On the escape from potentials with two exit channels

Abstract

The aim of this paper is to investigate the escape dynamics in a Hamiltonian system describing the motion of stars in a galaxy with two exit channels through the analysis of the successive intersections of the stable and unstable manifolds to the main unstable periodic orbits with an adequate surface of section. We describe in detail the origin of the spirals shapes of the windows through which stars escape.

Introduction

When we look up at the night sky, the stars appear fixed but, in fact, stars move through our galaxy although the changes in their positions on the sky are too small and slow to be appreciated over human timescales. These changes in the stars positions were first discovered in the eighteenth century by Edmond Halley, through the comparison of stellar catalogues from his time to a catalogue compiled by Hipparchus some two thousand years before. Nowadays, the motion of a star can be detected with a few years’ worth of high-precision astrometric observations. Some questions involving the nature of the motion of a star in a galaxy may arise: What kind of motions are possible in a galactic potential? How can we determine if a star can escape or not from the galaxy? The last question has focused the interest of many researchers during the last decades^{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}. The most of these efforts have been devoted to the numerical exploration of galactic systems by integrating sets of initial conditions regularly distributed in dense grids defined in an adequate surface of section. For each initial condition, the equations of motion must be integrated along a maximum time of integration or until the star leaves the potential. The maximum integration time is taken large enough to be sure that all orbits have enough time to escape. Even though this procedure gives information about the location of the basins of escape through the different exits in the surface of section considered, they can not describe the structure of the limiting curves of the regions of escape. In 1990, Contopoulos⁹ explored the phenomenon of escape in two conservative Hamiltonian systems representing the central part of a perturbed galaxy, with quadruple symmetry and four openings to infinity, and one axis symmetry and two openings to infinity, respectively. In both cases, there is an unstable periodic orbit (called “Lyapunov” orbit) located at every opening of the potential well. Contopoulos found that the asymptotic curves of the various unstable periodic orbits intersect at a complicated set of homoclinic and heteroclinic points, and govern the escape to infinity from the potential well. Navarro and Henrard¹² analyzed the structure of the windows of escape for the galactic system presenting four openings in the potential well, describing deeply the complex layered nature of the spirals obtained in Poincaré sections. There are “first-order” infinite spirals as those described by Contopoulos, but also “second order” spirals which are composed of an infinity of layers, “third order” spirals formed by an infinity of second order spirals, and so on. In 2017, Navarro¹³ investigated the shape of the windows of escape to infinity from a galaxy modeled by a bi-symmetrical harmonic oscillator perturbed with quartic terms, finding the same mechanism governing the escape from the potential well.

Even though the infinite layered structure of the spirals is well-known, it has never been unveiled with enough detail. The aim of this paper is to analyze the composition of this complex structure of infinite spirals to shed light on the geometry of the windows through which stars may escape from a galactic potential. This way leads to a better understanding of the properties of the escape of a particle from a galactic system. When the energy h of a star in a galaxy is larger than a certain critical value h_c, the star may escape to infinity. But not all stars with energy larger than the critical value escape from the galactic potential. There are many stars which remain trapped around a stable periodic orbit. In order to determine the escaping sets we must calculate the asymptotic manifolds to the Lyapunov orbits in the phase space. The intersection of the stable manifold with a surface of section originates a closed curve called a “limiting asymptotic curve”. Any orbit with initial condition inside this limiting curve escapes to infinity without intersecting again the surface of section. If we consider the successive intersections of the asymptotic manifold with the surface of section, we obtain a complicated structure, that we will describe in detail throughout this paper.

One way to study the problem of escapes from a galaxy is by using a simple model. We will analyze the following galactic potential,

$$W(x,y)=-\frac{1}{2}({x}^{2}+{y}^{2})+\mu x{y}^{2},$$

(1)

where μ > 0. This potential presents two channels of escape. In 2014, Zotos¹⁶ performed a numerical exploration of this system, locating the basins of escape towards the different windows and unveiling their relation with the times of escape of the orbits. The Hamiltonian corresponding to potential (1) reads

$$H=\frac{1}{2}({\dot{x}}^{2}+{\dot{y}}^{2})-W(x,y),$$

(2)

that is

$$H=\frac{1}{2}({\dot{x}}^{2}+{\dot{y}}^{2})+\frac{1}{2}({x}^{2}+{y}^{2})-\mu x{y}^{2}.$$

(3)

The equations of motion are given by

$$\begin{array}{ccccc}\ddot{x} & = & {W}_{x}(x,y) & = & -x+\mu {y}^{2},\\ \ddot{y} & = & {W}_{y}(x,y) & = & -y+2\mu xy.\end{array}$$

(4)

First, we will compute the stable and unstable manifolds to the periodic orbits located at the two exit channels of the potential well, in order to show that these structures, composed of infinite spirals, governs the rate of escape. We will demonstrate that there are different levels of infinitely winding spirals, each of these level embedded into the next one. It is also necessary to understand how the geometry of these different levels of spirals are interrelated to have a deep knowledge about the shape and size of the regions of the phase space leading to escape. We will proceed by calculating the intersections of the asymptotic manifolds with a surface of section, as it has been done previously by Contopoulos and coworkers^9,10 and Navarro and Henrard¹². In our investigation, we will take as surface of section the plane y = 0.

