Melnikov's method in String Theory

Melnikov's method is an analytical way to show the existence of classical chaos generated by a Smale horseshoe. It is a powerful technique, though its applicability is somewhat limited. In this paper, we present a solution of type IIB supergravity to which Melnikov's method is applicable. This is a brane-wave type deformation of the AdS$_5\times$S$^5$ background. By employing two reduction ans\"atze, we study two types of coupled pendulum-oscillator systems. Then the Melnikov function is computed for each of the systems by following the standard way of Holmes and Marsden and the existence of chaos is shown analytically.


Introduction
The gauge/gravity correspondence is a fascinating topic in the study of string theory. A certain class of it preserving a conformal symmetry, in which a string theory is defined on an anti-de Sitter (AdS) space and its dual is a conformal field theory (CFT), is called the AdS/CFT correspondence [1][2][3]. A prototypical example is a duality between type IIB string theory on the AdS 5 ×S 5 background and the four-dimensional N = 4 SU(N c ) super Yang-Mills theory in the large N c limit. Remarkably, the integrability structure exists behind it and hence one can check the conjectured relations in a rigorous way by employing various integrability techniques (For a big review, see [4]). In particular, the AdS 5 ×S 5 superstring is classically integrable in the sense of kinematical integrability [5]. The integrability exposed in this case is, however, rarely exceptional and such a good property is not equipped with general examples of the AdS/CFT (or gauge/gravity) correspondence.
Holographic interpretations in the gauge/gravity correspondence are usually concerned with curved string backgrounds and hence classical motions of a string are described by non-linear equations. But most of the non-linear equations are not integrable and therefore the cases. Section 4 is devoted to conclusion and discussion. In Appendix A, the definitions and useful identities of elliptic functions are summarized. Appendix B explains a detailed computation of Melnikov function for a spinning string ansatz.

Melnikov's method
In this section, we shall give a brief introduction of Melnikov's method 2 . This is an analytical method to argue the existence of classical chaos. The key ingredient in this method is the so-called Melnikov function and its simple zero points are related to a discrete dynamical system, Smale's horseshoe, which generates a certain class of chaos.
Let us concentrate on a continuous dynamical system composed of two particles which has the Hamiltonian represented by a direct sum like H = H 1 (q 1 , p 1 ) + H 2 (q 2 , p 2 ) . The four-dimensional phase space is spanned by the coordinates q i (i = 1, 2) and the conjugate momenta p i . We are concerned with evolution of q i and p i in time τ . It is also supposed that H 2 has a homoclinic orbit ( Fig. 1) with a hyperbolic saddle point p . The existence of the homoclinic orbit is surely crucial so as to apply Melnikov's method. In other words, this condition somewhat restricts the applicability of this method. This system is integrable because the Hamiltonian is separable and one can solve the equation of motion as a one dimensional Hamiltonian system. Then, once a certain small perturbation is added to the integrable system, the integrability is broken and orbits near the 2 For a concise review, for example, see [28].
homoclinic one are likely to be chaotic (c.f., KAM theorem [29][30][31]). When the perturbation is turned on, the homoclinic orbit opens up in general as in Fig. 2, while the saddle point survives up to a slight deviation of the location (i.e., the saddle is not resolved but just moves a little bit). Then there still remain a stable manifold W s (p) and an unstable one W u (p) which meats at the saddle point. The stable manifold is defined as the set of points that get infinitely close to the saddle point by time evolution. Similarly, the unstable manifold is the set of points getting closer to the saddle point by the inverse evolution 3 . In general, the two manifolds are separated after the system is perturbed, while each of them is closed at the saddle point and forms a closed loop called a homoclinic orbit (i.e., the degeneracy of the stable and unstable manifolds) in the non-perturbed system. The idea of Melnikov's method is to argue whether the stable manifold and the unstable manifold have a point of transverse intersection, i.e., a transverse homoclinic point. When the system is perturbed by a small interaction term ǫH int where ǫ is a small real parameter, the orbits on these manifolds are expressed as perturbative expansions from the original ones by an infinitesimal parameter ǫ . Then, one can define the separation between those manifolds, which is proportional to the Melnikov function.
In particular, in the case that at least one transverse homoclinic point exists, a certain class of chaos is generated by Smale's horseshoe (see Fig. 3). The saddle point expands small

