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Twist dynamics and Aubry-Mather sets around a periodically perturbed point-vortex

We consider the model of a point-vortex under a periodic perturbation and give sufficient conditions for the existence of generalized quasi-periodic solutions with rotation number. The proof uses Aubry-Mather theory to obtain the existence of a family of minimal orbits of the Poincar\'e map associated to the system.


Introduction
We study the advection of a passive particle in a two-dimensional ideal fluid. This phenomena can be described by the Lagrangian version of fluid mechanics: the particle moves according to a Hamiltonian system with the streamfunction playing the role of the Hamiltonian.
Given the vorticity ω of the incompressible fluid, the streamfunction is defined as a solution of the Poisson equation −∆Ψ = ω. From a physical point of view, a vortex is a zone of high vorticity. Mathematically, a vortex in the plane can be defined in different ways. It can be defined as a singularity of the vorticity or through a compact set of finite vorticity (vortex patch). See [1,7,8,10,18] for a summary on the various definitions.
We will be concerned with a point-vortex, defined as a Dirac delta of the vorticity. Under this definition, the streamfunction is the fundamental solution of the 2-dimensional Laplacian. These ideas were introduced in the seminal works of Helmholzt and Kirchoff in the XIXth century and nowadays point-vortices are studied as a branch of Fluid Mechanics with deep connections with Celestial Mechanics and Hamiltonian systems.
A point-vortex induces a streamfunction Ψ 0 = Γ 4π ln(x 2 + y 2 ) where Γ is the circulation (or strength) of the vortex and, up to a rescaling of unit, it can be set Γ = 2π. The solution of the corresponding Hamiltonian system are circular path around the orgin and describe the trajectories of a passive particle under the influence of the vortex. The frequency of rotation is inversely proportional to the radius of the path and tends to infinity as the radius tends to zero.
Physically, system (2) can be interpreted to model the advection of a particle under the action of a steady vortex placed at the origin and a periodic time dependent background flow.
The dynamics of advected particles in non-stationary flows including vortices have been intensively studied from different perspectives [4,6,9,23,24,27]. In particular, numerical tests suggest the presence of complex dynamics.
From an analytical point of view, in [22] the authors studied the stability properties of the vortex. More precisely, they proved that if the external streamfunction is analytic, then KAM theory applies and invariant curves of the Poincaré map exist close to the singularity. As a byproduct, there exist quasiperiodic solutions of the Hamiltonian systems with (sufficiently large) Diophantine frequencies. These can be seen as a reminiscent of the trajectories of the unperturbed Hamiltonian with Diophantine frequency.
In this paper, we will prove that, close to the singularity, quasi-periodic solutions exist for all frequency sufficiently large. Actually, our solutions will be a generalization of standard quasi-periodic solutions and in case of commensurable frequencies, we will get periodic solutions. These solutions exist also when KAM theory cannot be applied. Indeed, we will require very low regularity that prevents standard KAM theory from being applied.
To prove our result, we will apply a suitable version of Aubry-Mather theory [3,14] to the Poincaré map of system (2). A similar scheme have been used to describe the dynamics of different systems [11,13,20,25,28].
For each sufficiently large real number α, we will prove the existence of an invariant set M α (called Aubry-Mather set) with very interesting dynamical properties, among them each orbit in M α has rotation number α. For irrational rotation numbers, the corresponding Aubry-Mather sets are either curves or a Cantor sets. Solutions of system (2) with initial conditions in this set will be our generalized-quasi periodic solutions. In the rational case, the Aubry-Mather sets contain a periodic orbit.
In suitable variables, the Poincaré map will be an exact symplectic twist map of the cylinder. However, it will not be defined on the whole cylinder. Hence we cannot apply directly the result of Mather and we will prove an adapted version to this situation.
To apply our theorem, we will need to prove that the Poincaré map is exact symplectic and twist. The first property comes from the Hamiltonian character of the system. The twist condition is more delicate and relies on the behavior of the variational equation. We will give a proof following a perturbative approach. Here, we will ask that the perturbation has the origin as a zero of order 4.
From the point of view of dynamics of symplectic diffeomorphisms, we will describe some aspects of the dynamics around a singularity. In the integrable case, the flow can be continuously extended to the singularity, defining it as a fixed point. However, this extension is not C 1 . In the perturbed case, in general is not even possible to guarantee continuity of this extension. Since the flow is not regular, all the results coming from the theory of elliptic fixed points and transformation to Birkhoff normal form cannot be applied directly. We will overcome the problem of the singularity performing a change of variable that sends the singularity at infinity and has a regularizing effect. At this stage, the assumption of having the zero of order 4 in the perturbation play a fundamental role.
The paper is organized as follows. In Section 2 we state the problem and the main result. The definition of generalized quasi-periodic solution will be given in this section. In Section 3 we introduce the regularizing variables and the Poincaré map together with some preliminary estimates. In Section 4 we state and prove the suitable version of the Aubry-Mather theorem. In Section 5 it is proved the property of exact symplectic and Section 6 is dedicated to the proof of the twist property. The proof of the main result will be given in Section 7. Finally, we draw some conclusions in Section 8. Some technical lemmas are relegated to the Appendix.