Curves of Zero Velocity

The energy of the system is a constant of motion, so it is possible to obtain an equation relating the velocity of the particle to its position. If we consider a fixed value of the energy, we can construct the curves in the plane on which velocity vanishes. These are the so-called curves of zero velocity. Figure 1 shows some contours of the curves of zero velocity for several values of the energy. These curves describe the boundary between regions of possible motion and forbidden regions. Depending on the value of the energy, the system exhibit different types of behavior. There is a critical value of the energy (h_c) such that, for larger values of h, the curves of zero velocity are open and test particles may escape. For h < h_c, the curves are effectively closed and particles originated from the central potential region are confined there. If the energy of a star is larger than h_c, the curves of zero velocity present two openings (Fig. 1). At the openings we find two unstable periodic orbits, the so-called Lyapunov orbits. Due to the symmetries of the potential, the well opens up, for values of the energy larger than the critical value h_c, at two places along the lines defined by $y=\pm \sqrt{2}x$.

Figure 1

Curves of zero velocity for μ = 3, and different values of the energy: h₁ = 0.012, h₂ = 0.01504, h₃ = 0.062802 and h₄ = 0.11760. As the energy of the system grows, the windows of the potential well become wider. The guardian orbit is the almost straight line barring the opening.

We have computed the critical value by using the method described by Caranicolas¹⁸, obtaining

$${h}_{c}=\frac{1}{8{\mu }^{2}}.$$

The curves of zero velocity of the system (4) are given by the relation

$$f(x,y)=h-\frac{1}{2}({x}^{2}+{y}^{2})+\mu x{y}^{2}=0.$$

(5)

The points of the xy plane where the curve of zero velocity opens are the saddle-points of (5). To compute them, we must first solve the system

$$\frac{\partial f}{\partial x}=\mu {y}^{2}-x=0,\,\frac{\partial f}{\partial y}=2\mu xy-y=0.$$

(6)

The solutions of (6) are the critical points of (5). The saddle-points of (5) are the solutions of (6) satisfying the condition

$$S=(\frac{{\partial }^{2}f}{\partial {x}^{2}})(\frac{{\partial }^{2}f}{\partial {y}^{2}})-{(\frac{{\partial }^{2}f}{\partial x\partial y})}^{2} < 0.$$

In addition to the trivial solution x = y = 0, there are two saddle-points in the xy plane, given by

$$x=\frac{1}{2\mu },\,{y}^{2}=\frac{1}{2{\mu }^{2}}.$$

(7)

The substitution of the solution (7) into Eq. (5), leads to the critical value

$${h}_{c}=\frac{1}{8{\mu }^{2}}.$$

In our analysis, we will take μ = 3. The critical value of the energy associated to this value of μ is ${h}_{c}=1/72\simeq 0.0138888888$. For each larger value of h, there is a Lyapunov orbit bridging the opening, bouncing back and forth between the two “walls” of the pass (see Fig. 1). The most important property of Lyapunov orbits is that any orbit crossing them outwards moves always outwards and escape from the system. These unstable Lyapunov orbits can be considered as the guardians of the pass between the inner region and the outside region, as the asymptotic orbits to them compose the limits of the sets of escaping orbits.

The aim of this investigation is to clarify the properties of the escape for values of the energy close to the critical value h_c. To this end, we will analyze the geometry of the asymptotic curves to the Lyapunov orbits for a value of the energy close enough to h_c.

Computation of the Stable and Unstable Manifolds

As we are interested in the determination of the escaping sets, we must compute the asymptotic manifolds to the Lyapunov orbits. We have obtained the initial conditions of the upper Lyapunov orbit for a value of the energy slightly larger than the critical value through a geometric approach (Table 1). Then, we have applied the procedure by Deprit and Henrard¹⁹ to compute the family of periodic orbits taking the energy as parameter. The result of this continuation is shown in Fig. 2.

Table 1 Initial conditions of the periodic orbit.

Figure 2

Continuation of the family of unstable periodic orbits guarding the potential well.

As mentioned in the introduction, the stable asymptotic curves to the “guardian” Lyapunov orbits form the boundaries of the escape windows we are interested in. In order to compute the initial part of these asymptotic curves, we follow the scheme proposed by Deprit and Henrard²⁰. Let (x^*(t), y^*(t)) be a periodic orbit of equations (4), and T be its minimal period. As the system is conservative and Hamiltonian, two of its multipliers are equal to +1; the other two are of the form (ρ,1/ρ), where ρ is a nonzero complex number. If ρ is real and |ρ| ≠ 1, then Poincaré²¹ established that the periodic orbit is unstable, and there exists a real number a > 0 such that ρ = e^aT. We will refer to a as the characteristic exponent of the periodic orbit (x^*(t), y^*(t)).

