periodic vortex path given by z:→ B_R. Then, the stream function of the fluid confined in B_R and under the action of the vortex is given by 

    Ψ(t,ζ)=Γ/2π(lnζ-z(t)-lnζ-R^2/|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 second term models the influence of the solid circular boundary. It is useful to see ζ  as a complex variable, then the corresponding Hamiltonian system is 

    ζ̇^̇*̇ = Γ/2π i(1/ζ-z(t) - 1/ζ - R^2|
    z(t) |^2z(t)),

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. When z(t) is time-dependent, then the Hamiltonian ceases to be a conserved quantity and the analysis is more delicate. In <cit.>, 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). Hence, it is a natural question 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 <cit.>. Our intention is to advance on the comprehension of this difficult problem by analyzing the family of circular protocols z(t)=r_0exp(iθ_0 t). In this case, the change to a corotating frame ζ(t)=η(t)exp(iθ_0 t) transforms (<ref>) into the autonomous system

    η̇^* = iθ_0η^*+Γ/2π i(1/η-r_0 - 1/η -
    R^2r_0).

The hamiltonian structure is preserved, so the streamlines are just the level
curves of corresponding Hamiltonian, which is indeed a conserved quantity. This fact enables a complete bifurcation analysis of the relative equilibria, performed in Section <ref>, which correspond to periodic solutions of zero winding number for the original system. Moreover, in Section <ref> it is shown that the bifurcation scheme identified in Section <ref> is robust under small perturbations of the vortex path. Finally, the last section exposes some conclusions of the presented study.





§ PHASE PORTRAIT AND BIFURCATIONS
  

This section is devoted to the bifurcation analysis of the phase portrait of system (<ref>). Working on cartesian coordinates, the streamlines are level curves of the Hamiltonian function 

    Ψ(x,y)=-θ_0/2(x^2+y^2)+Γ/2πln√((x-r_0)^2+y^2/(x-R^2/r_0)^2+y^2).

Here (R,Γ,θ_0)∈(0,+∞)× (∖{0})^2 and r_0∈(0,R). From now on, for the sake of further simplicity, we denote 

    a(x):=a(x,r_0)=x-r_0,  b(x):=b(x,R,r_0)=x-R^2/r_0 and c:=Γ/2πθ_0.
 
Thus, system (<ref>) can be written in the (x,y)-variables as

    ẋ = ∂Ψ∂ y
    =-θ_0y+cθ_0y( 1a(x)^2+y^2 - 1b(x)^2+y^2),
    ẏ = -∂Ψ∂ x
    =θ_0x-cθ_0( a(x)a(x)^2+y^2 - b(x)b(x)^2+y^2).


Let ⊂^2 be the closed ball of center (0,0) and radius R. It is an immediate calculation to show that  is invariant by the flow of system (<ref>). Next result deals with the phase portrait of the system on . It will be shown that the position r_0 and angular velocity θ_0 of 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 ρ_0:=r_0/R and ϕ_0:=R^2/c=2π R^2θ_0/Γ. Thus, the parameter space of system (<ref>) turns Λ:={(ρ_0,ϕ_0)∈^2: 0<ρ_0<1  and ϕ_0≠ 0}. Moreover, let us define

    f(ρ_0,ϕ_0):=27ρ_0^2(ρ_0^2-1)+ϕ_0(2-3ρ_0^2-3ρ_0^4+2ρ_0^6-2(1-ρ_0^2+ρ_0^4)^3/2),

and

    ℬ:={(ρ_0,ϕ_0)∈Λ : ϕ_0f(ρ_0,ϕ_0)(ρ_0-1-ϕ_0/1+ϕ_0)(ρ_0-ϕ_0-1/1+ϕ_0)=0}.

The curve ℬ is the union of three curves, namely C_i, i=1,…,3 and splits the parameter space Λ into five connected components, ℛ_i, i=1,…,5, according with Figure <ref>.   