Statement of the problem and main result
Let us consider the perturbed Hamiltonian system given by (1)- (2). We suppose that the perturbation p(t, x, y) belongs to the following class Definition 1. Given ε > 0, consider the open disk around the origin D ε = {(x, y) ∈ R 2 : x 2 + y 2 < ε 2 We say that a continuous functions f : r.t the spatial variables (x, y) and all the partial derivatives are continuous w.r.t. (t, x, y). Now, given any N ∈ N we give the notion of zero of order N of a function f ∈ R k ε . The following definition will be of particular interest in the case N > k. Definition 2. Given a function f ∈ R k ε we say that the origin is a zero of order and satisfying the following properties.
• there exists a constant C such that, for all (t, x, y) ∈ R × D ε , Our result gives the existence of particular families of solutions: periodic and quasi-periodic solutions in a generalized sense. To define them, given a solution (x(t), y(t)) of (2), consider the functions having a relation with the standard polar coordinates. Actually θ(t) represents the angle in the clockwise sense, while r(t) is, up to a scaling constant, the inverse of the square of the radius.

Definition 3.
We say that the solution (x(t), y(t)), defined for t ∈ R • is non-singular if sup t∈R r(t) < ∞; • is bounded if there exists A > 0 such that inf t∈R r(t) > A; • has monotone argument if θ(t) is monotone; • has rotation number α if 1 2π lim t→∞ θ(t) t = α.

Remark 1.
A non-singular bounded solution with monotone argument rotates clockwise in a closed annulus around the origin. Moreover, the rotation number represents the average angular velocity.
We are ready to state the main result.
and Φ ξ (t, 2παt) = (r(t), θ(t)) ξ . If the function ξ → Φ ξ is continuous, then these solutions are classical quasi-periodic solutions with frequencies (1, α) in the sense of [26] (see also [21]). We will not guarantee the continuity, however, the function ξ → Φ ξ will have at most jump discontinuities and if ξ is a point of continuity then so are ξ + 2πα, ξ + 2π. Finally, the set Cl{(x(0), y(0)) ξ : ξ ∈ R} is either a curve or a Cantor set, recovering the classical definition of quasi-periodic solution in the case of having an invariant curve.

Some estimates on the solutions and the Poincaré map
Let us consider system (2) and, following section 4.1 of [22], consider the change of variables (x, y) = ϕ(θ, r) defined by These variables comes from applying first the Kelvin transform and subsequently the change to symplectic polar coordinates. System (2) transforms into where H(t, r, θ) = − 1 2 ln(2r) + h(t, r, θ) and h(t, r, θ) = p t, cos θ √ 2r , − sin θ √ 2r . System (6) is still a periodic planar Hamiltonian system with symplectic formλ = 1 4r 2 dr ∧dθ. Moreover, the change of variables ϕ transforms the domain R×D ε into the domain R × D with Let us write the Cauchy problem associated to system (6), in the following form: where, Since p ∈ R 3 ε , the vector field in (7) is continuous and C 2 in the spatial variables. This guarantees existence and uniqueness of the solution. Figure 1: Domains and evolution of two solutions over a period. (r(t), θ(t)) 1 represents the solution with (r 0 , θ 0 ) 1 ∈ Σ(a * ) and (r(t), θ(t)) 2 represents the solution with (r 0 , θ 0 ) 2 / ∈ Σ(a * ). Note that the solution (r(t), θ(t)) 1 remains in the domain D.