Poincaré showed that there are two one-parameter families of orbits associated to the unstable periodic orbit: a one-parameter family of orbits which tend asymptotically to the periodic orbit as the time t goes to +∞ (incoming asymptotic orbits), and a one-parameter family of orbits which tend asymptotically to the periodic orbit as t goes to −∞ (outgoing asymptotic orbits).

Thus, the incoming asymptotic orbits to a periodic orbit (x^*(t), y^*(t)) of period T are represented by the series

$$x(t,\varepsilon )={x}^{\ast }(t)+\varepsilon u(t),\,y(t,\varepsilon )={y}^{\ast }(t)+\varepsilon v(t),$$

(8)

where

$$\begin{array}{ccc}u(t) & = & \sum _{j\ge 1}\,{\varepsilon }^{j-1}{e}^{-jat}{x}_{j}(t),\\ v(t) & = & \sum _{j\ge 1}\,{\varepsilon }^{j-1}{e}^{-jat}{y}_{j}(t),\end{array}$$

(9)

for any integer j ≥ 1, the coefficients x_j(t) and y_j(t) are periodic with period T. The outgoing asymptotic orbits are represented by similar series, replacing a by −a. When ε is equal to zero, we recover the periodic orbit. Moreover, the asymptotic solutions (8) lie on the same energy manifold as the periodic orbit and, consequently, the series (9) represent an isoenergetic displacement of the periodic orbit.

Ths substitution of the series (8) into the differential equations (4) leads to

$$\begin{array}{ccc}\ddot{u} & = & {W}_{xx}^{\ast }u+{W}_{xy}^{\ast }v+\sum _{i\ge 1}\,{\varepsilon }^{i-1}\sum _{j\ge 0}\,{c}_{i}^{j}{(\frac{{\partial }^{i}W}{\partial {x}^{j}\partial {y}^{i-j}})}^{\ast }{u}^{j}{v}^{i},\\ \ddot{v} & = & {W}_{yx}^{\ast }u+{W}_{yy}^{\ast }v+\sum _{i\ge 1}\,{\varepsilon }^{i-1}\sum _{j\ge 0}\,{c}_{i}^{j}{(\frac{{\partial }^{i+j+1}W}{\partial {x}^{j}\partial {y}^{i-j}})}^{\ast }{u}^{j}{v}^{i}\mathrm{.}\end{array}$$

(10)

Here, f^* denotes the function f evaluated along the periodic solution, that is, f^*(t) = f(x^*(t), y^*(t)). Expanding u(t) and v(t) in powers of ε, the equation (10) lead to a recursive set of linear equations for the coefficients x_k(t) and y_k(t),

$$\begin{array}{ccc}{k}^{2}{a}^{2}{x}_{k}+2ka{\dot{x}}_{k}+{\ddot{x}}_{k} & = & {W}_{xx}^{\ast }\,{x}_{k}+{W}_{xy}^{\ast }\,{y}_{k}+{\Phi }_{k}(t),\\ {k}^{2}{a}^{2}{y}_{k}+2ka{\dot{y}}_{k}+{\ddot{y}}_{k} & = & {W}_{xy}^{\ast }\,{x}_{k}+{W}_{yy}^{\ast }\,{y}_{k}+{\Psi }_{k}(t),\end{array}$$

(11)

where Φ_k(t) and Ψ_k(t) are periodic functions corresponding to the contributions of the summations over j in (11) multiplied by e^jat. These contributions can readily be computed at order k and are periodic if the ${x}_{\ell }(t)$ and ${y}_{\ell }(t)$ (for $\ell < k$) are known and periodic. As a matter of fact, we do not need to know Φ_k(t) and Ψ_k(t) explicitly as functions of the time. As these coefficients have to be interpolated in Fourier series, it is enough to compute them at equally spaced instants along the period. Some details on the computation of the solution to (11) can be found in references ^12,13.

The Boundaries of the Escape Windows

We have integrated numerically and backward (using recurrent power series of order 20) some 1000000 orbits belonging to the stable manifold to the first quadrant Lyapunov orbit, until they cross the hyperplane y = 0 (with $\dot{y} > 0$). The initial conditions of this manifold have been computed by keeping the value of εexp(at) fixed at 0.0001 and choosing 1000000 equidistant values of the initial time t²².

In order to perform the analysis of the boundaries of the escape windows, we will introduce some nomenclature. Throughout the course of this paper, ϕ_U(t) and ϕ_L(t) denote the upper-right unstable periodic orbit, and the lower-right unstable periodic orbit, respectively. W_s,ν(ϕ) denotes the ν-th intersection of the stable manifold to the unstable periodic orbit ϕ with the surface of section y = 0, and Wu,ν(ϕ) denotes the ν-th intersection of the unstable manifold to the unstable periodic orbit ϕ with the surface of section y = 0. Here, $\nu \in {\mathbb{N}}$.

In the following, we will show the connection between the asymptotic manifolds to ϕ_U(t) and ϕ_L(t) for a value of the energy slightly larger than the critical value, h = 0.015040.