Melnikov's method in 4D systems -the standard case-
First of all, we will briefly introduce the reduction method of a four-dimensional system with q i (i = 1, 2) and the conjugate momenta p i to a two-dimensional system, as presented in [27]. The non-perturbed Hamiltonian H (0) is supposed to be a direct sum of two classically integrable systems like and the interaction between them is turned on as a small perturbation ǫ H int (q 1 , p 1 , q 2 , p 2 ) , where ǫ is an infinitesimal real constant. That is, the total Hamiltonian H is given by In the following, we assume that H 1 is periodic in q 1 and H 2 has a homoclinic orbit. The existence of the homoclinic orbit is crucial in Melnikov's method.

Introducing the action-angle variables
It is convenient to transform q 1 and p 1 into the angle-action variables Θ and I . Then the Hamiltonian H can be rewritten as H(Θ, I, q 2 , p 2 ) = H 1 (I) + H 2 (q 2 , p 2 ) + ǫ H int (Θ, I, q 2 , p 2 ). (2. 2) The Hamilton equations are given bẏ where the symbol "·" stands for the derivative with respect to τ . In addition, H int is also supposed to be periodic in Θ .
This dynamical system is essentially three-dimensional because the energy E is conserved.
Just for convenience, the unperturbed frequency is supposed to be positive, namely When ǫ is sufficiently small, Θ becomes a monotonically increasing function from the first equation in (2.3). Then, thanks to the τ -independence in the dynamical system, one can set Θ as a new time for the system by deleting τ .

Reduction method
The next task is to reduce the remaining degree of freedom I . By solving the energy conservation E = H(Θ, I, q 2 , p 2 ) , I is represented by Here L (0) and L (1) depend on E and hence these are also expressed as Then one can obtain a two-dimensional system which is composed of q 2 and p 2 with "time" Θ: Here the prime stands for the derivative with respect to Θ , and P (0) , P (1) , F (0) and F (1) are given by, respectively, Note that in the above computation we have used the expressions in (2.5) and the following relations:

Melnikov's method
We are now ready to introduce Melnikov's method.
To explain the role of the Melnikov function in this method, we employ the Poincaré map at Θ = Θ 0 with a perturbation parameter ǫ by P Θ 0 ǫ , where Θ 0 is a value modulo the period. By this Poincaré map, a point on a orbit at Θ 0 is mapped to another on the orbit at Θ 0 , modulo the period. Let us consider a small vicinity R around the transverse homoclinic point on the orbit. This can be mapped to a neighborhood of the saddle point U by finite numbers of forward or backward actions of P Θ 0 ǫ . Since these forward and backward actions can be connected at U and R , there is an integer N where (P Θ 0 ǫ ) N maps a point in U to another in U itself. This is Smale's horseshoe.
Therefore, what we have to do here is to show the existence of the transverse homoclinic point. To verify the condition to ensure its existence, let us calculate a separation between the stable and unstable manifolds passing through the saddle point, at Θ = Θ 0 . One can express a solution of (2.6) on the stable manifold as for Θ 0 ≤ Θ < ∞, and on the unstable manifold as Here the variables q lie on the normal to the unperturbed homoclinic orbit at Θ 0 for clarity. Then the separation at Θ 0 can be defined as where P (0) and F (0) are functionals of q . This is the projection to a vector (−F (0) , P (0) ) which is normal to the original homoclinic orbit. i.e., this vector is For convenience, let us introduce ∆ i (Θ, Θ 0 ) defined as Here P (1) and 2 (Θ − Θ 0 ) , and explicit Θ . Note that we have utilized the following relations: By employing the numerator in (2.11) , let us define the Melnikov function as Hence, as far as the leading order is concerned, the separation d(Θ 0 ) is proportional to the Melnikov function like By using ∆ i , the Melnikov function (2.14) can be rewritten as Here we have used the following relations: Note that near the saddle point, the derivatives of q 2 and p 2 are equal to zero. Hence Finally, the Melnikov function is represented by where q (0) 2 and p (0) 2 have been redefined as functions of τ , and { , } 2 is the Poisson bracket defined with q 2 and p 2 like According to Melnikov's method, the existence of simple zeros in M(τ 0 ) means that this system has a horseshoe for sufficiently small ǫ lying near the homoclinic orbit of (q 2 , p 2 ) on the energy surface H = E . More precisely speaking, the first example is a three-dimensional dynamical system 5 .

