Special functions created by Borel-Laplace transform of Hénon map

We present a novel class of functions that can describe the stable and unstable manifolds of the Henon map. We propose an algorithm to construct these functions by using the Borel-Laplace transform. Neither linearization nor perturbation is applied in the construction, and the obtained functions are exact solutions of the Henon map. We also show that it is possible to depict the chaotic attractor of the map by using one of these functions without explicitly using the properties of the attractor.


Introduction
The Hénon map [1] is a model that exhibits the same property as the Lorenz system [2]. It was developed to describe the atmospheric turbulence on the basis of the Navier-Stokes equation. Since then, numerous researchers have used this map as the simplest model to describe the chaotic behavior in various dissipative systems. The strange attractor, which appears as the closure of the unstable manifold of the Hénon map, is often considered a model of unpredictable motion as it is wellknown that the strange attractor's trajectory is nondeterministic. In this paper, we present an algorithm and a concrete functional form for describing the unstable manifold, as well as the stable manifold, of the Hénon map. We use the Borel-Laplace transform and asymptotic expansions to construct these functions. Hakkim and Mallick [3] calculated the separatrix splitting in a conservative system using the matched asymptotic expansions and Borel summation taking the standard map as an example. Tovbis et al. [4,5] and Nakamura and Hamada [6] discussed the relation between the Borel-Laplace transform and the Stokes phenomenon using the Hénon map by selecting parameters for which the system becomes Hamiltonian, and calculated the splitting angle. The works [7,8,9] are based on the idea of asymptotic expansions beyond all orders, which is used to capture the exponentially small effects.
In addition to these singular perturbative approaches, there exist some alternative methods to compute exponentially small terms. Voros [10] proposed the exact WKB method to solve the quantum oscillator andÉcalle [11,12,13] proposed the 2 CHIHIRO MATSUOKA AND KOICHI HIRAIDE resurgent analysis [14] to perform the resummation of divergent power series given by asymptotic expansions in differential equations. These methods enable us to calculate exponentially small terms without the perturbative approach. Gelfreich and Sauzin [15] applied the resurgent analysis to the Hénon map and calculated the splitting angle very accurately. An essential aspect of these works is that the systems considered can be reduced to a differential equation. In this paper, we consider the case of a system that cannot be reduced to a differential equation, i.e., a truly dissipative system. The purpose of our study is to construct special functions that can describe the stable and unstable manifolds in such systems and depict these manifolds with asymptotic expansions derived from the functions. We emphasize that no perturbative approaches, including linearizations, are used for the construction.
This paper is organized as follows. In Sec. 2, we recall the definition of the Borel-Laplace transform of the Hénon map. In Sec. 3, we present the special functions that describe the stable and unstable manifolds of the Hénon map. Using the asymptotic expansions of the functions described in Sec. 3, we depict the stable and unstable manifolds in Sec. 4.

Laplace transform of Hénon map
The Hénon map is a polynomial diffeomorphism f : where a and b are complex parameters [1]. The fixed points of this map are given as follows: It is obvious that when a fixed point P = (x f , y f ) is a saddle point, the two eigenvalues, α 1 and α 2 , of the derivative of f at P are the solutions of the quadratic equation α 2 + 2ax f α − b = 0, where α 1 and α 2 satisfy 0 < |α 1 | < 1 and |α 2 | > 1, respectively. We define the stable and unstable manifolds at the saddle point P by W s (P ) = {Q ∈ C 2 |f n (Q) → P as n → ∞} and W u (P ) = {Q ∈ C 2 |f n (Q) → P as n → −∞}, respectively. We will construct analytic functions that describe these manifolds by using the Borel-Laplace transform.
After shifting the fixed point P to the origin by x → x + x f and y → y + y f , we introduce a parameter t ∈ C that parameterizes the stable and unstable manifolds via the equation f [x(t), y(t)] = [x(t + 1), y(t + 1)]. Then, the Hénon map (2.1) yields the following difference equation where λ = −2ax f , and y(t) is given by y(t) = bx(t − 1). We remark that the boundary condition of x(t) in (2.3) is determined by selecting one of the two saddle points present among the fixed points given in (2.2).