In Fig. 3, we show two orbits: the first one, belonging to the stable manifold to the first quadrant periodic orbit ϕ_U(t), has been integrated backward until its intersection with the surface of section y = 0. The second one, belonging to the unstable manifold to the fourth quadrant periodic orbit ϕ_L(t), has been integrated forward until its intersection with the same surface of section y = 0.

Figure 3

An orbit belonging to the stable manifold to the first quadrant periodic orbit ϕ_U(t) integrated backwards until its intersection with the surface of section y = 0, and an orbit belonging to the unstable manifold to the first quadrant periodic orbit ϕ_L(t) integrated backwards until its intersection with the surface of section y = 0. Here, h = 0.01504.

Figure 4 shows the first intersection of the stable manifold to the periodic orbit ϕ_U(t) with the surface of section y = 0, that is, W_s,1(ϕ_U), as well as the first intersection of the unstable manifold to the periodic orbit ϕ_L(t) with the same surface of section, W_u,1(ϕ_L). We observe that these two sets, W_s,1(ϕ_U) and W_u,1(ϕ_L), do not intersect,

$${W}_{s,1}({\varphi }_{U})\cap {W}_{u,1}({\varphi }_{L})=\varnothing .$$

Figure 4

W_s,1(ϕ_U(t)) and W_u,1(ϕ_L(t)) for h = 0.01504. Both sets have an empty intersection, so we can not find any orbit coming from the lower-right infinity and escaping through the upper-right window after just one intersection with the surface of section.

The points belonging to W_s,1(ϕ_U) intersect the surface of section y = 0 with $\dot{y} > 0$ and $\dot{x} > 0$, while the points in W_u,1(ϕ_L) intersect y = 0 with $\dot{y} > 0$ and $\dot{x} < 0$. Orbits starting inside W_s,1(ϕ_U) are going to the upper-right infinity, while orbits starting inside W_u,1(ϕ_L) are coming from the lower-right infinity. As W_s,1(ϕ_U) ∩ W_u,1(ϕ_L) = $\varnothing$, we can not find any orbit coming from the lower-right infinity and escaping through the upper-right window after just one intersection with the surface of section.

On the one hand, the second intersection of the stable manifold to the periodic orbit ϕ_U(t) with the surface of section y = 0 takes place with a negative value of $\dot{y}$, so there is no intersection between W_s,2(ϕ_U) and W_u,1(ϕ_L). The same occurs between W_s,2ν(ϕ_U) and W_u,1(ϕ_L), for any $\nu \in {\mathbb{N}}$.

On the other hand, we have found that W_s,2ν+1(ϕ_U) ∩ W_u,1(ϕ_L) = $\varnothing$, for ν = 1, …, 6, and

$${W}_{s\mathrm{,15}}({\varphi }_{U})\cap {W}_{u\mathrm{,1}}({\varphi }_{L})\ne \varnothing \mathrm{.}$$

In Fig. 5, we show some of these sets, and we can observe that the sets W_s,3(ϕ_U), W_s,7(ϕ_U) and W_s,11(ϕ_U) do not intersect W_u,1(ϕ_L). However, the intersection between W_s,15(ϕ_U) and W_u,1(ϕ_L) is not empty. This fact means that we can find orbits coming from the lower-right infinity and escaping through the upper-right window after fifteen intersections with the surface of section y = 0. We show an example of such an orbit in Fig. 6. We can also conclude that there are no orbits coming from the lower-right infinity and escaping through the upper-right window intersecting the axis y = 0 a number of times smaller than 15.

Figure 5

W_s,3(ϕ_U(t)) and W_u,1(ϕ_L(t)) (upper-left panel), W_s,7(ϕ_U(t)) and W_u,1(ϕ_L(t)) (upper-right panel), W_s,11(ϕ_U(t)) and W_u,1(ϕ_L(t)) (lower-right panel), and W_s,11(ϕ_U(t)) and W_u,1(ϕ_L(t)) (lower-right panel), for h = 0.01504. In the first three cases, the asymptotic curves to ϕ_U and ϕ_L have an empty intersection. However, W_s,11(ϕ_U(t)) ∩ W_u,1(ϕ_L(t)) ≠ $\varnothing$, so we can find orbits coming from the lower-right infinity and escaping through the upper-right window after fifteen intersections with the surface of section y = 0. We can also conclude that there are no orbits coming from the lower-right infinity and escaping through the upper-right window intersecting the axis y = 0 a number of times smaller than 15.

Figure 6

An orbit coming from the lower-right infinity and escaping through the upper-right window after fifteen intersections with the surface of section y = 0.

Now, we will focus our attention in the origin of the infinite spirals around W_s,3(ϕ_U) that we can observe in the upper-left panel of Fig. 5. To this purpose, we consider the intersections between the stable and unstable manifolds to the upper-right periodic orbit. In Fig. 7, we show the intersection between W_s,2(ϕ_U) and W_u,1(ϕ_U) in the surface of section y = 0 (with $\dot{y} < 0$), for h = 0.01504. We observe that W_s,2(ϕ_U) ∩ W_u,1(ϕ_U) ≠ $\varnothing$, that is, the two “rings” intersect. The intersection points correspond to homoclinic orbits to the upper-right periodic orbit ϕ_U(t). These four homoclinic orbits intersect two times the surface of section y = 0. In Fig. 8 (left panel), we show one of these homoclinic orbits to ϕ_U(t). Orbits starting inside both rings (colored in dark grey in Fig. 7) are coming from the upper-right infinity and are going to the upper-right infinity. They just pass through the center of the galaxy, intersecting the surface of section y = 0 two times. In Fig. 8 (right panel), we show one of these orbits. Moreover, orbits starting from one of the two crescents, colored in blue in Fig. 7, have a previous intersection with the surface of section y = 0, and of course escape in the future by the upper-right opening of the potential.