Examples of computing Melnikov functions
This case has not been covered in the previous subsection, but the computation scheme is almost the same. A central difference is that the reduction process is not necessary. It would really be instructive to study this case and to capture the essential idea of the Melnikov function.

1) a three-dimensional dynamical system
We consider a pendulum system with a periodic external force described by the Hamiltonian, and it depends on time τ explicitly. Then the Hamilton equations are given bẏ Here ǫ is an infinitesimal constant. Note that these equations can be regarded as those of reduced systems like (2.6) by replacing τ with Θ . This point is also concerned with the footnote 5.
When the perturbation is turned off (i.e., ǫ = 0) and the energy E is set to 1, this system exhibits a separatrix solution given by The pendulum simply oscillates around the stable fixed point q = 0 below E = 1 , while the pendulum rotates all the way above E = 1 .
Then by substituting the reduced equations (2.20) and the homoclinic solution (2.21) into the formula (2.17) , the Melnikov function can be computed as It has nontrivial simple zeros at τ 0 = π/2 + πl (l ∈ Z) . Hence, when E is close to 1 , the pendulum behaves in a weird manner -it switches oscillation and rotation randomly. This is nothing but the origin of chaotic motion. Melnikov's method tells us that this chaotic motion is generated by Smale's horseshoe.
2) a four-dimensional dynamical system As the second example, let us look at a simple pendulum-oscillator system with a small perturbation. This is a four-dimensional dynamical system composed of q 1 , q 2 and the conjugate momenta p 1 and p 2 . By taking ǫ H int = ǫ (q 2 − q 1 ) 2 /2 as a perturbation term, the perturbed Hamiltonian is given by Here ǫ is an infinitesimal parameter again and ω is a frequency of the oscillator.
When the perturbation is turned off (i.e., ǫ = 0), the system is completely separable: the one is a simple oscillator and the other is a pendulum. The variables of the oscillator q 1 and p 1 can be transformed into the action and angle variables, I and Θ . Then the Hamiltonian can be rewritten as (2.24) For the unperturbed system, the oscillation can be written as Θ = ωτ , and the separatrix solution of the unperturbed subsystem with (q 2 , p 2 ) is the same as in the previous.
When the perturbation is turned on, the pendulum and oscillator systems begin to interact each other weakly, and the pendulum feels the motion of the oscillator through the interaction term. This situation is quite similar to the dynamical system (2.20). Thus, as in Sec. 3.1, the reduction method leads to a two-dimensional dynamical system to which Melnikov's method can be applied.
In the present case, one can straightforwardly employ the formula (2.17). By putting (2.17), the Melnikov function can be evaluated as .
Hence this system also exhibits chaotic motions generated by Smale's horseshoe.