Let (ρ_0,ϕ_0)∈Λ. The set Λ∖ℬ corresponds to regular parameters of system (<ref>). On each connected component, the phase portrait is the following:


  * If (ρ_0,ϕ_0)∈ℛ_1 then the dynamics on  is a global vortex at (r_0,0) (see Figure <ref>).


  * If (ρ_0,ϕ_0)∈ℛ_2 then the system has a vortex at (r_0,0), a center at (x_c^*,0) with x_c^*∈(-R,0) and two hyperbolic saddles (x_s^*,± y_s^*) at ∂ with a saddle connection inside  (see Figure <ref>).


  * If (ρ_0,ϕ_0)∈ℛ_3 then the system has a vortex at (r_0,0), a center at (x_c^*,0) with x_c^*∈(-R,0) and a hyperbolic saddle at (x_s^*,0) with x_s^*∈(r_0,R) (see Figure <ref>).


  * If (ρ_0,ϕ_0)∈ℛ_4 then the system has a vortex at (r_0,0), a center (x_c^*,0) and a hyperbolic saddle (x_s^*,0) satisfying 0<x_c^*<x_s^*<r_0 (see Figure <ref>).


  * If (ρ_0,ϕ_0)∈ℛ_5 then the dynamics on  is a global vortex at (r_0,0) (see Figure <ref>).

Moreover, the set ℬ corresponds to bifurcation parameters of system (<ref>). On each curve the phase portrait is the following:


  * If (ρ_0,ϕ_0)∈ C_1 then the system has a vortex at (r_0,0) and a degenerated saddle at (-R,0) (see Figure <ref>). 


  * If (ρ_0,ϕ_0)∈ C_2 then the system has a vortex at (r_0,0), a center at (x_c^*,0) with x_c^*∈(-R,0) and a degenerated saddle at (R,0) (see Figure <ref>).


  * If (ρ_0,ϕ_0)∈ C_3 then the system has a vortex at (r_0,0) and a cusp at (x_p^*,0) (see Figure <ref>), where 

    x_p^*:=x_p^*(R,r_0)=R^2+r_0^2-√(R^4-R^2r_0^2+r_0^4)/3r_0.






The bifurcation diagram of the phase portrait of system (<ref>) on  with respect to the variables (ρ_0,ϕ_0) is depicted in Figure <ref>. There are eight possibilities (see Figure <ref>) which always presents a vortex at (r_0,0) and:


  * If ϕ_0>0 and 0<ρ_0<1-ϕ_0/1+ϕ_0 then it has no critical points in . Then the dynamics in  is a global vortex.

  * If ϕ_0>0 and ρ_0=1-ϕ_0/1+ϕ_0 then it has exactly one critical point in : a degenerated saddle at (-R,0).

  * If ϕ_0>0 and ρ_0>max{1-ϕ_0/1+ϕ_0,ϕ_0-1/1+ϕ_0} then it has exactly three critical points in : a center (x_C^*,0) with x_C^*∈(-R,0) and two hyperbolic saddles (x_SB^*,± y_SB^*) in ∂ with a saddle connection inside .

  * If ϕ_0>0 and ρ_0=ϕ_0-1/1+ϕ_0 then it has exactly two critical points in : a center (x_C^*,0) with x_C^*∈(-R,0) and a degenerated saddle at (R,0).

  * If ϕ_0>0 and ρ_0<ϕ_0-1/1+ϕ_0 then it has exactly two critical points in : a center (x_C^*,0) with x_C^*∈(-R,0) and a hyperbolic saddle at (x_S^*,0) with x_S^*∈(r_0,R).

  * If ϕ_0<0 and η(ρ_0,ϕ_0)>0 then it has no critical points inside .

  * If ϕ_0<0 and η(ρ_0,ϕ_0)=0 then it has exactly one critical point inside : a cusp at (x_SN^*,0), where  x_SN^*=x_SN^*(R,r_0)=R^2+r_0^2-√(R^4-R^2r_0^2+r_0^4)/3r_0.