Figure 7

W_s,2(ϕ_U) and W_u,1(ϕ_U) for h = 0.01504. There are four intersection points related to homoclinic orbits to the upper-right guardian periodic orbit ϕ_U(t).

Figure 8

Homoclinic orbit (left panel) to ϕ_U(t) intersecting the surface of section y = 0 two times, and orbit coming from the upper-right infinity, and going to the upper-right infinity, after two intersections with the surface of section y = 0 (right panel).

The third intersection, shown in Fig. 9, is composed of two “tongues” which spiral around the stable manifold to the first quadrant guardian orbit, W_s,1(ϕ_U). These two tongues are images of the two crescents colored in blue in Fig. 7. The spirals are infinite, but of course we have computed (and shown in Fig. 9) only a part of them. As we have mentioned above, W_s,3(ϕ_U(t)) does not intersect W_u,1(ϕ_U) or W_u,1(ϕ_L). Thus, all the orbits starting from these two tongues have a previous intersection with the surface of section y = 0, and will leave the galaxy by the upper-right window after two crossings with the axis y = 0. As a consequence of this fact, the fourth intersection of the stable manifold to the upper-right periodic orbit ϕ_U(t) with the surface of section y = 0, is also composed of two tongues, which spiral around W_s,2(ϕ_U). In Fig. 10, we show the trace of the fourth intersection of the stable manifold to ϕ_U(t) with the surface of section. As in the previous intersection, orbits starting inside these tongues will leave the galaxy by the upper-right window after three crossings with the axis y = 0. As before, W_s,4(ϕ_U) is composed of two “tongues” which spiral around the stable manifold to the first quadrant guardian orbit. These two tongues are images of the two crescents colored in blue in Fig. 9.

Figure 9

The third intersection W_s,3(ϕ_U) is composed of two tongues which spiral around the stable manifold to the first quadrant guardian orbit, W_s,1(ϕ_U). These two tongues are images of the two crescents colored in blue in Fig. 8.

Figure 10

Trace of the fourth intersection of the stable manifold to ϕ_U(t), W_s,4(ϕ_U), with the surface of section.

In Fig. 11, we show the intersection between W_s,4(ϕ_U) and W_u,1(ϕ_U). We observe that W_s,4(ϕ_U) intersects W_u,1(ϕ_U) at an infinite set of points. Each of these points of intersection between both manifolds corresponds to a homoclinic orbit to ϕ_U(t) which intersects the surface of section y = 0 four times. Orbits starting inside W_1,u(ϕ_U), and inside one of the two tongues of W_s,4(ϕ_U), come from infinity by the upper-right window and leave the galaxy by the same window after four crossings of the axis y = 0. Orbits starting inside the tongues but outside the grey area enclosed by W_u,1(ϕ_U) have a previous intersection with the surface of section. We have colored these parts of the tongues in different colors in order to follow these orbits backward easily.

Figure 11

Intersection between W_s,4(ϕ_U) and W_u,1(ϕ_U), for h = 0.01504.

In Fig. 12, we give an scheme of the structure of the part of W_s,4(ϕ_U) outside W_u,1(ϕ_U), in order to clarify the destiny of the different components of the tongues when we follow them backward in time. First, we will focus our attention in the left component of the manifolds, and the same analysis will be applied to the right component. We observe that we can distinguish two parts in the left component: the tongue named A and colored in dark green in Fig. 12, and an infinite set of “bridges”, each of them inside the previous one. We have numbered these bridges as shown in Fig. 12: A₁, A₂, A₃, …. This set of bridges are alternately colored in light and medium blue, depending on the tongue of W_s,4(ϕ_U) they belong (see Fig. 10). The tongue A intersects W_u,1(ϕ_U) at two different points, corresponding to a pair of homoclinic orbits to ϕ_U. These two orbits intersect the surface of section y = 0 four times. On the other hand, each bridge A_ν, for any ν ∈ N, intersects W_u,1(ϕ_U) at four points, each of them corresponding to a homoclinic orbit to ϕ_U, which intersects four times the surface of section y = 0. This means that there exists an infinite set of initial conditions corresponding to homoclinic orbits to ϕ_U. Each of these set of homoclinic orbits intersects four times the surface of section y = 0.

Figure 12

Scheme of the structure of W_s,4(ϕ_U) outside W_u,1(ϕ_U), for h = 0.01504.