Melnikov's method -a non-direct sum case-
So far, we have considered Melnikov's method in the standard case. For later purpose, however, we need to generalize it slightly. The method we discuss here is a generalization of the reduction method in [27].
In this subsection, we are interested in the Melnikov function in more intricate dynamical systems. In general, one of H i 's may depend on the other one. Namely, we have in our mind the Hamiltonian of the following type where the entire H int is supposed to be periodic in τ hereafter. Then a system composed of q 1 and p 1 continues to feel a potential due to the energy of H 2 , even though the perturbation is turned off, i.e., ǫ = 0 . In such a system, one can formally change q 1 and p 1 to I and Θ (keeping q 2 and p 2 unchanged) . But I and Θ fail to be a canonical pair.
The Hamilton equations are given bẏ (2.27) Now we have to take account of an additional equation coming from the explicit timedependence of the Hamiltonian:Ḣ For a moment, let us consider the case with ǫ = 0 . Then the variables of the system become separable. When we concentrate on H 1 by setting H 2 as a constant h , the actionangle variables can be written as Here I and Θ satisfy and inversely, In general, Ω depends on H 2 and is assumed to be a positive constant. Furthermore, suppose that a ratio of Ω and the period of H int is rational for a reason explained later. Then, τ can be expressed as and H int is periodic in Θ to the leading order in ǫ . Since the τ -dependence appears only in H int , τ can be replaced by Θ/Ω as far as the quantities are computed up to the first order of ǫ . When ǫ is sufficiently small,Θ becomes positive; hence Θ may be regarded as a new time coordinate for the system again.
Now it is a turn to reduce the degrees of freedom from q i , p i , τ, H to q 2 , p 2 , H .
Let us first remove τ in H int by τ = Θ/Ω and the derivative with respect to τ byΘ d dΘ . The next step is to rewrite functions of q 1 and p 1 in terms of I, Θ and H 2 . Namely, the concerned quantities are rewritten as follows: , (2.33) Finally, one can eliminate I through the relation with the help of the following implicit function: The resulting expressions are given by where we have introduced the following quantities: (2.37) Now three variables: q 2 , p 2 and H are concerned with our analysis. Here there are some comments. The first is that H is a slow variable compared to the other two. The second is that P (1) and F (1) are periodic in Θ because the ratio of Ω and the period of H int has been taken to be rational. This periodicity is required in Melnikov's method; hence the rational condition has been supposed just below (2.32) . Finally when H satisfies appropriate conditions, the reduced system (2.37) exhibits a horseshoe for sufficiently small ǫ , as we will show below.
Because the energy H is not conserved, one needs to check whether the Poincaré map would map a volume element on a surface with constant energy to the same surface or another surface but close to the original. Let us assume that there is N where (P Θ 0 ǫ ) N has a horseshoe. N is related to ǫ, and the limit ǫ → 0 makes 1/N go zero. The average change of energy by ( Then, (P Θ 0 ǫ ) N maps the energy surface to itself to leading order in ǫ. This means that there exists a map from a vicinity near the saddle point to itself under this condition 6 .
Before defining the Melnikov function in this case, one needs to remove the explicit Θdependence in P (0) and F (0) . Since H and H 2 are nearly constant as far as the Poincaré section is concerned, one can expand those in ǫ as  Then, functions periodic in Θ are periodic in T as well if 2 , H (0) ), Θ) is periodic in Θ. One finds the equations of motion of q 2 and p 2 finally become where we have introduced the following quantities: Because this is a one-dimensional system, to which one can apply Melnikov's method, the Melnikov function is straightforwardly defined as This function is proportional to the separation of the stable and unstable manifolds at τ = τ 0 near the surface of H = H (0) because ∆H = 0 is assumed. Hence, Melnikov's method says that the existence of simple zeros in M(τ 0 ) indicates that this system has a horseshoe for sufficiently small ǫ lying near the homoclinic orbit of (q 2 , p 2 ) and near the energy surface H = H (0) .

A brane-wave deformation of AdS 5 ×S 5
In this section, we shall consider an application of Melnikov's method in the context of String Theory. In other words, we are concerned with an appropriate string background to which Melnikov's method is applicable.
An example of such backgrounds is a brane-wave type deformation of the AdS 5 ×S 5 background constructed in [26]. The background is composed of the metric and the fiveform field strength ds 2 = ds 2 AdSpp + ds 2 S 5 , where a satisfies a = (l + 2)/2 with l an integer greater than 0. The dilaton is constant and the other fluxes are zero. Here the metric of round S 5 is expressed as a U(1) fibration over CP 2 , where χ is the local coordinate on the Hopf fibre and ω is the one-form potential for the Kähler form on CP 2 . The metric of CP 2 and ω are given by where Σ a (a = 1, 2, 3) are defined as The remaining task is to fix an arbitrary function f (µ, θ) . A remarkable point is that the metric (3.1) is a pp-wave type deformation; hence the only non-trivial equation of motion is the (+, +)-component of the equations of motion for the metric. Thus all we have to do is to solve the following constraint: Here the overall constant can be absorbed by rescaling η ; hence we have simply set it to 1.
We will study the motion of a string propagating on the background presented above.
To show classical chaos in the bosonic sector, we will not touch on the fermionic sector. In addition, to make our analysis simpler, we will adopt two ansätze. In the following, we will compute Melnikov's function under each of the ansätze.