  * If ϕ_0<0 and η(ρ_0,ϕ_0)<0 then it has exactly two critical points inside : a center (x_C^*,0) and an hyperbolic saddle (x_S^*,0) satisfying 0<x_C^*<x_S^*<r_0.







For the sake of simplicity we first begin the proof by showing that the only critical points in  that do not lie on the line {y=0} are the hyperbolic saddles (x_s^*,± y_s^*) at ∂ of case (b) on the statement. To this end, assuming y≠ 0, from equations in (<ref>) we have that ẋ=0 if and only if

    -1+c(1/a(x)^2+y^2-1/b(x)^2+y^2)=0.

Since a(x)^2<b(x)^2 for all x<R, if ϕ_0<0 then c<0 and so the left hand side of the previous equality is negative. Then assume ϕ_0>0. In this case, ẋ=0 if and only if

    y^2=-1/2(a(x)^2+b(x)^2)+1/2√((b(x)^2-a(x)^2)(4c+b(x)^2-a(x)^2)).

Substituting the previous equality on the expression of ẏ in (<ref>) and equaling zero one gets the equation

    (2x-a(x)-b(x))(a(x)+b(x))+√((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-r_0 and b(x)=x-R^2/r_0, the previous equation has the unique solution 

    x^*_s=R^2+r_0^2/2r_0-c/2r_0(1-r_0^2/R^2)

and so

    (y^*_s)^2=1/4(2(c^2+R^4)/R^2-(c-R^2)^2/r_0^2-(c+R^2)^2r_0^2/R^4).

It is a computation to show that x_s^*∈(-R,R) if and only if ρ_0>max{1-ϕ_0/1+ϕ_0,ϕ_0-1/1+ϕ_0} (and so if and only if (ρ_0,ϕ_0)∈ℛ_2) and (x_s^*)^2+(y_s^*)^2=R^2. 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 (<ref>). In the case of (x_s^*,y_s^*) the determinant of the Jacobian matrix is

    det(DX(x_s^*,y_s^*))=(cR-R^3-(c+R^2)r_0)(cR-R^3+(c+R^2)r_0)θ_0^2/c^2(R^2-r_0^2)

which is negative if and only if ρ_0>max{1-ϕ_0/1+ϕ_0,ϕ_0-1/1+ϕ_0}. Then, (x_s^*,y_s^*) is a hyperbolic saddle. The same argument is valid for (x_s^*,-y_s^*). Moreover, since ∂ is an invariant curve of system (<ref>) and (x_s^*,y_s^*)∈∂, ∂ is the stable manifold of one saddle (and unstable of the other). The correspondent unstable (stable) manifold cuts transversally the disk of radius R due to the hyperbolicity of the saddles and so the connection between the saddles follows by Poincaré-Bendixon's theorem. 

The previous argument shows that out of case (b) on the statement, all the critical points of system (<ref>) lie on {y=0}. Let us prove now the remaining cases of the result. Let us first consider ϕ_0>0. This correspond to statements (a)-(c) and (f)-(g).
We can assume with no loss of generality that θ_0>0 and Γ>0. The case with θ_0<0 and Γ<0 follows by reversion of time. Notice that the hypothesis ϕ_0>0 implies c>0. System (<ref>) has critical point at (x^*,0) inside the disk of radius R if and only if the function

    F(x):=θ_0(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^*∉{r_0,R^2/r_0} (those correspond to singularities on the Hamiltonian function and so no critical points). Thus, system (<ref>) has a critical point at (x^*,0) in  if and only if

    x^*a(x^*)b(x^*)=c(r_0-R^2/r_0)=:λ=λ(r_0,R,c).