Remark 3.
The change of variables ϕ has the effect to transform the phase space from the plane to the cylinder. The singularity is moved from the origin to r → ∞.
In this sense, the change of variables has a regularizing effect since the functions F, G in (7) are bounded for r → ∞. The fact that the origin is a zero of order 4 plays a fundamental role in this discussion. See estimate (10) in the following Lemma 3.1 for more details.
Since the domain D is not invariant, we need to control the growth of the solutions. For this purpose, given a > r * we introduce the set Σ(a) =]a, ∞[×T ⊂ D and prove the following lemma, whose meaning is illustrated in Fig. 1.

Lemma 3.1. Let us assume that the origin is a zero of order 4 of the function
, the corresponding solution of (7) is well defined on t ∈ [0, 1] and (r(t), θ(t)) ∈ D for all t ∈ [0, 1]. Moreover, the following estimate holds for some K > 0.
Proof. Since the origin is a zero of order 4 of p, there exists a constant C > 0 such that Then, from the definition of F and G, we have for any t ∈ R and (r, θ) ∈ D.
Finally, integrating the second equation of (7), we deduce that Here we have employed (10) and the above estimates on r(t).
Now, let us introduce the Poincaré map P as where (r(t; r 0 , θ 0 ), θ(t; r 0 , θ 0 )) is the solution with initial condition (r(0), θ(0)) = (r 0 , θ 0 ). Lemma 3.1 together with existence and uniqueness of the solutions of problem (7) guarantee that the Poincaré map is well defined. Due to the regularity of the vector field of (6), P ∈ C 2 (Σ(a * )), concretely is a diffeomorphism of a section of the cylinder. The proof of the theorem will be a consequence of a suitable version of the so called Aubry-Mather theory applied to the previous Poincaré map. The following section is dedicated to the statement and proof of this result.