A reduction ansatz: an oscillating string
In this subsection, we will consider an oscillating string ansatz.
Let us perform a coordinate transformation [33]: Note here that the points with mx + = nπ (n ∈ Z) are singular. After this transformation, the system gets restricted and a certain part of the deformed AdS background is magnified.
The resulting metric is given by Note here that the undeformed AdS 5 is also deformed to a pp-wave type metric.
In the following, we are concerned with a string theory defined on this background. We will concentrate on the bosonic part and study the classical action of Polyakov type with the light-cone gauge.
For simplicity, let us take the following ansatz: Here n ∈ Z is a winding number along the φ-direction. Note that this ansatz is consistent with the original equations of motion.
From now on, let us set a = 3 for simplicity and the function f (µ, θ) is given in (3.4).
This is just because the case with a = 2 is integrable under the ansatz (3.7) and hence the case with a = 3 is the simplest and non-integrable, though the case with a > 3 would be non-integrable as well.
Then the light-cone Hamiltonian is simplified as 7 This is nothing but a coupled pendulum-oscillator system. It should be remarked that the Hamiltonian (3.8) has the explicit τ -dependence due to the singular coordinate transformation (3.5) . The unperturbed Hamiltonian is where H 2 is the Hamiltonian of a pendulum system, The perturbation term is given by where η is assumed to be infinitesimal.
The resulting Hamiltonian is expressed in terms of (I, Θ) as follows: 8 Here we have introduced the energy E . From the equation of motion of (I, Θ) , Now Z is written as a function of τ through the transformation (3.12) , In order to calculate the Melnikov function, the separatrix solution of the pendulum system H 2 is necessary. If one takes then P (0) and F (0) in (2.43), in this system, are given by 17) and the equations of motion for the pendulum system with time T by which can easily be solved by µ = 2 Arctan tanh nL 2 4 T .
By integrating (3.16) , the following solution is obtained, The new period (2.42) becomes T period = π/ √ 2h . Now, with T (τ ) , the separatrix solution can be rewritten as Note that its energy h is h (0) = n 2 L 4 /8 .
One must check the energy condition (2.39) . The equation of the energy iṡ Its average on the homoclinic orbit is given bẏ Here H lc is an approximation in the region sufficiently near We also use the following expression, which is sufficiently small.
Then, the change of the energy by a Poincaré map with N shifts of time by the period 2π/m is estimated as sin(2mτ 1 ) , (3.22) at N → ∞ . In the above calculation, we have omitted the subleading order in A and used an approximate relation The equation (3.22) suggests that the total energy generally changes monotonically. Eventually, we can expect this change from the beginning because the frequency of the factor oscillating explicitly in the perturbed Hamiltonian coincides with the frequency of unperturbed Z . Hence, it possibly provides a "resonance" source. The condition (2.39) is however satisfied when If a string with this condition starts its trajectory from the vicinity near the homoclinic point, then the energy will not change drastically when the string comes back to the vicinity near the homoclinic point. Here µ ) . This Poisson bracket is reduced to Then the Melnikov function is calculated as To derive the above expression, we have used the following approximation of T and τ This Melnikov function has simple zeros at least when A is sufficiently small for any τ 1 . Thus the result indicates the existence of chaos generated by a Smale's horseshoe for trajectories satisfying ∆H lc = 0 , that is, Although such trajectories are quite rare, trajectories with ∆H lc ∼ 0 remain in a small range of the energy after performing each of the Poincaré maps if the observation time is taken to be sufficiently short.