Notice that, since ϕ_0>0, λ<0. The cubic polynomial P(x):=xa(x)b(x) has zeros at x=0, x=r_0 and x=R^2/r_0. P(x) is negative if x∈(-∞,0)∪(r_0,R^2/r_0) and it is positive if x∈(0,r_0)∪(R^2/r_0,+∞), and the local maximum and minimum are, respectively,

    x_M=R^2+r_0^2-√(R^4-R^2r_0^2+r_0^4)/3r_0,   x_m=R^2+r_0^2+√(R^4-R^2r_0^2+r_0^4)/3r_0.

On the other hand, since ϕ_0>0 then λ=λ(r_0,R,c) varies from zero to -∞. Thus P(x)-λ=0 has always a solution x^*∈(-∞,0), it has a double solution x^*=x_m if P(x_m)=λ and two solutions in (r_0,R^2/r_0) if P(x_m)<λ. Let us study when this solutions correspond to critical points in . We point out that x_m>R so at most two zero of P(x)-λ lie in (-R,R). It is a computation to show that P(R)-λ>0 if ρ_0>ϕ_0-1/1+ϕ_0, P(R)-λ=0 if ρ_0=ϕ_0-1/1+ϕ_0 and P(R)-λ<0 if ρ_0<ϕ_0-1/1+ϕ_0. On the other hand, P(-R)-λ>0 if ρ_0<1-ϕ_0/1+ϕ_0, P(-R)-λ=0 if ρ_0=1-ϕ_0/1+ϕ_0 and P(-R)-λ<0 if ρ_0>1-ϕ_0/1+ϕ_0.
Thus, if (ρ_0,ϕ_0)∈ℛ_1, no roots of P(x)-λ are inside [-R,R] and so the result in (a) holds. If (ρ_0,ϕ_0)∈ C_1, the unique zero of P(x)-λ in [-R,R] is x=-R. This correspond to a critical point of system (<ref>) at (-R,0). Moreover, it is a degenerated saddle since ∂ is an invariant curve of system (<ref>) so (f) is proved. If (ρ_0,ϕ_0)∈ℛ_2, P(x)-λ has only one zero x_C^*∈(-R,0). Then, on account of the previous discussion about the hyperbolic saddles (x_s^*,± y_s*) on ∂, result in (b) is proved.
If (ρ_0,ϕ_0)∈ C_2, P(x)-λ has two zeros: x=x_c^*∈(-R,0) and x=R. The critical point (R,0) correspond to a degenerated saddle since ∂ is an invariant curve of system (<ref>). Then (g) holds.
Finally, if (ρ_0,ϕ_0)∈ℛ_3, P(x)-λ has two zeros: x=x_c^*∈(-R,0) and x=x_s^*∈(r_0,R). In order to end with the case ϕ_0>0 it only remains to prove that x_c^* and x_s^* are a center and a hyperbolic saddle, respectively.

The Jacobian matrix associated to system (<ref>) with y=0 is given by

    DX(x,0)=(
    
    0    θ_0(-1+c( 1a(x)^2-1b(x)^2))
    θ_0(1+c( 1a(x)^2-1b(x)^2))     0
    ).

Notice that θ_0(1+c( 1a(x)^2-1b(x)^2))>0 for all x∈(-R,R). On the other hand, setting x=x^* a critical point of (<ref>), we have

    c(1/a(x^*)^2-1/b(x^*)^2)-1
       =x^*(1/a(x^*)+1/b(x^*))-1
       =(x^*)^2-R^2/(x^*-r_0)(x^*-R^2/r_0),

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)=(
    
    0    θ_0(x^*)^2-R^2/(x^*-r_0)(x^*-R^2/r_0)
    θ_0(2+(x^*)^2-R^2/(x^*-r_0)(x^*-R^2/r_0))     0
    ).