A generalized Aubry-Mather theorem
We denote by C = R × T, T = R/2πZ the cylinder and consider the strip Σ := (a, b) × T and the corresponding liftΣ : We denote the lift by where such that f ′ never vanishes. Without loss of generality we fix f ′ > 0.
We suppose that Φ is exact symplectic with respect to the form Remark 4. Note that the function S(r, θ) is defined in the cylinder, hence the lift S(r, x) must be a 2π-periodic function in the variable x such that We also suppose that Φ is twist, that is Suppose additionally that the following uniform limits (w.r.t. x) exist Note that α ± (x) are 2π-periodic C 2 functions and define The main result of this section deals with the existence of special orbits of the diffeomorphism Φ. To state the Theorem, we recall that a sequence (x n ) n∈Z of real numbers is increasing if x n < x n+1 for all n ∈ Z and we say that any two translates are comparable if for any (s, q) ∈ Z 2 only one of the following alternatives holds x n+q + 2πs > x n ∀n, x n+q + 2πs = x n ∀n, x n+q + 2πs < x n ∀n.
We are now ready to state the main result of this section: Theorem 4.1. With the previous setting, suppose that W + − W − > 8π and fix α such that 2πα ∈ (W − + 4π, W + − 4π). Then x n+q = x n + 2πs ∀n ∈ Z; • if α ∈ R \ Q there exists a compact invariant subset M α ⊂ Σ (and a corresponding subsetM α ⊂Σ ) with the following properties: -each orbit (r n , x n ) n∈Z ∈M α is such that the sequence (x n ) is increasing and any two translates are comparable, -the set M α is either an invariant curve or a Cantor set.
The following corollary gives an equivalent interpretation of the result and has been proven in [11].
where φ is monotone (strictly if α ∈ R \ Q ) and η is of bounded variation.
The proof of Theorem 4.1 will make use of the generating function. We introduce it in the following iv) a sequence (r n , x n ) n∈Z is an orbit ofΦ iff for all n ∈ Z Proof. By the twist property, α + (x) > α − (x) ∀x ∈ R, so that we can consider the open connected set (see figure 2) From the periodic property of the functions α ± (x) this set is invariant under the translation T 2π,2π (x, x 1 ) = (x + 2π, x 1 + 2π).
By the twist condition we can solve the implicit function problem and obtain a unique C 2 function R(x, x 1 ) : B −→ (a, b) such that and, by implicit differentiation, Moreover, uniqueness implies that R(x + 2π, x 1 + 2π) = R(x, x 1 ). Analogously we get with R 1 (x + 2π, x 1 + 2π) = R 1 (x, x 1 ). Hence, the map (11) is equivalent to Now, we use the exact symplectic condition and define the generating function This maps is clearly C 2 (B), and, a posteriori, we will get C 3 regularity. From the periodicity conditions of S and R, one can prove the periodicity condition ii).
To prove point iii), we use (12), (14) to get that for all (x, x 1 ) ∈ B, so that the twist condition and the monotonicity of f imply To prove the last point, a similar computation as (15) gives for all (x, x 1 ) ∈ B, Equations (15)- (16), together with the regularity of f, R, R 1 have the consequence that h ∈ C 3 (B), proving point i).
Less formally (15)-(16) also imply that the map Φ can be expressed implicitly: It means that an orbit (r n , x n ) n∈Z is such that for every n ∈ Z .
Remark 5. The equation is known as discrete Euler-Lagrange equation.
The usual Mather's theorem (see Theorem 4.2), gives sufficient conditions on the generating function in order to get orbits with rotation number. In particular it is required h ∈ C 2 (R 2 ) and properties ii) y iii) of Lemma (4.1) should hold in the whole plane. For this reason we need the following extension lemma. A version of this lemma is stated in [15, chapter 8] and for the sake of completeness, we report here a detailed proof (see also [12,13]).
Define the following set and let h : W −→ R be a C r+1 function such that: Then there existsh ∈ C r (R 2 ) such that: Proof. The domain W is invariant under the translation T 2π,2π , in consequence the cocient set W/(2πZ) 2 is compact so that Consider a C ∞ real valued function χ : R 2 → [0, 1] such that χ(x + 2π, x 1 + 2π) = χ(x, x 1 ) and Let's define the function Then by the definition of χ we have that D ∈ C r−1 (R 2 ) and D(x + 2π, x 1 + 2π) = D(x, x 1 ). Moreover, In particular, with the hypotheses on h we have : Now, let us consider the Cauchy problems for the wave equation (with periodic boundary conditions): The change of variable conjugates system (17) to the classical wave equation where, denoting x(t, y) = (B ± ) −1 (y − t), x 1 (t, y) = t + y, v(t, y) = u (x(t, y), x 1 (t, y)) , f (t, y) = − 4 (B ± ) ′ (x(t, y)) D (x(t, y), x 1 (t, y)) , .
Using the terminology introduced in Theorem 4.1, we recall some of the conclusions of Mather theory  (Mather [5, 15]). Consider a C 2 function h : R 2 → R such that h(x + 2π, x 1 + 2π) = h(x, x 1 ) and ∂ 2 xx 1 h ≤δ < 0 for all (x, x 1 ) ∈ R 2 . Fix α ∈ R. Then (i) if α = s/q ∈ Q there exists an increasing sequence (x n ) n∈Z and an homeomorphism of the circle g α such that g α (x n ) = x n+1 and |x n − x 0 − 2πnα| < 2π for every n ∈ Z, -∂ 1 h(x n , x n+1 ) + ∂ 2 h(x n−1 , x n ) = 0 and x n+q = x n + 2πs for every n ∈ Z; (ii) If α ∈ R \ Q there exists a set M α of increasing sequences x = (x n ) n∈Z such that -if x ∈ M α then ∂ 1 h(x n , x n+1 ) + ∂ 2 h(x n−1 , x n ) = 0 for every n ∈ Z, any two translates are comparable and |x n − x 0 − n2πα| < 2π for all n ∈ Z, -there exists a Lipschitz homeomorphism of the circle g α with rotation number α and a closed set A α ⊂ R such that x ∈ M α iff x 0 ∈ A α and g n α (x 0 ) = x n for all n, -the set Rec(g α ) ⊂ A α of recurrent points of g α is either the whole R or a Cantor set.

