## 4. Vertical motion aroundOur 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 . ## 4.1. Families of 3-D periodic orbitsWith 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 and their type of stability/instability are calculated. Owing to the symmetry of the problem only the orbits need to be discussed.
In a first approach we carry out a numerical exploration of the
family of 3-D periodic orbits for different values of
close to the transition, more specifically,
, 0.60, 0.621, 0.622, keeping
. For any value of fixed,
we obtain the corresponding family of periodic orbits, beginning at
with , continuing with
increasing up to a maximum value, and
decreasing down to , 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 the orbits look like twisted and bended
eights, the extremities being bent toward the galaxy centre. At higher
amplitudes and higher 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 Fig. 4 shows the evolution of the stability parameters , as the amplitude varies for different bar masses. We see that there are different critical orbits announcing resonances: five with (the last one is the terminating orbit of the family) and one (two for ) with ; 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.
Then, for a fixed we focus our attention on the evolution of the stability at the beginning of the family, when varies in the interval (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 0.60, and 0.621, that is , the periodic orbits are always stable, while for , they are complex unstable (the complex and are not plotted in the figure) for a certain interval and become stable afterwards, that is, when decreasing , there is a minimum value of for which the curves and coincide and become complex for smaller . Now, we analyze the precise evolution through the marginal value,
for increasing from 0.621 to 0.622. We show in
Fig. 5, that
So, we can conclude that the effect of the change of stability (S-
) for the Lagrangian points
on the stability character of the periodic
orbits around them is to delay the transition, and for increasing
the stability of the periodic orbits is given
by the following patterns:
## 4.2. Orientation of the bifurcationOnce 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 The transition to complex instability is the Hamiltonian version of
the Hopf bifurcation in dissipative systems. Considering the
Now the orientation of the bifurcation for the family of periodic
orbits around given in the diagram above is
considered. We examine both the "vertical" evolution with variable
and fixed, and the
"horizontal" one with variable and
fixed. The same procedure follows in the two
cases. Let us fix for instance, ; from the
bifurcation diagram, when varying we have two
transitions from stability to complex instability, corresponding to
and (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 and
close to (the same
happens for a complex unstable orbit with close
to and ). If we plot the
consequents of the corresponding orbit on the Poincaré section
, they are not confined at all and the orbit
escapes. Thus, the bifurcation unfolds on the stable side, i.e., it is
of
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
© European Southern Observatory (ESO) 1998 Online publication: June 2, 1998 |