The structure of the right component of the part of W_s,4(ϕ_U) outside W_u,1(ϕ_U) is similar. There is a tongue called B and colored in light green, and an infinite set of bridges (B₁, B₂, …) alternately colored in light and medium blue, as in the left component. The tongue B intersects W_u,1(ϕ_U) at two points, corresponding to a pair of homoclinic orbits to ϕ_U. These two homoclinic orbits intersect the surface of section y = 0 four times. On the other hand, each bridge B_ν, for any ν ∈ N, intersects W_u,1(ϕ_U) at four different points, each of them corresponding to a homoclinic orbit to ϕ_U, which intersects four times the surface of section y = 0. Thus, there exists a second infinite set of initial conditions corresponding to homoclinic orbits to ϕ_U, each one of them intersecting four times the surface of section y = 0.

Let us introduce here the notation T to denote the function that maps an initial condition in the surface of section y = 0 to the previous intersection of the corresponding orbit with the surface of section.

As commented above, the orbits with initial conditions inside A, B, A_ν and B_ν, for any ν ∈ N, have a previous intersection with the surface of section y = 0. Following these orbits backward, we compute the fifth section (see Fig. 13). The intersection W_s,5(ϕ_U) is composed by two simple tongues (colored in light and dark green) plus two complex ones. The simple ones are pre-images of the crescents A and B, shown in Fig. 12 and, thus, we denote them by T(A) and T(B), respectively. The complex tongues are composed of “subtongues” embedded inside each other, as “russian dolls”, following the same architecture than the bridges we have described in Fig. 12. There are an infinity of these subtongues infinitely winding around W_s,3(ϕ_U), the third intersection of the stable manifold to ϕ_U with the surface of section y = 0.

Figure 13

Trace of W_s,5(ϕ_U), the fifth intersection of the stable manifold to ϕ_U(t) with the surface of section y = 0, for h = 0.01504.

In Fig. 14, we show a detail of the spiral structures we have found and shown in Fig. 13. Orbits starting inside the black region (corresponding to initial conditions inside W_s,3(ϕ_U)) escape after two intersections with the surface of section y = 0.

Figure 14

Scheme of a detail of the structure of W_s,5(ϕ_U), for h = 0.01504.

The orbits which start inside the complex tongues have a different fate depending on the parity of the number of subtongues in which they are embedded. Those starting from the colored area (Figs 13 and 14) are mapped on the pieces of the tongues analyzed in Fig. 12 which are not inside the grey area, and thus escape after four crossings of the surface of section. In Fig. 14, we have indicated the corresponding parts of the tongues in Fig. 12. There, T(A₁) denotes the pre-image of the bridge A₁, T(A₂) the pre-image of the bridge A₂, and so on.

As W_s,5(ϕ_U) (y = 0, $\dot{y} > 0$) does not intersect W_u,1(ϕ_U) (y = 0, $\dot{y} < 0$) or W_u,1(ϕ_L), the sixth intersection between the stable manifold to ϕ_U and the surface of section y = 0 (that is, W_s,6(ϕ_U)) has the same structure than W_s,5(ϕ_U), as we can observe in Fig. 15. Here, we find T²(A), the pre-image of the tongue T(A), which belongs to W_s,5(ϕ_U), T²(B), the pre-image of T(B), and also the sequences T²(A₁), T²(A₂), … and T²(B₁), T²(B₂), …. In Fig. 16, we show the intersection between W_s,6(ϕ_U) and W_u,1(ϕ_U). In this figure, we observe that some parts of W_s,6(ϕ_U) stay outside W_u,1(ϕ_U). As in the previous intersection of the stable manifold, the orbits which start inside the complex tongues have a different fate depending on the parity of the number of subtongues in which they are embedded. Those starting from the colored area (light and medium blue) are mapped on the pieces of the tongues (with equal color) analyzed in Figs 13 and 14, and thus escape after five crossings of the surface of section.

Figure 15

Trace of W_s,6(ϕ_U), the sixth intersection of the stable manifold to ϕ_U(t) with the surface of section y = 0, for h = 0.01504.

Figure 16

Intersection between W_s,6(ϕ_U) and W_u,1(ϕ_U), for h = 0.01504.

The structure of the intersection between W_s,6(ϕ_U) and W_u,1(ϕ_U) is quite complex, and it must be analyzed carefully. To that end, we have constructed a simplified scheme of what we can observe in Fig. 16. In order to simplify this analysis, we have divided it into four groups: G₁, G₂, G₃ and G₄.

G₁ (Fig. 17) is composed of a simple tongue, colored in dark green and named T_A, plus an infinite set of bridges which tends to W_s,4(ϕ_U) (colored in black in the figure). The tongue T_A is the part of T²(A) outside W_u,1(ϕ_U). The infinite set of bridges tending to W_s,4(ϕ_U) is formed by bridges of alternating colors (light and medium blue) depending on their origin in W_s,4(ϕ_U) (see Fig. 12). In Fig. 17, C₁ represents the part of T²(A₁) outside W_u,1(ϕ_U) and, in general, C_ν denotes the part of T²(A_ν) outside W_u,1(ϕ_U), for any ν ∈ N. Each bridge of this infinite sequence intersects W_u,1(ϕ_U) at four points, which are the initial conditions of four homoclinic orbits to ϕ_U. Each of these homoclinic orbits intersects six times the surface of section y = 0.