Remark 6.
We recall that the homeomorphism g α satisfies g α (x+2π) = g α (x)+2π for all x ∈ R and the set Rec(g α ) is defined as the set of accumulation points of {g n α (x) + 2πk : (n, k) ∈ Z 2 } and is independent on the choice of the point x ∈ R. Moreover, the condition |x n − x 0 − n2πα| < 2π, ∀n ∈ Z implies that 1 2π lim n→∞ x n n = α, and α is called the rotation number of the orbit.
We are now ready for the proof of our result.
Proof of Theorem 4.1. We apply Lemma 4.1 and get the generating function h defined on the set

and satisfying the corresponding properties o)-iv).
Consider the case in which both W + , W − are finite. Since W + − W − > 8π, for every α such that we can choose ε > 0 such that

Now, consider the set
and apply Lemma 4.2 with B ± (x) := x + W ± ∓ ε that clearly are diffeomorphisms. We can extend the function h to the whole R 2 getting a functionh satisfying the conditions in Theorem 4.2 and such thath = h in W. By applying Theorem 4.2 we obtain sequences (x n ), such that From this inequality we obtain: that means that for every n ∈ Z, (x n+1 ,x n ) ∈ W. But sinceh = h in W, Hence, in case of rational α we definẽ such that (x n ,r n ) ⊂ Σ is the (s, q)-periodic orbit of Φ. In the irrational case, the setM α is given bỹ Note that the Lipschitz regularity of f −1 plays a role at this stage. In case −∞ < W − < W + = ∞ is enough to choose α such that 2πα > W − + 4π and fix M > 2πα + 8π. Fix ε such that and apply extension lemma with B − (x) = x + W − + ε and B + (x) = x + M, in order to get the same result.
The other cases are similar.

Exact symplectic properties of the Poincaré map
Fix r ≥ 2 and a C r+1 function f : (a, b) −→ R such that f ′ (r) never vanishes and consider the associated differential formλ = df (r) ∧ dθ = f ′ (r)dr ∧ dθ on (a, b) × R. In local coordinates, the corresponding time dependent Hamiltonian system takes the form Suppose that H : R × (a, b) × R → R is continuous in t and C r+1 in the phase variables (r, θ) and the following periodicity hold H(t + 1, r, θ) = H(t, r, θ) and H(t, r, θ + 2π) = H(t, r, θ).
By the periodicity in θ we have the phase space is the cylinder (a, b) × T.
Let us now prove that We start with Note that the term ∂ r H(t, r(t), θ(t))[∂ r 0 r(t)] is canceled. Now, using the first equation in (19), and integrating by parts the last term, we obtain Replacing in (20) and using the second equation in (19), we get Analogously,

