Chains of rotational tori and filamentary structures close to high multiplicity periodic orbits in a 3D galactic potential
This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis and Zachilas 1994] in order to visualize them. As examples we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.
Recently, Katsanikas and Patsis  (hereafter KP11) studied the structure of the phase space in the neighborhood of simple periodic orbits in a 3D autonomous Hamiltonian system of galactic type. In the present paper we extend this work and we investigate the orbital behavior in the neighborhood of periodic orbits with high order multiplicity. Especially, we study the orbital behavior in the neighborhood of a stable 2-periodic and a stable 7-periodic orbit and in the neighborhood of the accompanying simple unstable periodic orbits.
In Cartesian coordinates , if we consider our surface of section to be defined by , a “2-periodic” orbit is one that closes after 2 intersections with the surface of section when . This is a periodic orbit of multiplicity 2. In the same way, in general, an “-periodic” orbit is a periodic orbit of multiplicity . Here we study two cases with and . For the visualization of the 4D surfaces of section we use the method of color and rotation [Patsis and Zachilas 1994]. With this method we plot the consequents in a 3D subspace of the 4D surface of section and every consequent is colored according to its location in the 4th dimension. For a description of this method, the meaning of viewing angles etc., see KP11.
Our Hamiltonian is of the form
where is the potential we used in our applications, i.e.:
This potential is the same as in KP11, i.e. a triaxial double Miyamoto disk rotating around its short axis with angular velocity =60 .
In our units, distance =1 corresponds to 1 kpc. The velocity unit corresponds to 209.64 . For the Jacobi constant Ej=1 corresponds to 43950 . For the rest of the parameters we have used the following values: . The parameters determine the geometry of the disks, while are scaling factors.
The trajectories are calculated numerically by using a 4 order Runge-Kutta scheme, with a constant time step that secures a 13 digits precision. A typical integration for consequents with a Core 2 Duo CPU/2.2GHz computer takes about 3.6 minutes of real time.
The calculation of the linear stability of a periodic orbit is based on the method of Broucke  and Hadjidemetriou . By this method we calculate the stability indices b1, b2 and the quantity (following the notation of Contopoulos and Magnenat ). Depending on the values of the stability indices and that of , a periodic orbit can be stable (S), simple unstable (U), double unstable (DU) and complex unstable (). For definitions see Contopoulos and Magnenat . For a generalization of this method to systems with higher than 3 degrees of freedom see Skokos .
2 Spaces of Section
The stable and simple unstable orbits of the 2-periodic case belong to the 3D families of 2-periodic orbits “” (an initially stable family) and “” (an initially simple unstable family). They are bifurcations of a planar 3/1 family on the equatorial plane. The morphology of the orbits of the families and is depicted in Figs. 1 and 2 respectively.
We call the 3D 7-periodic families (an initially stable family) and (an initially simple unstable family). They are bifurcations of the family x1v1 [Skokos et al 2002a,b] that is associated with the vertical resonance 2/1 in our galactic system. In Figs. 3, 4 we show the morphology of the orbits of the families and respectively.
2.2 4D spaces of section in the neighborhood of the 2-periodic orbits
Phase space structure close to s
We consider first the stable 2-periodic orbit at Ej=. We trace it at the initial conditions . In the neighborhood of we observe two tori, when we perturb its initial condition , by belonging to the intervals and . All these perturbations of the initial conditions are isoenergetic displacements on the surfaces of section. For example if we take we observe two tori in Fig. 5 around the two points (black dots) of the stable 2-periodic orbit. On the two tori we observe a smooth color succession. This means that the succession of the colors of the consequents on a given structure follows the succession of the colors on the color bar at the right side of the figure. The consequents are colored according to the value of the coordinate, that is not used in the 3D spatial projection. In all diagrams of this paper the color is given according to the coordinate. The color values in the color bars are normalized to the [0,1] interval corresponding to . For the details of the method the reader may refer to KP11. The smoothness of the color variation can be checked by looking at the color bar, on the right of Fig. 5, where we see that the color variation is between neighboring shades. The colors on the upper left torus vary from red to orange to yellow and to green, while the colors on the other torus vary from green to light blue to blue. This means that the fourth value of the consequents has a smooth distribution in the 4th dimension. If we perturb the initial conditions with larger values of perturbations, for example , we observe again two tori with smooth color variation for 1000 consequents as in the previous case. However, if we continue the integration of the orbits we observe that the consequents start to deviate from two tori and they form a cloud in the 3D projection of the 4D surface of section around them (Fig. 6). In the cloud the color is mixed and this means that the points are far away in the 4h dimension. The dynamical behavior according to which an orbit stays close to an invariant torus for some time and then diffuses in phase space is typical of sticky orbits [Contopoulos and Harsoula 2008]. In Table 1 we give the values and the direction of the perturbations for which we observe tori. Just beyond this interval we encounter sticky orbits as the one in Fig. 6.
