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Heteroclinic orbits and transport in a perturbed integrable Suris map

Explicit formulae are given for the saddle connection of an integrable family of standard maps studied by Y. Suris. When the map is perturbed this connection is destroyed, and we use a discrete version of Melnikov's method to give an explicit formula for the first order approximation of the area of the lobes of the resultant turnstile. These results are compared with computations of the lobe area.

Suris [2,3] showed that f δ is integrable with integral I δ (θ, r) = cos πr + δ cos π(2θ − r) , i.e., I δ • f δ = I δ . Contours of I δ are shown in Fig. 1. The map is integrable for any δ; however, we will consider the case 0 < δ < 1, as the topology of the saddle connections changes at δ = 1. For 0 < δ < 1 the map f δ has hyperbolic fixed points at z a = (− 1 2 , 0) and z b = ( 1 2 , 0), that are connected by two saddle connections, forming the upper and lower separatrices of the fixed point resonance. Standard maps are examples of monotone twist maps (for a review, see [3]) defined by the geometrical twist condition that vertical lines tilt to the right upon iteration, ∂θ ′ ∂r > 0. Such maps have Lagrangian generating functions, S(θ, θ ′ ), which generate the map implicitly through the equation dS(θ, θ ′ ) = r ′ dθ ′ − rdθ, or equivalently where the subscripts indicate derivatives with respect the first and second arguments of S. The twist condition implies that ∂ 1 ∂ 2 S < 0. We assume that our twist map has zero net flux, which is equivalent to S(θ + 1, θ ′ + 1) = S(θ, θ ′ ). For maps of the standard form, (1), the generating function is 1 The potential can also be written where the dilogarithm is defined by dilog(x) ≡ x 1 log(z) 1−z dz.
We will let S δ denote the generating function for the Suris map f δ . Since the Suris map is integrable and has twist, Birkhoff's theorem [3] implies that the saddle connection between the equivalent points z a and z b is the graph of a function r = χ(θ), for θ ∈Ū = [− 1 2 , 1 2 ]. The dynamics (1) restricted to the saddle connection defines a circle diffeomorphism Our goal is to show that, when the Suris map is perturbed, the stable and unstable manifolds of the continuation of the fixed points intersect transversely. We use a version of Melnikov's method [4] for twist maps [5,6,7,8,9,10,11]. The system is perturbed by adding to the original generating function any C 2 periodic function P (θ, θ ′ ). We will assume for simplicity that P does not move the fixed points. Then for small enough ǫ, S δ + ǫP is a twist generating function, and the corresponding map f δ,ǫ has hyperbolic fixed points at z a and z b . As was shown in [9], if the Melnikov potential, given by has a nondegenerate critical point in U, then the manifolds W u (z a , f δ,ǫ ) and W s (z b , f δ,ǫ ) intersect transversely for ǫ small enough (the same conclusion is valid for W s (z a , f δ,ǫ ) and W u (z b , f δ,ǫ )). It is known that the series for L converges absolutely and uniformly to a C 2 function on compact subsets of U. As an example below, we consider the case P (θ, θ ′ ) = cos 2 πθ. Thus the potential L provides sufficient conditions for the transversal destruction of the saddle connection, just as in the classic applications of the Melnikov integral. In §2 we formulate a slightly stronger version of the previous result (Theorem 2). The Melnikov potential also has a physical interpretation: the difference between values of L at neighboring critical points is the area of the turnstile lobe that they delineate. The area of the lobe gives a coordinate independent measure of the separation of the perturbed stable and unstable manifolds as well as the flux from one region to another (cf. [3]).
The sum for the order ǫ approximation for the lobe area is calculated explicitly in §3 and compared with numerical results in §4. We summarize our results, as Theorem 1 (Main Theorem). Let S δ , given by (4) with potential V δ (2), be the generating function for the integrable Suris map f δ . Then S δ,ǫ = S δ + ǫ cos 2 πθ generates a twist map f δ,ǫ . For all 0 < δ < 1 and ǫ small enough there are two neighboring, primary heteroclinic orbits from z a = (− 1 2 , 0) to z b = ( 1 2 , 0). The lobe defined by these two orbits has an area given by where ϑ 4 (z, q) is the Jacobi theta function.
Note that the multipliers of the hyperbolic fixed point of f δ are ν and 1/ν, so that ν is a natural parameter to use (see e.g., [11]). The series Γ(ν) is rather intriguing. It is an analytic function of ν on the interval {0 < ν < 1}, and approaches zero rapidly as ν increases. In fact, though (7) implies that Γ(ν) is strictly positive, it approaches zero exponentially fast: −π 2 log(1/ν) as ν → 1 − , or equivalently δ → 0 + . However, note that (6) is only valid for fixed δ as ǫ → 0. If these parameters both approach zero, for example according to some relation δ = δ(ǫ), then the analysis becomes much more difficult. In particular when O(ǫ) term in (6) is exponentially small in δ, it could easily be dominated by terms that are formally higher order in ǫ. 2 Melnikov potential for twist maps In this section we review the derivation of the Melnikov potential for twist maps [9]. We begin with a C 2 Lagrangian generating functionS(θ, θ ′ ) that satisfies the twist condition and has zero net flux. It gives a map of the annulus implicitly through (3). Alternatively, let the action W of a sequence [θ] = {θ i , θ i+1 , . . . , θ j }, be Then, an orbit of the map that begins at θ i and ends at θ j corresponds to a critical point of W under variation with respect to the interior points [3]. The corresponding momenta are then defined through (3) as r t = −∂ 1S (θ t , θ t+1 ). Thus, for example, a point (θ a , r a ) is a fixed point of the map if and only if θ a is a critical point ofS(θ, θ), and r a is defined through (3).
The distance between the perturbed stable and unstable manifolds of any twist map with a saddle connection can be computed using the variational principle [12,13]. We begin with a mapf generated byS. Suppose thatf has two hyperbolic fixed points z a = (θ a , r a ) and z b = (θ b , r b ), and there is a saddle connection defined by the graph of a function χ(θ) on the interval U = (θ a , θ b ) between these points. A diffeomorphism h :Ū →Ū is induced by the restriction of the map to the saddle connection: Let P be a C 2 function with zero net flux. Then the functioñ generates a twist map f ǫ for small enough ǫ. Since hyperbolic points are nondegenerate critical points of the action [14], the perturbed map will have nearby hyperbolic fixed points for small enough ǫ. A simple case occurs when θ a is a critical point ofP (θ, θ) as well as ofS(θ, θ) since it is then a critical point ofS ǫ (θ, θ) as well. Thus the fixed points ofS ǫ will have unchanged configurations, but their momenta will be modified according to (3). The action gives useful formulae for the areas of regions for twist maps [15]. We will use one such relation to obtain the Melnikov potential: a relation between the graph χ and the action of orbits on the stable and unstable manifolds of a hyperbolic fixed point [16]. Let z 0 be a point on the unstable manifold of a fixed point z a that is close enough to z a so that the segment of W u (z a ) containing z 0 is given by a graph (θ, χ(θ)). Let (θ t , χ(θ t )), t ≤ 0, be the preorbit of this z 0 . Defining the backward action difference as then the unstable manifold is defined by the graph of the function A corresponding formula for the forward action of an orbit on the stable manifold yields a formula for the graph of an initial segment of the stable manifold, χ s , of z b : The difference between these two actions leads to the Melnikov-like formula for the transversal intersection of these manifolds. To summarize: Theorem 2. LetS be the generating function for a twist mapf that has two hyperbolic fixed points z a and z b with a saddle connection given by r = χ(θ) for θ ∈ U = (θ a , θ b ). Let f induce a diffeomorphism h(θ) on the connection. LetS ǫ =S + ǫP generate the twist mapf ǫ , such that the perturbationP has the following properties Then for ǫ > 0 small enough: i) the perturbed map has two hyperbolic fixed points near z a and z b ; ii) the Melnikov potential converges absolutely and uniformly to a C 2 function on U; and iii) if L has a nondegenerate critical point on U, then the unstable and stable manifolds of the two fixed points intersect transversely.
We can relate the Melnikov potential to the area of a lobe in the stable and unstable manifolds using the action formula of [17]. Suppose p and q are two neighboring heteroclinic orbits, for example as shown in Fig. 2. Then the area lobe is determined by the difference in action between the orbits of p and of q [17] The area A is the signed area below the segment of W s between q and p and above that of W u . Since the action is stationary on an orbit and since the actions of p and q are equal when there is a saddle connection, it is easy to see that the area of the lobe is determined, to lowest order, by the difference between two critical values of the Melnikov potential L (cf. [12,13]). To summarize: LetS ǫ ,f ǫ , U, h and L be defined as in Theorem 2. Assume that θ p and θ q are two points in U such that Then the heteroclinic points p and q continue to heteroclinic points of f ǫ , and the stable and unstable manifolds of f ǫ enclose a lobe with area