Consequently, taking x^*=x_c^*∈(-R,0) we have (x_c^*)^2-R^2<0 and (x_c^*-r_0)(x_c^*-R^2/r_0)>0. Therefore (x_c^*)^2-R^2/(x_c^*-r_0)(x_c^*-R^2/r_0)<0 and so (DX(x_c^*,0))>0. This implies that (x_c^*,0) is a center. On the other hand, taking x^*=x_s^*∈(r_0,R), (x_s^*)^2-R^2/(x_s^*-r_0)(x_s^*-R^2/r_0)>0 and so (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 ϕ_0<0. This corresponds to statements (d), (e) and (h). In this situation we can assume with no loss of generality that θ_0>0 and Γ<0. The opposite case follows by reversion of time. Notice that the hypothesis ϕ_0<0 implies c<0. Consequently, on account of the equation (<ref>) critical points can only belong to {(x,y)∈^2:y=0}. Similarly as before, system (<ref>) has a critical point at (x^*,0) inside the disk of radius R if and only if (<ref>) is satisfied. Notice that, since ϕ_0<0, in this case λ=λ(r_0,R,c) varies from zero to +∞. Thus, on account of R<R^2/r0, if λ stays above of the maxima of P(x) inside (0,r_0) then P(x)-λ=0 has a unique zero which is larger than R. This happens when f(ρ_0,ϕ_0)<0. If f(ρ_0,ϕ_0)=0 then the maximum of P(x) inside (0,r_0) contact λ and gives the cusp (x_p^*,0) with x_p^*=x_M. Finally, if f(ρ_0,ϕ_0)>0 then the maximum of P(x) is greater than λ and so P(x)-λ has two real roots inside (0,r_0): namely x_c^* and x_s^*, satisfying 0<x_c^*<x_M<x_s^*<r_0. It only remains to prove the stability of such critical points. This follows from the expression in (<ref>) of the Jacobian matrix associated to system (<ref>) with y=0. We point out that, since a(x)^2<b(x)^2 for all x∈(0,r_0) and c<0, we have 

    θ_0(-1+c( 1a(x)^2-1b(x)^2))<0.

On the other hand, setting x=x^* a critical point of system (<ref>), on account of F(x^*)=0 we have

    1+c(1/a(x^*)^2-1/b(x^*)^2)=2+(x^*)^2-R^2/(x^*-r_0)(x^*-R^2/r_0)=3r_0(x^*)^2-2(R^2+r_0^2)x^*+R^2r_0/(r_0-x^*)(R^2-r_0x^*).

The previous expression is positive if x^*∈(0,x_M) and it is negative if x^*∈(x_M,r_0). This proves that x_c^* is a center and x_s^* is an hyperbolic saddle and ends with the proof of (d), (e) and (h).







§ PERIODIC PERTURBATIONS AND LOCAL CONTINUATION OF PERIODIC ORBITS


Given an autonomous planar Hamiltonian system 

    η̇= J∇ℋ(η),

it is interesting to ask about the existence of periodic solutions of the non-autonomous planar Hamiltonian system

    η̇= J∇ H(t,η;ϵ),

which are small T-periodic perturbations of (<ref>) meaning that H(t,η;0)≡ℋ(η). A. Fonda, M. Sabatini and F. Zanolin prove in <cit.> 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;ϵ) such periodic orbits exist.

More precisely, consider ℋ:𝒜→ twice continuously differentiable and 𝒜⊆^2 a period annulus such that the inner and outer components of its boundary are Jordan curves. Assume that 𝒜 is not isochronous, that is the period of the periodic orbits in 𝒜 covers an interval [𝒯_min,𝒯_max], with 𝒯_min<𝒯_max. Then consider H:×𝒜×(0,ϵ_0)→, whose gradient with respect to the second variable, denoted by ∇ H(t,η;ϵ), is continuous in (t,η;ϵ), locally Lipschitz continuous in η and T-periodic in t for some T>0. Under these assumptions, the authors in <cit.> prove the following result:


Given two positive integers m and n satisfying

    𝒯_min<mT/n<𝒯_max,

there is an ϵ̅>0 such that, if ϵ≤ϵ̅, then system (<ref>) has at least two mT-periodic solutions, whose orbits are contained in 𝒜, which make exactly n rotations around the origin in the period time mT.