Figure 17

Detail of the structure of group G₁ of the intersection between W_s,6(ϕ_U) and W_u,1(ϕ_U), for h = 0.01504.

G₂ (Fig. 18) is composed of a simple tongue (colored in light green, and named T_B), plus an infinite set of simple bridges and “rings”. The crescent T_B is part of T²(B). The infinite set of simple bridges and rings tends to W_s,4(ϕ_U), and it is composed of an infinite set of simple bridges (as those which appear in W_s,4(ϕ_U)) and a set of a new kind of structure called ring. We define a ring as an infinite set of bridges tending to a simple bridge belonging to a part of W_s,4(ϕ_U). In Fig. 18, we have represented a detail of this structure. The first ring, named R₁, is a continuation of the group G₃ (see Fig. 16). Figure 19 gives a detail of its composition. We observe that R₁ is formed by two infinite sequences of bridges that accumulate at B₁, which is one of the bridges belonging to the part of W_s,4(ϕ_U) outside W_u,1(ϕ_U). We have also indicated the origin of each of the bridges in R₁, by writing the name of the bridges of G₃ (D₁, D₂, D₃, …) they come from. Each of the bridges belonging to one of these two sequences intersects W_u,1(ϕ_U) at four points. Each of these points is associated to the initial conditions of a homoclinic orbit to ϕ_U, which intersects six times the surface of section y = 0.

Figure 18

Detail of the structure of group G₂ of the intersection between W_s,6(ϕ_U) and W_u,1(ϕ_U), for h = 0.01504.

Figure 19

Detail of the composition of ring R₁, belonging to the group G₂ in the part of W_s,6(ϕ_U) outside W_u,1(ϕ_U).

Following with the analysis of G₂, we find a bridge which corresponds to a part of T²(A). We have named this bridge E₁ in Fig. 18. This bridge intersects W_u,1(ϕ_U) at four points, each of them corresponding to a homoclinic orbit to ϕ_U that intersetcs six times the surface of section y = 0.

Then, we find a second ring, named R₂, inside the bridge E₁. The composition of R₂ is similar to that of R₁. This second ring is a continuation of the infinite set of bridges which form G₁. A detail of its structure is shown in Fig. 20. R₂ is formed by two infinite sequences of bridges that accumulate at B₂, which is one of the bridges belonging to the part of W_s,4(ϕ_U) outside W_u,1(ϕ_U). We have also indicated the origin of each of the bridges in R₂, by writing the name of the bridges of G₁ (C₁, C₂, C₃, …) they come from. Each of the bridges belonging to one of these two sequences intersects W_u,1(ϕ_U) at four points. Each of these points is associated to the initial conditions of a homoclinic orbit to ϕ_U, which intersects six times the surface of section y = 0.

Figure 20

Detail of the composition of ring R₂, belonging to the group G₂ in the part of W_s,6(ϕ_U) outside W_u,1(ϕ_U).

Inside R₂, there is a bridge which corresponds to a part of T²(B). We have named this bridge E₂ in Fig. 18. This bridge intersects W_u,1(ϕ_U) at four points. Each of these points corresponds to a homoclinic orbit to ϕ_U that intersetcs six times the surface of section y = 0.

The union of these four structures (R₁, E₁, R₂ and E₂) is a new kind of object, named “belt”. Inside E₂, we find a set colored in brown in Fig. 18, containing an infinite sequence of belts infinitely spiraling around W_s,4(ϕ_U). Each belt intersects W_u,1(ϕ_U) at an infinite set of points, each of them corresponding to a homoclinic orbit to ϕ_U. As the homoclinics described before, all these orbits intersect six times the surface of section.

The structure of G₃ is shown in Fig. 21. G₃ is composed of an infinite set of bridges which tends to W_s,4(ϕ_U) (colored in black in the figure). The infinite set of bridges tending to W_s,4(ϕ_U) is formed by bridges of alternating colors (medium and light blue) depending on their origin in W_s,4(ϕ_U) (see Fig. 12). In Fig. 21, D₁ represents the part of T²(B₁) outside W_u,1(ϕ_U) and, in general, D_ν denotes the part of T²(B_ν) outside W_u,1(ϕ_U), for any ν ∈ N. Each bridge of this infinite sequence intersects W_u,1(ϕ_U) at four points, which are the initial conditions of a homoclinic orbit to ϕ_U. Each of these homoclinic orbits intersects six times the surface of section y = 0.

Figure 21

Detail of the structure of group G₃ of the intersection between W_s,6(ϕ_U) and W_u,1(ϕ_U), for h = 0.01504.