Phase space structure close to u
Close to , at Ej= , we trace also the simple unstable 2-periodic orbit (Fig. 5). Our code finds it at initial conditions . In its neighborhood we have found two types of dynamical behavior. We can see the first type if we perturb for example the initial conditions by . In this case we observe in Fig. 7 a “ribbon” that connects the points of the simple unstable 2-periodic orbit. If we apply the method of color and rotation we observe in Fig. 8 a smooth color variation from red to blue. This means that the “ribbon” is a 4D structure. After 5300 intersections we observe in Fig. 9, that the consequents leave the “ribbon” and they scatter in the phase space. This orbit is also sticky in the sense that remains close to a given phase space structure for some time and then diffuses in the 4D space of section.
The second type of dynamical behavior can be seen in the neighborhood of the simple unstable 2-periodic orbit for Ej= . If we add for example a perturbation in the z-direction in the initial conditions of we observe a filamentary surface in the 3D projection of the 4D surface of section (Fig. 10). This filamentary structure connects the points of and forms four loops. The two of them surround the two tori that are around the two points of s. In the regions, close to the points of we have two self-intersections of the filamentary structure. In Fig. 10 we observe two more self-intersections indicated with arrows and the formation of 4 loops. The loops surround the periodic points and two more points representing a periodic orbit symmetric to with respect to the equatorial plane , while the arrows represent an orbit symmetric to . The dynamics in the neighborhood of this second set of periodic orbits is similar to the one close to and u. Now we apply the method of color and rotation in order to study the distribution of the consequents in the 4th dimension (Fig. 11). We observe, that we have also in this case a smooth color variation from red to blue and this means that the consequents are on the filamentary structure in the 4D space of section. For more than 4700 intersections the points deviate from this structure and form a cloud of points around it. Therefore in this case we encounter again the phenomenon of stickiness. Table 2 gives the range of the perturbation of the initial conditions for which we find the “ribbons” or the “filaments” connecting the points of u, for . For larger values of the perturbations we find clouds of points as the one we presented in Fig. 9.
2.3 4D spaces of section in the neighborhood of 7-periodic orbits
The structures observed close to the 2-periodic orbits, have been found in the neighborhood of every -periodic orbit we have studied in our system. The results are qualitatively similar, but the filamentary structure that joins the points of the simple unstable orbits becomes complex as increases. Below we give one more example of the phase space structure close to a high multiplicity orbit, this time around a 7-periodic one.