The authors also emphasize the following immediate consequence:


For any positive integer N there is a ϵ̅_N>0 such that, if ϵ<ϵ̅_N, then system (<ref>) has at least N periodic solutions, whose orbits are contained in 𝒜.


Our propose in this section is to illustrate this situation in the case when system (<ref>) has a non-degenerated center inside . This occurs for parameters (ρ_0,ϕ_0)∈ℛ_2∪ℛ_3∪ℛ_4, corresponding to the phase portraits (b), (c) and (d) in Figure <ref> and Theorem <ref>. Let us denote by  the period annulus of the center. The inner boundary of  is the center itself, namely p, whereas the outer boundary of  is formed by saddle connections. In both cases the outer boundary have 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.


Let (ρ_0,ϕ_0)∈ℛ_2∪ℛ_3∪ℛ_4 and let p=(x,0) be the non-degenerated center of system (<ref>). Then the period of the associated linearized system at p is

    T_0(x)=2π/√(θ_0^2ν(x/R)(2+ν(x/R)))

where ν(x):=x^2-1(x-ρ_0)(x-1/ρ_0).



From the expression of the Jacobian matrix of the system in (<ref>) we have that the eigenvalues associated to the center are

    λ_±=±ω i=±√(θ_0^2 x^2-R^2/(x-r_0)(x-R^2/r_0)(2+(x^2-R^2)/(x-r_0)(x-R^2/r_0)))i

where ω denotes the frequency of the linearized center. Thus, setting ρ_0=r_0/R, and using that the period of the linearized center is T_0=2π/ω the result holds.


Let us now consider the periodically perturbed stirring protocol z_ϵ(t)=r_ϵ(t)exp(iθ_0 t) on system (<ref>) where r_ϵ(t) is a smooth T-periodic perturbation of r_0. More concretely, r_ϵ(t)=r_0+ϵ f(t)+g(t;ϵ) with f and g(·;ϵ) T-periodic analytic functions and g(t;ϵ) tending to zero uniformly on t∈ as ϵ tends to zero. The same change to a corotating frame than the presented at the beginning of this paper transforms (<ref>) into a periodic Hamiltonian system with Hamiltonian function


    Ψ(t,x,y;ϵ)=-θ_0/2(x^2+y^2)+Γ/2πln√((x-r_ϵ(t))^2+y^2/(x-R^2/r_ϵ(t))^2+y^2).



Let (ρ_0,ϕ_0)∈ℛ_2∪ℛ_3∪ℛ_4.
For any positive integer N there is a ϵ̅_N>0 such that, if ϵ<ϵ̅_N, then the Hamiltonian system u̇ = J∇Ψ(t,u;ϵ) has at least N periodic solutions contained in D_R. Particularly, the flow induced by system (<ref>) with T-periodic protocol z_ϵ(t) has infinity many periodic trajectories with zero winding number.



The spirit of this proof is to use Theorem <ref> in a certain period annulus where the regularity hypothesis are satisfied. On the one hand, since the outer boundary of the whole period annulus of the center in system (<ref>) is a saddle connection, the period of the periodic orbits tends to infinity as they approach the outer boundary. On the other hand, setting p=(x^*,0) the center itself, Lemma <ref> states that the period tends to

    T_0(x^*)=2π/√(θ_0^2ν(x^*/R)(2+ν(x^*/R)))

as the orbits tends to p. The analyticity of the period function ensures then that for any M>0 large enough there exists a period annulus, namely 𝒜_M, such that the period of its orbits covers [T_0(x^*),M].