The twist property for the vortex problem
In this section we consider the Poincaré map P associated to system (7). We recall the notation The following theorem will clearly imply the twist condition (13), Theorem 6.1. Suppose that p ∈ R 2 ε and the origin is a zero of order 4. Then To prove the theorem, let us fix a solution (r(t; θ 0 , r 0 ), θ(t; θ 0 , r 0 )) of problem (7). Note that, since p ∈ R 2 ε , the vector field of system (7) is C 1 in the variables (r, θ) so that the solution is unique and we can consider the associated variational equation Ẏ = M(t, r(t; r 0 , θ 0 ), θ(t; r 0 , θ 0 ))Y , where M(t; r, θ) = ∂ (F (t, r, θ), 2r + G(t, r, θ)) ∂ (r, θ) is the Jacobian of the vector field in (7). We denote the matrix solution and by the definition of the Poincaré map, In the integrable case p = 0, the Jacobian matrix is A = 0 0 2 0 and the solution of the corresponding variational equation is that shows that the Poincaré map of the unperturbed problem is twist.
To prove the result in the non integrable case, we will follow a perturbative approach. More precisely, we will prove that the solution remains close to that of the integrable case over a period t ∈ [0, 1]. For this purpose we begin considering the following splitting. To simplify the notation we will denote a solution of (7) by (r(t), θ(t)) where we have dropped the dependence on the initial conditions. Lemma 6.1. Under the hypothesis of Theorem (6.1), we have the following splitting: Moreover, Proof. Since the origin is a zero of order 4 for p, we can split the perturbation as where T 4 is a homogeneous polynomial of degree 4. From system (7) so that p induce the following splitting on F where, using the homogeneity of T 4 w.r.t the variable r, Therefore we write where b 12 = b 12 (t, r, θ) and c ij = c ij (t, r, θ) defined as and c 11 := ∂ rF (t, r, θ) c 12 := ∂ θF (t, r, θ) c 21 := ∂ r G(t, r, θ) c 22 := ∂ θ G(t, r, θ) Concerning the matrix C, one can first explicitly write the expression of the entries c ij recalling thatF has just been introduced and G is defined in (8). Note that they depend on the derivatives up to second order of the functions w.r.t the variables (r, θ). These derivatives are estimated in Lemma A.1 of Appendix A and can be used to get the following estimate r 3/2 |c 11 | + r 1/2 |c 12 | + r 2 |c 21 | + r |c 22 | ≤ K with K independent on r, θ. We evaluate the entries c ij on a solution (r(t), θ(t)) and we remember that from Lemma 3.1 we have that for r 0 > a * |r(t) − r 0 | ≤ K 1 , ∀ θ 0 ∈ T and t ∈ [0, 1].
Let us define the function β(t) := θ(t) − 2r 0 t. The estimate (9) on the angular evolution gives a bound of β ∞ . To get a bound of˙ β ∞ we observe thaṫ And again from (9) and (10), we havė . q(s, cos(2r 0 s + β(s)), sin(2r 0 s + β(s)))ϕ(s)ds Let us write the variational equation (22) as We claim that the solution Y(t; r 0 , θ 0 ) converge uniformly, as r 0 → ∞, to the solution Y int (t; r 0 , θ 0 ) of the integrable case (23). We prove it applying Lemma A.3 of Appendix to the family of matrices M(t, r 0 , θ 0 ) = B(t, r 0 , θ 0 ) + C(t, r 0 , θ 0 ) as r 0 → +∞. From Lemma 6.1 we have that M(t, r 0 , θ 0 ) is uniformly bounded and C(t, r 0 , θ 0 ) converge uniformly to 0 so that it converge also in the weak* topology. To study the convergence of the matrix B(t, r 0 , θ 0 ) it is enough to consider the term b 12 . From (24) and using the density of C ∞ in L 1 we have Hence, we can apply Lemma A.3 and get from which (21) follows evaluating in t = 1.
The function ξ → Φ ξ (a, b) introduced in Remark 2 has the same regularity of the functions φ, η that can have at most jump discontinuities. Moreover, from properties (26), (27), if ξ is a point of continuity, so are ξ + 2π and ξ + 2πα.
Finally, these solutions have rotation number α, actually, where

Conclusions
This paper can be seen as an example of the study of twist dynamics around a singularity in Hamiltonian systems. As a paradigmatic example we choose the point-vortex model. In suitable variables we applied a version of Aubry-Mather theory to get similar results as in the case of exact area-preserving maps of the annulus [14]. We suppose that the origin was a zero of order 4 for the perturbation. This condition played a role in the regularizing change of variable ϕ. For this reason, it seems unclear how to weaken this assumption.
Our result leaves open the distinction between classical and generalized quasiperiodic solutions. This relies on the nature of the corresponding Mather set with irrational rotation number. Actually it can be either a invariant curve or a Cantor set. A possible future development of the present work could be finding conditions that break invariant curves.
The following result is a lemma of Riemann-Lebesgue type.
The coefficients q k can be expressed in terms of the functions α j,h and belong to C 1 (R/Z), q k (t) = 1 2π 2π 0 q(t, cos θ, sin θ)e −ikθ dθ.
The condition (28) implies that q 0 (t) vanishes everywhere and so the integral I(t) we want to estimate can be expressed as the sum q k (s)e ikβ(s) e ikλs ϕ(s) ds.
Since we have excluded k = 0 these integrals can be estimated by a standard procedure in the theory of oscillatory integrals, see for instance [2]. After integrating by parts I k (t) = 1 ikλ q k (t)e ikβ(t) e ikλt ϕ(t) − q k (0)e ikβ(0) ϕ(0) − t 0 q k (s)e ikβ(s) ϕ(s) ′ e ikλs ds . Therefore, Finally, we state the following lemma concerning the uniform convergence of the solution of a linear ODE whose time-dependent coefficients are bounded and converging weak* in L ∞ . Similar results can be found in [16] and [19]. For the proof we will follow the lines of the proof of Lemma 2.1 in [19].