Phase space structure close to s1
We apply a perturbation both in the -direction and -direction in the initial conditions of the stable 7-periodic orbit of the family , and , for Ej= . At this Jacobi constant we find at initial conditions . The perturbed orbit has been integrated for consequents. In the 3D projection of the 4D surface of section we observe seven tori surrounding the points with the initial conditions of (Fig. 12). In this projection we observe, that two of the tori intersect each other. To check their position in the 4th dimension we apply the method of color and rotation (Fig. 12). By taking into account the values of all tori in the 4th dimension, , and scaling the colors according to =, we find that to each small torus correspond one or mostly two primary colors. We observe that in the case of the two tori that intersect each other in the 3D projection we have the meeting of different colors at their intersection. This means that this intersection does not exist in the 4D space but only in the 3D projection, as expected. Next we apply the method of color and rotation only to one torus, which we call T1 and which is indicated with an arrow in Fig. 12. In Fig. 13 we observe a smooth color variation on its surface, which obviously corresponds to a different range than that in Fig. 12, and we see a succession of colors from red to orange, to yellow, to green, to light blue, to blue and finally to violet. Comparing this result with this found in KP11 we realize that we have a morphology typical for rotational tori [Vrahatis et al 1997, KP11]. The only difference is that we do not observe in the present case the transition of the color sequence from the external to the internal surface of the torus, as observed in cases around stable simple periodic orbits in KP11 (compare with figure 11 in KP11).
If we perturb the initial conditions of the periodic orbit of by we see in Fig. 14 seven tori without any intersection of these tori in the 3D subspace. These tori have also a smooth color variation (Fig. 14). In Table 3 we give the range of the perturbation, for which we find tori in the neighborhood of s1.
Phase space structure close to u1
Now we investigate the properties of orbits in the phase space in the neighborhood of the simple unstable periodic orbit . The orbit we study is at and is found with initial conditions . Firstly we add a perturbation in the initial conditions in the -direction . In Fig. 15 we see that the consequents that are depicted with red color double bow in the 3D projection of the 4D surface of section that surrounds the seven tori that are around the points of . If we color the consequents according to the value of their fourth dimension we see in Fig. 16 that we have a smooth color variation from red to violet but at the two intersections A and B we have the meeting of different colors (e.g. blue with orange). This means that we have different values of the fourth dimension at the two intersections and these intersections are only projection effects and not true intersections in the 4D space of section. The consequents depart from the structure we give in Fig. 16, after 25000 intersections.
If we perturb the same orbit by at Ej= . We observe in Fig. 17, that the consequents (with red) form a filamentary structure that connects the points of the simple unstable 7-periodic orbit and surrounds the seven tori of the stable 7-periodic orbit. In Fig. 17 with the numbers 1 until 7 we indicate the 7 points of in the subspace. At these regions the filaments, that have been formed by evolving the orbit in time, cross each other in the 3D projection of the 4D surface of section (Fig. 17). We observe, that besides the crossings at the numbered regions, where we have the initial conditions of u1, we have 7 more crossings of the filamentary structure by itself and the formation of 7 new loops, like those surrounding the tori around the points of s1. We have again here, like in the 2-periodic orbit in section 2.2.2, the presence of the symmetric family, with respect to the equatorial plane.
In Fig. 18 we observe a smooth color variation along the consequents that build the filaments. Starting from 1 we have a color succession from green to light blue, then to blue at 2, after that to light blue, then to green at 3, etc. At the regions, where we have the self crossing of the filaments, i.e. close to the points 1, 2,…7, we observe that the regions are characterized by just one color only. For example at 1 the shade is green, at 2 blue etc. This means that the fourth coordinate of the consequents at these regions has this time the same value and the intersections in the 3D projections are real intersections in the 4D space. We also observe that the color evolves along both branches that depart from, or arrive at the points of the unstable periodic orbit. Both branches show the same color evolution between two successive crossings.
The smooth succession of the colors along the filaments is observed as long as the consequents participate in the filamentary structure. However, if we consider more than about 6500 consequents in the neighborhood of the orbit, the points diffuse, occupying a large volume in the phase space and finally they form a cloud in the 3D projections that surrounds the structure we observe in Fig. 18. The dimensions of this cloud becomes about 15 times larger in the x direction than the structure depicted in Figs. 17 and 18. This is determined by the space in which the particles are allowed to move for this Ej. The cloud of points can be seen in Fig. 19. The red configuration in the central region of the diagram is the structure we observe in Fig. 18. By giving colors to the consequents according to their distribution in the 4th dimension we have realized that in the cloud the distribution of points in the 4th dimension is mixed (Fig. 20). In Table 4 we summarize the range of perturbation for which the consequents remain on the filamentary structure.
