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Astron. Astrophys. 334, 829-839 (1998)

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4. Vertical motion around [FORMULA]

Our aim in this section is to describe the effect of the transition stability-complex instability on the family of 3-dimensional periodic orbits starting at the Lagrangian points [FORMULA].

4.1. Families of 3-D periodic orbits

With standard Newton-type techniques improved by a least-squares approach to treat numerical degeneracies (see e.g., P84, P85b) the vertical periodic orbits starting at the Lagrangian points [FORMULA] and their type of stability/instability are calculated. Owing to the symmetry of the problem only the [FORMULA] orbits need to be discussed.

[FIGURE] Fig. 3. Some members of the family of 3-D periodic orbits starting at [FORMULA] (top) and [FORMULA] (bottom) and ending on the nearly circular retrograde family in the galaxy plane at a period-doubling bifurcation. The tube thickness increases proportional to the Hamiltonian value h. The semi-transparent ellipsoid corresponds to the model Ferrer's bar

In a first approach we carry out a numerical exploration of the family of 3-D periodic orbits for different values of [FORMULA] close to the transition, more specifically, [FORMULA], 0.60, 0.621, 0.622, keeping [FORMULA]. For any value of [FORMULA] fixed, we obtain the corresponding family of periodic orbits, beginning at [FORMULA] with [FORMULA], continuing with increasing [FORMULA] up to a maximum value, and decreasing down to [FORMULA], where the family ends on the nearly circular retrograde family in the galaxy plane, at a 2-periodic (period doubling) bifurcation. Fig. 3, top, illustrates the evolution of shape of the orbits along the family. Near [FORMULA] the orbits look like twisted and bended eights, the extremities being bent toward the galaxy centre. At higher amplitudes and higher h the two eight lobes lean more and more toward the plane until they merge with the retrograde family at a period doubling bifurcation.

Following Broucke (1969), the linear stability of a 3D periodic orbit is given by 3 real parameters, equivalent but more economic to discuss than the 4 complex eigenvalues that would characterize the linear stability of a periodic orbit in an autonomous 3D Hamiltonian system. The stability parameters [FORMULA], [FORMULA] and [FORMULA] are explained in detail in, e.g., P85a. We just recall that a periodic orbit is stable whenever the two [FORMULA] are real and in the interval [FORMULA]. If [FORMULA] is real and negative, the [FORMULA] must be complex and we have complex instability.

Fig. 4 shows the evolution of the stability parameters [FORMULA], [FORMULA] as the amplitude varies for different bar masses. We see that there are different critical orbits announcing resonances: five with [FORMULA] (the last one is the terminating orbit of the family) and one (two for [FORMULA]) with [FORMULA] ; those orbits are candidates to possible bifurcations of other families, which we have not followed, our aim being to give a first order description of the 3D dynamics.

[FIGURE] Fig. 4. Stability parameters of the family of 3-D periodic orbits around [FORMULA] for [FORMULA], 0.60, 0.621, and 0.622. In the frame [FORMULA] the curve [FORMULA] gets very close to the value 2, but is not strictly tangent over a finite interval as one might suppose from the plot.

Then, for a fixed [FORMULA] we focus our attention on the evolution of the stability at the beginning of the family, when [FORMULA] varies in the interval [FORMULA] (which would correspond to speeds transverse to the plane up to approximately 100 km/s).

From Fig. 4 we see that for the values of [FORMULA] 0.60, and 0.621, that is [FORMULA], the periodic orbits are always stable, while for [FORMULA], they are complex unstable (the complex [FORMULA] and [FORMULA] are not plotted in the figure) for a certain interval and become stable afterwards, that is, when decreasing [FORMULA], there is a minimum value of [FORMULA] for which the curves [FORMULA] and [FORMULA] coincide and become complex for smaller [FORMULA].

Now, we analyze the precise evolution through the marginal value, for [FORMULA] increasing from 0.621 to 0.622. We show in Fig. 5, that already before achieving the marginal value, there is an interval of stable periodic orbits at low amplitude, an interval of complex unstable ones at an intermediate amplitude, and an interval of stable ones at high amplitudes (for instance Fig. 5, when [FORMULA]).