SPECIAL FUNCTIONS OF HÉNON MAP 3
Now, we determine the solution of (2.3) in the form of a Laplace integral, given by where the path γ depends on the position and form of singularities on X. Substituting (2.4) into (2.3), we obtain the following integral equation for the Borel transform, X(ζ), of x(t): where * denotes convolution defined by A is defined by A(ζ) = e −ζ − λ − be ζ , and C is a constant satisfying the relation A(0)X(0) = C. We assume that X(ζ) can be expanded as a Taylor series in a neighborhood of the origin, and we defineX by Substituting (2.6) and (2.7) into (2.5), we have AX + 2aa 0 * X = W, (2.8) where W = W 0 − aX * X and W 0 = −aa 2 0 ζ − a 0 A + C. In order to develop an algorithm to determine the solutionX(ζ), we formally expandX(ζ) and W (ζ) using a parameter σ as and We then substitute these expansions into (2.8). Then, we have for each order of σ, whereX n (n = 1, 2, · · · ) is given as the solution to (2.8): In (2.10), the zeros of A(ζ), given by for k ∈ Z, are singularities of the functionX n , where θ + = arg α 1 and θ − = arg α 2 . We call these singularities, derived from the zeros of A(ζ), first singularities. The singularities ζ + k and ζ − k coincide with each other and appear on the imaginary axis for a = 1 and b = −1 (|α 1 | = |α 2 | = 1, θ + = θ − ), i.e., the Hénon map is reduced to a Hamiltonian system [4,5,6,15]. For the dissipative case, where in the Hénon attractor [1] appears, the parameters are assumed to be a = 1.4 and b = 0.3, for which θ + = 0 and θ − = ±π. When |α 1 | < 1 (|α 2 | > 1), i.e., for the stable (unstable) manifold, all singularities appear on the Re(ζ) > 0 (Re(ζ) < 0) plane. The two functions describing the stable and unstable manifolds are independent of each other. We remark that new singularities can be created by convolution when Re(ζ ± k ) = 0. After some lengthy calculations, we can prove that from the relation C = A(0)X(0).

Borel transform of Hénon map
The algorithm for obtaining the solutionX is as follows. In order to simplify the discussion, we select a first singularity ζ 1 ≡ ζ ± 0 . Taking into account that ζ = 0 is a regular point ofX, we perform the analytic continuation ofX starting from the origin. First, we expand A, F 0 , and W 0 in (2.10) in a neighborhood of the origin and calculate the integral inX 1 . In this calculation, ζ is bounded by ζ 1 : |ζ| < ζ 1 . Then, using thisX 1 , we calculate the convolution W 1 from the relation given by (2.9). Substituting W 1 into (2.10), we obtainX 2 . Performing these calculations repeatedly, we obtain the solution up toX n (n = 1, 2, · · · ) in a neighborhood of the origin. Next, expanding A, F 0 , and W 0 in a neighborhood of the first singularity ζ 1 , we repeat the same calculations forX n and W n (n = 1, 2, · · · ). The obtainedX can be continued analytically up to |ζ| < 2|ζ 1 |. Further, we repeat these iterative calculations in neighborhoods of the higher-order singularities ζ = N ζ 1 (N = 2, 3, · · · ) derived from ζ 1 . All singularities N ζ 1 (N ≥ 2) are created by convolution.
If a, b ∈ R, then the coefficients b when N = 1.
In order to perform the Laplace transform of X(ζ) = a 0 +X(ζ), we have to uniformize the multivalued function X and construct a univalued function. We do not discuss the uniformization method in detail here. The resulting Laplace transform x(t) is given by the following theorem. and X R (ζ, N ) is the uniformized function of X(ζ), which depends on the N -th singularity N ζ 1 . The argument θ in the integral (3.6) is the angle connecting the origin and N ζ 1 .
The Laplace transformed function x(t) in (3.6) is an entire function, i.e., this function does not have singularities on the t-plane, except at infinity. We emphasize that the Laplace transform (3.6) is performed over (−∞, ∞), and not [0, ±∞). The estimate (3.5) guarantees the existence of this infinite integral, i.e., due to the existence of the term |α 1 | N log N (|α 1 | < 1) in (3.5), the exponential growth at t = −∞ when Re(ζ) > 0 (t = ∞ when Re(ζ) < 0) is suppressed by taking N → ∞. This fact enables us to depict the stable and unstable manifolds with the asymptotic expansion; as shown in the next section.