The Suris Map
We will use the results of the previous section withf = f δ ,S = S δ ,S ǫ = S δ,ǫ ,f ǫ = f δ,ǫ and the particular perturbationP (θ, θ ′ ) = cos 2 πθ. We give formulas for the lobe area of the perturbed Suris map. To find the saddle connection, we must find the diffeomorphism h induced by the dynamics of the Suris map f δ on the connection (8). We will consider h defined only on the set U = {− 1 2 < θ < 1 2 }. We first give some properties of h.
On the other hand, This implies (13) for the upper signs. The substitution θ → −θ in equation (14), gives the lower signs. It is easy to see that if θ ∈ U, then (14) is positive and therefore 0 < θ − h ν (θ) < 1. We conclude that for all θ ∈ U, To finish the proof we take tangent of the second difference and use the sum formula for tangent to obtain .
Since this last expression is − tan( π 2 V ′ δ (θ)), this completes the proof. With the help of lemma 4, we can give a description of the intersection of the saddle connection between (− 1 2 , 0) and ( 1 2 , 0). This connection is given by the functions χ ± , and is shown as the heavy curves in Fig. 1.
Lemma 5. Let f δ be the twist map generated by S δ . Then (− 1 2 , 0) and ( 1 2 , 0) are hyperbolic fixed points for f δ and there are two saddle connections between them given by the graphs Proof. Using Lemma 4, the map can be written in the form Now since θ = − 1 2 is an stable fixed point for h ν , this graph gives the left going saddle connection. In the same way setting r = χ + (θ) = θ − h ν (θ) gives Since the map conjugates to h ν −1 on this graph, this is clearly the right going saddle connection.