A reduction ansatz: a spinning string
In this subsection, we consider a spinning string ansatz.
In order to obtain a desired ansatz, we convert the Poincaré coordinate of the AdS part to the global coordinate by performing the following transformation: 1 z = L(cosh ρ sin t + sinh ρ cos ξ cos η) , L z x ± = L √ 2 (cosh ρ cos t ± sinh ρ cos ξ sin η) , L z x 1 = L sinh ρ sin ξ sin ζ , L z x 2 = L sinh ρ sin ξ cos ζ . (3.27) As in the previous subsection, we will study the bosonic part of the Polyakov action.
As a remark, we will take here the spatial direction of the string-world-sheet to be infinite (i.e., an infinitely long string), rather than the usual closed string. With the gauge condition t = κτ , let us consider the following ansatz: Because we are considering an infinite string, there is no need to impose a boundary condition on µ(σ) . The relevant part of the metric is rewritten as For simplicity, we concentrate on the case with a = 2 hereafter. The function f (µ, θ) is given in (3.4) . Note here that this case is not integrable under the ansatz (3.28) in comparison to the previous ansatz (3.7) .
Then the Lagrangian density is given by where the prime here is defined as the derivative with respect to σ . L has been absorbed into κ , σ , ω i , and η so that ρ and µ are canonically normalized. Also, we impose a condition on κ and ω 1 like so that the system should be bounded.
If one regards σ as time, "Hamiltonian," which governs the σ-dependence of ρ and µ , can be defined as Thanks to the condition (3.31) , the second term is positively definite. The classical solution should satisfy the Virasoro constraints, Note that this condition (3.33) is exactly the same as the equation for a Noether charge for the translation along the σ-direction in (3.30) where the charge is κ 2 /2 . Thus there is not any contradiction as far as the total "energy" is κ 2 /2 . Moreover, as far as the Virasoro constraints are satisfied, the ansatz (3.28) is consistent with the original equations of motion.
In the following, we treat σ like time and use the symbol "τ ", instead of σ .
This system is regarded as a coupled pendulum-oscillator system. The Hamiltonian of an oscillator system is given by and that of a pendulum system is While the latter system is a usual pendulum one, the former is oscillated in a squared hyperbolic sine potential. The interaction term is written as where η is assumed to be infinitesimal. When η = 0 , the Hamiltonian system is a direct sum of two systems where one is an oscillator and the other has a homoclinic orbit. Hence to compute the Melnikov function, we can follow the standard method described in Sec. 2.1.
The solution of the oscillator system with its energy E is where the function am(x, k) is the Jacobi amplitude function, defined as the inverse of the elliptic integral of the first kind F (x, k) as The last expression in (3.37) is obtained as an approximation when E is sufficiently small.
Let us compute the Melnikov function by using the separatrix solution of the pendulum system, The energy of this solution is ω 2 2 /8 . As the total energy is κ 2 /2 , the value of E is estimated as (4κ 2 − ω 2 2 )/8 , if the interaction term is negligible.
The present case does not require the energy condition because the system is not disturbed by an external force. Hence the Melnikov function defined in (2.17) is given by where H where we have used the following identities 9 : Note here that due to the condition (3.31) , ω 1 is real and s < 1 .
The final step is to perform the integration with respect to τ . It is really complicated and messy; hence the detailed calculation is explained in Appendix B. After a lengthy calculation, the Melnikov function is given by where we have introduced the following quantities and the functions g 1 and g 2 are defined as, respectively, For the definition of elliptic functions, see Appendix A.
The factor sd(ω ′ 1 τ 0 ) cd(ω ′ 1 τ 0 ) in the first term of (3.42) has non-trivial zeros periodically at τ 0 = Kl/ω ′ 1 , where l is integer. Then g ′′ 1 ((2k +1)π/ω 2 ) also depends on τ 0 and has a non-zero value at general τ 0 . Although Im[g 2 (τ 0 + ((2l + 1)K + i(2k + 1)K ′ )/ω ′ 1 )] in the second term is not periodic in τ 0 , its absolute value decreases exponentially as τ 0 goes to infinity. Then it gets smaller than the maximum of the absolute value of the first term when τ 0 is sufficiently large. Furthermore, when E is small enough, the second term is negligible, and it is easier to see simple zeros as explained in Appendix B. Thus the Melnikov function has simple zeros at sufficiently large τ 0 , and hence this system exhibits classical chaos associated with Smale's horseshoe.