G₄ (Fig. 22) has a similar structure to that of G₂. It is composed of an infinite set of belts. First, in the outer part of the structure, we find a bridge which corresponds to a part of T²(A). We have named this bridge E′₁ in Fig. 22. This bridge intersects W_u,1(ϕ_U) at four points. Each of these points corresponds to a homoclinic orbit to ϕ_U that intersetcs six times the surface of section y = 0. Inside E′₁, we find the ring R′₂, with the same origin than R₂, as R′₂ is also a continuation of the infinite set of bridges that forms the group G₁. Thus, it has the same composition than R₂ (Fig. 20): two infinite sequences of bridges that accumulate at B₂, which is one of the bridges belonging to the part of W_s,4(ϕ_U) outside W_u,1(ϕ_U). Each of the bridges belonging to one of these two sequences intersects W_u,1(ϕ_U) at four points, and each of these points is associated to the initial conditions of a homoclinic orbit to ϕ_U, which intersects six times the surface of section y = 0.

Figure 22

Detail of the structure of group G₄ of the intersection between W_s,6(ϕ_U) and W_u,1(ϕ_U), for h = 0.01504.

Inside R′₂, there is a bridge corresponding to a part of T²(B). We have named this bridge E′₂ in Fig. 18. This bridge intersects W_u,1(ϕ_U) at four points. Each of these points corresponds to a homoclinic orbit to ϕ_U that intersetcs six times the surface of section y = 0.

Then, and inside E′₁, there is a ring, named R′₁, with the same structure than R₁ (Fig. 19). R′₁ is formed by two infinite sequences of bridges that accumulate at B₁, which is one of the bridges belonging to the part of W_s,4(ϕ_U) outside W_u,1(ϕ_U). Each of the bridges belonging to one of these two sequences intersects W_u,1(ϕ_U) at four points, associated to the initial conditions of a homoclinic orbit to ϕ_U, which intersects six times the surface of section y = 0.

As it occurs in G₂, the union of these four structures (E′₁, R′₂, E′₂ and R′₁) repeats infinitely spiraling around W_s,4(ϕ₄). Each belt intersects W_u,1(ϕ_U) at an infinite set of points, each of them corresponding to a homoclinic orbit to ϕ_U. As the homoclinics described before, all these orbits intersect six times the surface of section.

Orbits starting inside W_1,u(ϕ_U), and inside one of the colored areas of W_s,6(ϕ_U), come from infinity by the upper-right window and leave the galaxy by the same window after six crossings of the axis y = 0. Orbits starting inside the colored areas but outside the grey area enclosed by W_u,1(ϕ_U) have a previous intersection with the surface of section. Following these orbits backward, we compute the seventh section (see Fig. 23). We have colored in black the tongues belonging to W_s,5(ϕ_U), and in dark grey the two tongues belonging to W_s,3(ϕ_U). Orbits with initial conditions inside the dark grey region (corresponding to W_s,3(ϕ_U)) escape after two intersections with the axis y = 0, and orbits starting inside the black region (corresponding to initial conditions inside W_s,5(ϕ_U)) escape after four intersections with the surface of section y = 0. Finally, orbits starting in the colored regions escape after six intersections with the surface of section.

Figure 23

Trace of W_s,7(ϕ_U), the seventh intersection of the stable manifold to ϕ_U(t) with the surface of section y = 0, for h = 0.01504.

The structure of the intersection W_s,7(ϕ_U) becomes more intrincately complex than that of W_s,6(ϕ_U). In general, we observe that for any tongue belonging to W_s,5(ϕ_U), there is an infinite sequence of tongues in W_s,7(ϕ_U) tending to it. This is due to the fact that each bridge belonging to W_s,6(ϕ_U) that intersects W_u,1(ϕ_U) originates a tongue infinitely spiraling around W_s,5(ϕ_U) in W_s,7(ϕ_U). Thus, each ring in W_s,6(ϕ_U) generates an infinite sequence of tongues tending to the corresponding tongue in W_s,5(ϕ_U) when we obtain the seventh intersection of the stable manifold to ϕ_U, in the same manner than we find an infinite sequence of bridges tending to a simple bridge in the ring structure. We have illustrated this mechanism in Fig. 24.

Figure 24

A ring structure in the part of W_s,6(ϕ_U) outside W_u,1(ϕ_U) (left) and a detail of its composition when we follow it backward until the seventh intersection with the surface of section (right). There are two infinite sequences of tongues infinitely accumulating at a tongue belonging to W_s,5(ϕ_U), which is colored in black in the right sketch.

Conclusions

In this paper, we study the properties of the escape of orbits from a potential presenting two channels of escape through the analysis of the asymptotic surfaces to the Lyapunov periodic orbits in the openings of the potential well. Depending on the value of the energy, the system exhibit different types of behavior. There is a critical value of the energy (h_c) such that, for larger values of h, the curves of zero velocity are open and test particles may escape. For h < h_c, the curves are effectively closed and particles originated from the central potential region are confined there. For each value of h larger than h_c, there is an unstable periodic orbit located at every opening of the potential well. The asymptotic curves of the various unstable periodic orbits intersect at a complicated set of homoclinic and heteroclinic points, and govern the escape to infinity from the potential well. In our investigation, we have analyzed the geometry of these manifolds through the study of their successive intersections with the surface of section y = 0, in order to show the shape and the size of the windows through which test particles escape from the potential well.