Proof of the Main Theorem
We now can sketch the proof of the main theorem. The analysis of the infinite series for the Melnikov potential relies on some summation formulae for elliptic integrals.
Proof. Let P (θ, θ ′ ) = cos 2 πθ. It is clear that P satisfies the conditions of Theorem 2. Let L be the Melnikov potential, (5). According to Theorem 2 a sufficient condition for transversal intersection of the perturbed manifolds is that L has a nondegenerate critical point on the interval U = {− 1 2 < θ < 1 2 }. The graph of L(θ) over U is shown in Fig. 3. To proceed, we defineL(z) = L( 2 π arctan(z)). A little algebra givesL(z) Since z = tan( π 2 θ) is monotone increasing, we need only show that there is a nondegenerate critical point ofL. We are going to show that z q = 0 is a local maximum ofL and First, we writeL as Now, since α t (−z) = α −t (z), this implies thatL is even, and soL ′ (0) = 0. Differentiation givesL This expression defines a function Γ 0 , which can be rewritten by expanding the denominator and rearranging the sums as This sum can be obtained from Fourier expansions of Jacobi elliptic functions which are written in terms of the nome, q(k) = exp −π K ′ K , where K(k) is the complete elliptic integral, K ′ = K(k ′ ) and k ′ 2 = 1 − k 2 [18]. Recall that q is a monotone increasing diffeomorphism of k on (0, 1). A Fourier expansion for elliptic functions [18, formula 912.01] can be differentiated to show that where ϑ 4 (z, q) is a Jacobi theta function [18, formula 1051.01]. ThereforeL ′′ (0) < 0, for all 0 < ν < 1. This shows that θ q = 0 is a nondegenerate local maximum of L for all 0 < ν < 1.
Using Theorem 3, we conclude that a lobe of area is enclosed by the stable and unstable manifolds, where θ p and θ q are given above. Finally, we use (15,16) to obtain Subtracting these two, and expanding the series we obtain where Γ(ν) was defined in (7). Finally, this result can be explicitly written as upon differentiating the Fourier series [18,908.51] once. We have therefore obtained (6).
The explicit formula for the first order approximation of the area is compared with numerical computations in Sec. 4.

Exponentially small behavior
In this subsection we investigate the asymptotics of the sum, (7), as δ → 0. Our result is summarized as: Lemma 6. Let Γ(ν) be defined by (7), then Proof. We use the expression (17) in terms of the elliptic integral K and the nome q.
Recall that q → 1 − as k → 1 − and k ′ → 0 + . The needed asymptotic forms are [18, formulae 112.01 and 112.04] as k → 1 − . Putting these into (17) gives the promised result. We again note that (18) does not imply that the area itself is necessarily exponentially small: (6) is only valid for fixed δ as ǫ → 0. Nevertheless we will see in §4 that (18) agrees remarkably well with our numerical calculations of the lobe area.

Anti-Integrable Limit
It is much easier to obtain an expression for the lobe area for large large ǫ. Our main purpose is to have an expression that is valid in the regime of large ǫ where numerical calculations are difficult. This expression is easy to obtain using the "anti-integrable limit" [19]. For simplicity, in this section we assume that P depends only upon θ in this section. In this case the anti-integrable limit is obtained by scaling the action by ǫ −1 , to getŜ = S(θ, θ ′ )/ǫ = P (θ) + ǫ −1 S δ , and then setting ǫ −1 = 0. The point is that when ǫ ≫ 1 > δ the points on the two heteroclinic orbits are all found in a neighborhood of the critical points of the potential P = cos 2 (πθ), i.e., at θ = m/2. In the anti-integrable limit, an orbit consists solely of a sequence of configuration points sitting at these critical points. Thus the two heteroclinic orbits are given by the sequences The action is stationary on an orbit, thus to first order in the small parameter ǫ −1 , the change in the orbit with ǫ can be ignored in the action difference to give .