3 Lyapunov Characteristic Numbers
In this paper we study two kinds of orbits. The first kind is represented by tori in the 4D surface of section in the neighborhood of stable 2-periodic and 7-periodic orbits. The second kind is represented initially by structures confined in phase space (double bow, ribbon, “filamentary” structures etc.) and then by clouds in the 4D surfaces of section. These orbits are located in the neighborhood of simple unstable 2-periodic and 7-periodic orbits. In this section we calculate the “finite time” Lyapunov Characteristic Number (LCN) for these two types of orbits.
The “finite time” Lyapunov Characteristic Number is defined as:
where and are the distances between two points of two nearby orbits at times t = 0 and t respectively (see e.g. Skokos )
Firstly, for the first kind of orbits, we computed the maximal Lyapunov Characteristic Number (mLCN). For example, for the orbit that is represented in the 4D surface of section in Fig. 14, the value of LCN(t) decreases like 1/t and tends to zero as we can see in Fig. 21.
Then, we calculated the “finite time” Lyapunov Characteristic Number () for the second type of orbits, for example for the orbit in Figs. 18 and Fig. 19. In Fig. 18 we have for the first 6500 consequents a filamentary structure. During this period the LCN(t) of the orbit decreases to a value (Fig. 22a). Beyond that point the orbit is represented by a cloud of points in the 4D surface of section. During this phase, the LCN(t) fluctuates as time increases and finally increases and tends to level off around (Fig. 22b).
In this paper we studied the phase space structure in the neighborhood of stable and simple unstable periodic orbits in a rotating 3D galactic potential. We presented the dynamical behavior of two sets of stable-simple unstable orbits. The first was in the neighborhood of periodic orbits of multiplicity , and the second in the neighborhood of periodic orbits of multiplicity . For less than 6000 consequents we observed a direct correspondence between the standard configuration, which we encounter in a resonance zone in 2D autonomous Hamiltonian systems, and the structure of phase space in our 4D spaces of section, namely a succession of elliptic and hyperbolic points. This means that also in the case we study we observed a succession of stable and simple unstable points in the 4D spaces of section. For larger integration times the consequents diffuse and occupy a larger volume of the phase space. We found similar behavior for all cases of -periodic orbits we studied in this system.
The following are the main conclusions from our work:
In the neighborhood of the stable -periodic orbit we found tori surrounding its initial conditions. We found smooth color variation along these tori in two ways. Along a particular torus and also along all tori, considering them as one object. This depends on the scale of the fourth coordinate, which gives the colors to the consequents. These tori are rotational tori in the terminology introduced by Vrahatis et al. . Similar structures can be seen in the paper by Martinet & Magnenat . However, we did not observe in the present study the color transition from the external to the internal side of the torus on the individual tori as in some cases in KP11.
Integrating an orbit close to a simple unstable -periodic orbit to obtain a few thousands of consequents, we found that they form a filamentary structure with smooth color variation in the 4th dimension. The filaments that are formed in this way connect the points of the simple unstable -periodic orbit and surround the seven tori around the points of the stable -periodic orbit. The filamentary structures appear either as “ribbons” or as “bows” in the 4D spaces of section.
In the regions close to the points of the simple unstable periodic orbit two branches of a filamentary structure meet and their consequents have the same color. This shows, that we have at these points self-intersections of the structures in the 4D space of section.
The above described situation is a direct extrapolation of the typical case of a 2D autonomous Hamiltonian system with the chain of stability islands and the chaotic zone, which connects the points of the corresponding unstable periodic orbit. We note that in this 3D case the unstable orbit is simple unstable and we have a smooth color variation along the filaments when we consider the first few thousands of the intersections in the space of section.
We encountered cases, where the consequents that remained on the filaments and reinforced this structure for a few thousand intersections, diffused later in the 4D space. The diffusion in the 4D space, was characterized by mixing of colors.
Acknowledgments We thank Prof. Contopoulos for fruitful discussions and valuable comments. MK is grateful to the “Hellenic Center of Metals
Research” for its support in the frame of the current research.
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