Let us apply Theorem <ref> in 𝒜_M. To this end, it is enough to show that ∇Ψ(t,η;ϵ) is continuous in (t,η;ϵ)∈×𝒜_M×(0,ϵ_0) for some ϵ_0>0, locally Lipschitz continuous in η=(x,y)∈𝒜_M and T-periodic in t. From the expression in (<ref>),

    ∇Ψ(t,η;ϵ)=(
    [                -yθ_0+cθ_0 y(1(x-r_ϵ(t))^2+y^2-1(x-R^2/r_ϵ(t))^2+y^2); θ_0x-cθ_0(x-r_ϵ(t)(x-r_ϵ(t))^2+y^2-x-R^2/r_ϵ(t)(x-R^2/r_ϵ(t))^2+y^2) ]).

The previous vector is continuous for all (t,(x,y);ϵ)∈×^2×(0,ϵ_0) whereas (x,y)∉{(r_ϵ(t),0),(R^2/r_ϵ(t),0)}. Since r_ϵ(t)=r_0+o(ϵ) and R^2/r_0>R we can take ϵ small enough to ensure that R^2/r_ϵ(t)>R. On the other hand, by Theorem <ref> (b)-(d), d_H(𝒜_M,r_0)>d_H(𝒜_M,γ)>0 where γ denotes the saddle connection that forms the outer boundary of the period annulus and d_H denotes the Haussdorf distance of non-empty compact subsets of ^2. Thus, by continuity, there exists ϵ_0>0 small enough such that d_H(𝒜_M,r_ϵ(t))>d_H(𝒜_M,γ)>0 for all ϵ<ϵ_0. This implies that ∇Ψ(t,η;ϵ) is continuous for all (t,η;ϵ)∈×𝒜_M× (0,ϵ_0) as we desired. Moreover, for fixed (t,ϵ)∈×(0,ϵ_0), since d_H(𝒜_M,r_ϵ(t))>d_H(𝒜_M,γ)>0 then ∇Ψ(t,η;ϵ)∈ C^1(𝒜_M). Consequently, ∇Ψ(t,η;ϵ) is locally Lipschitz continuous in 𝒜_M.
Then we can apply Theorem <ref> and, particularly, Corollary <ref> to show that for any positive integer N there exists 0<ϵ̅_N<ϵ_0 such that if ϵ<ϵ̅_N then system η̇=J∇Ψ(t,η;ϵ) has at least N periodic solutions in 𝒜_M⊂ D_R. Finally, by construction of 𝒜_M those periodic solutions have zero winding number with respect to the vortex.






§ CONCLUSIONS


The main result of Section <ref> has a natural reading for the underlying physical model. Note that ρ_0 is the ratio between the path and domain radii respectively, while ϕ_0 measures the relation between the path angular speed and the vortex strength. The sign of ϕ_0 indicates if the sense of rotation of the vortex and the path is the same or opposite. For example, fixing the positive parameters R,Γ,r_0 and leaving θ_0, for small positive θ_0 there are no equilibria. Then, a first bifurcation point is θ^*_0=Γ/2π R^2R-r_0/R+r_0, where a degenerate saddle appears at (-R,0). A second bifurcation point appears at θ^**_0=Γ/2π R^2R+r_0/R-r_0. For θ_0∈ ]θ^*_0,θ^**_0[, there is a center and two hyperbolic saddles in the border of the domain connected by an heteroclinic. They travel along the border until they collide at θ_0=θ^**_0 into a degenerate saddle, that enter into the domain as an hyperbolic saddle for values above θ^**_0. On the other hand, for negative values of θ_0, corresponding to opposite rotating sense of the vortex and the stirring protocol, we identify a typical saddle-node bifurcation. 

In the identified bifurcation scheme, saddles are connected by heteroclinic or homoclinic orbits that constitute barriers for the flux transport. Around the centers, the particles rotate with different periods, and this fact makes possible an application of a suitable result for perturbed hamiltonians, proving that for a perturbed vortex path there exist infinitely many periodic solutions that do not rotate around the vortex. The problem to identify more general classes of vortex protocols that generate this kind of periodic orbits with zero winding number is still open.

9


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