Conclusion and discussion
In this paper, we have considered an application of Melnikov's method in the context of the gauge/gravity correspondence. In particular, we have presented a string background to which Melnikov's method is applicable. This is a brane-wave type deformation of the AdS 5 ×S 5 background. By employing two reduction ansätze, we have studied two types of coupled pendulum-oscillator systems. Then the Melnikov function has been computed for each of the systems and the existence of chaos has been shown analytically. From more technical aspects, it is significant to consider a generalization to include the friction. Recall that an external force was induced with the oscillating ansatz. If the friction could be introduced as well, one can realize the system with the energy injection and dissipation. Such a system is not a conserved system but dissipative. The chaos in the conserved systems is well described by the KAM theorem [29][30][31] and its characteristic is profoundly understood. The chaos in the dissipative systems, however, has a richer structure such as strange attractors. It is really intriguing to realize strange attractors on the stringtheory side and consider the physical interpretation of its gauge-theory counter part.
It would also be nice to look for other string backgrounds to which Melnikov's method is applicable. We have just presented one example here. It is interesting to consider some applications of Melnikov's method for black hole solutions in relation to the information loss process. It may be a good issue to clarify the relation between the event horizon and the Melnikov function.
We hope that Melnikov's function would provide a new tool in studying the gauge/gravity correspondence.
Then the Melnikov function is given by where we have introduced the following new quantities: The integration in (B.2) can be evaluated by summing up the contributions coming from the poles of A(ω 2 τ ) and B(ω ′ 1 (τ + τ 0 )) . In the following, let us evaluate the poles from A and B , respectively.

1) the contributions coming from the poles of A
First of all, we shall evaluate the contributions coming from the poles of A .
Hence, to compute the residue, one needs to evaluate the second-order derivative of B(x) .
Because A(ω 2 τ ) is an odd function and the integration range runs from −∞ to ∞ , only the odd part of B(ω ′ 1 (τ + τ 0 )) can contribute to the integral. Hence the calculation of derivatives of B(x) is simplified to some extent. In addition, by employing a relation (A. 3) of the Jacobian elliptic function nd , one finds that the odd part of B(x + x 0 ) is given by 4s 2 nd(x 0 ) sd(x 0 ) cd(x 0 ) nd(x) sd(x) cd(x) (1 + s 2 s ′2 sd(x 0 ) sd(x)) 4 nd(x 0 ) nd(x) + s 4 sd(x 0 ) cd(x 0 ) sd(x) cd(x) .
Note here that all of the second arguments of the elliptic functions are given by s (but those are omitted just for simplicity).
2) the contributions coming from the poles of B The next task is to evaluate the contributions coming from the poles of B .
The Jacobian elliptic function nd(x, s) has two single poles inside the fundamental unit cell. Two periods of nd(x, s) are given by K and 4iK ′ , where K = K(s) and K ′ = K(s ′ ) .
Note here that s < 1 and s ′ < 1 by construction in the present case; hence K and K ′ are real numbers.
Depending on the value of the residue, there are two kinds of poles. The first residue is given by −i/s ′ for the poles at x (l,k) 1 ≡ (2l + 1)K + i(4k + 1)K ′ (k, l ∈ Z) .
Hence the asymptotic behaviors near these poles are given by Now that B(x) has quadruple and double poles at x = (2l +1)K +i(2k +1)K ′ , it is necessary to compute the third-order and first-order derivatives of A(x) in order to evaluate the residue around these poles.

Doing the integration
Let perform the τ -integration in (B.2). To perform this integral, we need to consider a contour integral. Generally, the contour depends on the behavior of the integrand. The contour is closed on the upper-half plane with a semicircle in the counterclockwise direction when the integrand of (B.2) goes to zero in the limit τ → ∞ . Similarly, the contour should be closed on the lower-half plane with a semicircle in the clockwise direction depending on the behavior of the integrand in the limit τ → ∞ . By denoting these contours by C u and C l , respectively, the Melnikov function (B.2) can be rewritten as M(τ 0 ) = κ 2 ω 2 8 Cu + C l dτ A(ω 2 τ )B(ω ′ 1 (τ + τ 0 )) .

Approximation works well
When E is sufficiently small, one can readily derive an approximate form of the Melnikov function. The leading term is given by Now that the τ -integration can be performed easily, the resulting expression is given by Then it is obvious that M(τ 0 ) has non-trivial simple zeros at Thus the conclusion that classical chaos exists does not change even with a small energy approximation.
Note here that the approximation is made for the integrand before performing the τintegration. One can also reproduce the same expression by expanding the exact formula (3.42) in terms of small E . That is, the order of the τ -integration and the small E expansion is irrelevant to the final result.