For the Suris map, this yields
We compare this result with the numerical calculations in §4.

Numerical comparison
In this section, we compare the theoretical results with numerical computations of the lobe area. The task is to find the actions of the two homoclinic points p and q. For this task we use the symmetry of the Suris map. The action difference between the orbits of q and p gives the lobe area, (11). The Suris map is reversible: it is conjugate to its inverse by an involution R: R 2 = I and Rf δ R = f −1 δ , where the reversor is  [20]. Thus to find a heteroclinic orbit it is sufficient to do a one dimensional search for points on the unstable manifold that intersect one of these fixed sets.  Figure 4: Finding some heteroclinic orbits using symmetry. Shown are the fixed sets Fix(f t R), for t = 0, 1, 2 and a rough example of initial guess z 0 , leading to t c = 1.
To find these orbits numerically we move along W u (z a ) to the first points that intersect Fix(R) and Fix(f R), respectively. For example, to find q ∈ FixR, let z 0 = (θ 0 , r 0 ) = z a + ηE u where E u is the unstable eigenvector, and η is a small parameter to be chosen below. Let t c + 1 be the first time for which the iterate of z 0 is beyond Fix(R). Now choose a point z(θ) on the line from z 0 to z 1 = f δ (z 0 ), parameterized by the initial angle θ. By construction we are guaranteed that the function Z(θ) = π 1 f tc δ (z(θ)) has a zero for θ ∈ [θ 0 , θ 1 ]. We use a root finding method (Brent's method) to determine this zero to some precision, say ρ. The choice of precision influences the original value for η, as well as the number of iterates until a crossing. Assuming W u is smooth, the point z 0 will be O(η 2 ) away from W u . After t c iterates, however, this error will decrease by the factor λ −tc where λ is roughly the unstable multiplier of the fixed point. There is no sense in having this error smaller or larger than the precision of our root finder, so we set ρ ∼ λ −tc η 2 . On the other hand, since we start a distance η from the fixed point, and wish to go a distance O(1) to find the first crossing of the symmetry line, we have ηλ tc = O(1). Thus, it is appropriate to set η ∼ ρ 1/3 . To find the second homoclinic point, p ∈ Fix(f δ R), we repeat the above analysis, using crossing of Fix(f δ R) to determine t c , etc. The lobe area is given by the difference in action between these two orbits, from (11).
For our computations, using IEEE double precision arithmetic, we set ρ = 10 −19 . These computations give apparently accurate results providing A >> 10 −14 . Subsequent to our obtaining these results, Delshams and Ramírez-Ros used extended precision arithmetic to obtain lobe areas as small as 10 −4200 [1]. In Fig. 5 we show a comparison of the result of (6) with the numerical results on a log scale. The analytical result agrees well with the numerical results when ǫ = 10 −5 . We show the same data on a linear scale in Fig. 6. We plot the sum (7) only for δ > 0.15, since it is indistinguishable from asymptotic formula (18) for smaller values. Moreover, the asymptotic formula agrees with the ǫ = 10 −5 computation within 1% up to δ = 0.8. In Fig. 6, the anti-integrable results evaluated at ǫ = 1 are also shown. We are unable to obtain numerical results for such a large ǫ, as the multiplier of the fixed points is too large.

Conclusion
The perturbed Suris map studied here depends on two parameters, the Suris parameter δ and the perturbation strength ǫ. We obtained the lobe area for the fixed point resonance of this map for small ǫ to O(ǫ) and for large ǫ to O(ǫ −1 ). In the small epsilon case, the numerical computations indicate that the asymptotic formula (18) for δ → 0 is a good approximation even up to δ = 0.8. A similar Melnikov analysis is possible for other standard maps [7,11]. In particular [9] showed that there exists a large class of standard maps with saddle connections. Interestingly these maps are not all integrable. The use of the Melnikov potential is not restricted to twist maps; in fact, it can be applied to any exact symplectic map [13].
The method can also be applied to any higher dimensional twist map that has a saddle connection of the type described in this paper (see [13,10]). Integrable examples of such maps have been found in [21,22,23], and the Melnikov method has been applied to applied to the four dimensional McLachlan map [13]. Similar techniques are also applicable to volume preserving maps [24]. The study of perturbations of twist maps with saddle connections in higher dimensions is important because it could help in the development of a higher dimensional theory of transport.