Asymptotic expansions and stable and unstable manifolds
In this section, we describe the stable and unstable manifolds of the Hénon map by using the asymptotic expansion of x(t), as given in (3.6). The asymptotic expansion of x(t) is provided by the following theorem.
Theorem 5. The asymptotic expansion of x(t), as given in (3.6), has the form where the + and − signs correspond to the stable and unstable manifolds, respectively.
Due to the estimate (3.5), the asymptotic expansion (4.1) converges for all t ∈ C except for the origin t = 0; therefore, this expansion is not formal. To depict the stable and unstable manifolds, we apply the following property of the dynamical system: where the upper (lower) sign in ∓ (or ±) corresponds to the stable (unstable) manifold and n is a large positive integer. When Re(t) > 0 and Re(t) − n < 0, the curve (x, y) = [x(t − n), bx(t − n − 1)] describes the stable manifold. However, when Re(t) > 0 and Re(t) − n > 0, the curve [x(t − n), bx(t − n − 1)] tends to the fixed point P due to the relation lim n→∞ f n [x(t)] → P . When Re(t) < 0 and Re(t) + n > 0, the curve (x, y) = [x(t + n), bx(t + n − 1)] describes the unstable manifold. On the other hand, when Re(t) < 0 and Re(t) + n < 0, the curve [x(t + n), bx(t + n − 1)] tends to the fixed point P since lim n→−∞ f n [x(t)] → P . We remark that the asymptotic expansion given by (4.2) is not the one obtained by the shift t → t ∓ n in the expansion of x(t) in (4.1). Figures 1 and 2 show the stable and unstable manifolds obtained by using the asymptotic expansion (4.2). Here, ζ 1 = − log |α 1 | and ζ 1 = − log |α 2 | − πi for These values suggest that x(t − n) in (4.2) (stable manifold) is real when t is real, while x(t + n) (unstable manifold) is complex when t is real. Regarding the argument θ in the Laplace transform (3.6), θ = 0 for the stable case and θ = tan −1 (−π/ log |α 2 |) for the unstable case. The coefficient b In Fig. 1, we can take the parameter t to be real. The blue and red solid lines depict the region 19990.5 ≤ t ≤ 20000 in (x, y) = [x(t − n), bx(t − n − 1)] and −20010 ≤ t ≤ −20000 in (x, y) = [x(t + n), bx(t + n + 1)], respectively. The latter cannot be depicted continuously from the former when t is real. For the blue and red lines, 40000 and 5000 plot points, respectively, are considered. We remark that (x, y) = [x(t − n), bx(t − n − 1)] ((x, y) = [x(t + n), bx(t + n + 1)]) is continuous with respect to real t (see the discussion for the unstable case below). The blue and red lines in the first quadrant in Fig. 1 (a) extend up to the order of (x, y) ∼ (10 5 , 10 10 ), while the (blue) line in the second quadrant extends to approximately (x, y) ∼ (−7.34, 61.8). The oscillatory motion exhibited by the blue line spans these two regions.The value of y in the first quadrant increases with −(t − n), and hence, tends to infinity as t − n → −∞. The red line in the first quadrant is expected to tend to infinity without the oscillation.
For the unstable case, x(t + n) is not real-valued when t is real. Therefore, in order to depict the unstable manifold, we cut the manifold (x, y) = [x(t + n), bx(t + n − 1)] at points for which the imaginary part of (x, y) becomes zero and project the resulting section onto R 2 . The algorithm for calculating the points for which Im(x, y) = 0 is as follows. We first note that from the asymptotic expansion (4.2), we can easily see Im(x) = 0 when t is integer. Taking this into account, we first divide the complex t-plane into rectangles such that an integer is set to one of the vertices. Then, we identify the points t such that the distance from the real axis satisfies the condition s(t) ≡ x i (t) 2 + y i (t) 2 = 0, starting from the neighborhood of one integer to the next integer in turn, where x i (t) = Im[x(t + n)] and y i (t) = Im[bx(t + n − 1)]. One integer is divided further into fine grid points. We divide one integer into 10 6 parts, i.e., there exist 10 6 grid points between one integer and the next integer in the real t direction. We also set 10 6 grid points between two adjacent integers in the positive imaginary t direction. Figure 2 shows the unstable manifold obtained by substituting the points t satisfying s(t) = 0 into [x(t + n), bx(t + n − 1)]. In this example, the only points t that satisfy s(t) = 0 within the precision are integers, and the other points are selected such that s(t) takes minimum values within a rectangle having integers as vertices. The minimum points t m (m = 1, 2, · · · ) obtained with this method are periodic and are given by t m = m( t r + i t i ), where t r = 2.5 × 10 −5 and t i = 1.2 × 10 −4 . That is, the minimum points appear on the line with slope t i / t r = 4.8.
For Fig. 2 (a), the integers that satisfy −20000 ≤ Re(t) ≤ −19980 are taken. We set t(∈ C) as t = t r + it i , and detect the points which satisfy |s(t)| ≤ 10 −5 (we call these minimum points) over the region −20000 ≤ t r ≤ −19980 and 0 ≤ t i ≤ 1. In order to carry out this detection, one integer is divided into 10 6 grid points, i.e., the above rectangular region (−20000 ≤ t r ≤ −19980 and 0 ≤ t i ≤ 1) is composed of 20 × 10 6 × 10 6 grid points. The number of points detected exceeds 10 5 , and all of these points form the unstable manifold. If we adopt all these points to depict the unstable manifold, the points are too 'dense' and the fine structure in the unstable manifold 'collapses'. Therefore, we 'thin' these minimum points to 21407 (see below), following the rule described in the next paragraph. Mathematically, these manipulations including the method in next paragraph are nonessential. When we depict the points (Re(x), Re(y)) satisfying such s(t), Fig. 2 is obtained.
For integers with 0 ≤ |Re(t) + n| ≤ 10, we select m = 100 in t m , while we select m = 10 and m = 1 when 11 ≤ |Re(t) + n| ≤ 16 and 17 ≤ |Re(t) + n| ≤ 20, respectively. Due to the round-off error, the minimum points t m are difficult to compute as |Re(t)+n| increases and t approaches the next integer within an interval. Figure 2 (a) is generated with 21407 points. The unstable manifold cannot be visualized by interpolating the points.
The solutions of the Hénon map x(t) are independent when a path in the Laplace transform is fixed and there exist countable infinite paths for one set of (a, b). Once the unstable manifold is described using the function, we can calculate the entropy in the system from the length of the curves in the unstable manifold [16]. Therefore, the function that we obtained here can be used to study the thermodynamics of dissipative systems.