[FIGURE] Fig. 5. Stability parameters of the family of 3-D periodic orbits around [FORMULA] for [FORMULA], 0.62165, 0.62168, and 0.62170.

So, we can conclude that the effect of the change of stability (S- [FORMULA]) for the Lagrangian points [FORMULA] on the stability character of the periodic orbits around them is to delay the transition, and for increasing [FORMULA] the stability of the periodic orbits is given by the following patterns: stable, stable-complex unstable-stable, complex unstable-stable. In a schematic way, the propagation of the transition is shown by the transition diagram given in Fig. 6. It is visible that the concerned mass interval is tiny: 0.001. Above the critical mass value [FORMULA] all the periodic orbits with transverse velocities less than [FORMULA] are complex unstable.

[FIGURE] Fig. 6. Transition from stability to complex instability in a diagram [FORMULA]. The complex unstable regime is marked by [FORMULA], the stable one by S.

4.2. Orientation of the bifurcation

Once we have obtained the transition diagram, we focus our attention on the effect of the bifurcation on the central periodic family. On one hand, as far as the bifurcating manifolds from the transition orbit are concerned, we know that, if k denotes the rotation number of the transition orbit, there bifurcate periodic orbits or 2D invariant tori depending on whether k is rational or irrational respectively (see P85a; Heggie 1985).

The transition to complex instability is the Hamiltonian version of the Hopf bifurcation in dissipative systems. Considering the orientation of the bifurcation, the bifurcating structures may be "direct" (they unfold on the unstable side), or "inverse" (they unfold on the stable side). According to the orientation, the effect on the complex unstable central orbit is completely different: in the direct case it confines the chaotic orbits for some time, which may be long, or, in the inverse case, it allows an immediate escape (see P85ab for details).

Now the orientation of the bifurcation for the family of periodic orbits around [FORMULA] given in the diagram above is considered. We examine both the "vertical" evolution with variable [FORMULA] and [FORMULA] fixed, and the "horizontal" one with variable [FORMULA] and [FORMULA] fixed. The same procedure follows in the two cases. Let us fix for instance, [FORMULA] ; from the bifurcation diagram, when varying [FORMULA] we have two transitions from stability to complex instability, corresponding to [FORMULA] and [FORMULA] (see Fig. 5). To determine whether the associated bifurcations are direct or inverse, we take initial conditions close to the ones of a complex unstable periodic orbit with [FORMULA] and [FORMULA] close to [FORMULA] (the same happens for a complex unstable orbit with [FORMULA] close to [FORMULA] and [FORMULA]). If we plot the consequents of the corresponding orbit on the Poincaré section [FORMULA], they are not confined at all and the orbit escapes. Thus, the bifurcation unfolds on the stable side, i.e., it is of inverse type.

[FIGURE] Fig. 7. a [FORMULA] projections of the consequents of an orbit in the Poincaré section [FORMULA], [FORMULA] belonging to the stable region. b Envelope of the stable invariant tori around the central family obtained with the anti-dissipative method ([FORMULA]). The same initial conditions are taken in both cases: [FORMULA], [FORMULA], [FORMULA], [FORMULA].

It is interesting to see this effect on the invariant tori surrounding the stable central family. In order to reach the last invariant torus, we use the anti-dissipative procedure described in P85ab for symplectic mappings and galactic potentials. It consists in perturbing the equations of motion by an anti-dissipative term (i.e., by dilating phase-space volumes with time) proportional to a factor D slightly larger than 1 but conserving the initial Hamiltonian value h. We take initial conditions of an orbit belonging to the stable region; for [FORMULA], the Hamiltonian case, the orbit lies on a torus (see Fig. 7 a). For [FORMULA], the consequents on the section [FORMULA] explore larger and larger tori at constant h until they reach the last one. Then there is a sudden escape as visible on the last outer points in Fig. 7 b. An upper bound of the extent of the invariant tori is provided by the envelope of the accumulated points just before the escape. Fig. 7 thus confirms that the Hopf-like bifurcation unfolds on the stable side of the varying parameter [FORMULA], i.e., it is of inverse type.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998