33,517 research outputs found
A method to find unstable periodic orbits for the diamagnetic Kepler Problem
A method to determine the admissibility of symbolic sequences and to find the
unstable periodic orbits corresponding to allowed symbolic sequences for the
diamagnetic Kepler problem is proposed by using the ordering of stable and
unstable manifolds. By investigating the unstable periodic orbits up to length
6, a one to one correspondence between the unstable periodic orbits and their
corresponding symbolic sequences is shown under the system symmetry
decomposition
Instabilities and stickiness in a 3D rotating galactic potential
We study the dynamics in the neighborhood of simple and double unstable
periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic
type. In order to visualize the four dimensional spaces of section we use the
method of color and rotation. We investigate the structure of the invariant
manifolds that we found in the neighborhood of simple and double unstable
periodic orbits in the 4D spaces of section. We consider orbits in the
neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis
(the rotational axis of our system). Close to the transition points from
stability to simple instability, in the neighborhood of the bifurcated simple
unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the
asymptotic curves of the unstable manifold surround regions of the phase space
occupied by rotational tori existing in the region. For larger energies, away
from the bifurcating point, the consequents of the chaotic orbits form clouds
of points with mixing of color in their 4D representations. In the case of
double instability, close to x1v2 orbits, we find clouds of points in the four
dimensional spaces of section. However, in some cases of double unstable
periodic orbits belonging to the z-axis family we can visualize the associated
unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky
to this surface for long times (of the order of a Hubble time or more). Among
the orbits we studied we found those close to the double unstable orbits of the
x1v2 family having the largest diffusion speed.Comment: 29pages, 25 figures, accepted for publication in the International
Journal of Bifurcation and Chao
The structure and evolution of confined tori near a Hamiltonian Hopf Bifurcation
We study the orbital behavior at the neighborhood of complex unstable
periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a
transition of a family of periodic orbits from stability to complex instability
(also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable
periodic orbits move out of the unit circle. Then the periodic orbits become
complex unstable. In this paper we first integrate initial conditions close to
the ones of a complex unstable periodic orbit, which is close to the transition
point. Then, we plot the consequents of the corresponding orbit in a 4D surface
of section. To visualize this surface of section we use the method of color and
rotation [Patsis and Zachilas 1994]. We find that the consequents are contained
in 2D "confined tori". Then, we investigate the structure of the phase space in
the neighborhood of complex unstable periodic orbits, which are further away
from the transition point. In these cases we observe clouds of points in the 4D
surfaces of section. The transition between the two types of orbital behavior
is abrupt.Comment: 10 pages, 14 figures, accepted for publication in the International
Journal of Bifurcation and Chao
Planar resonant periodic orbits in Kuiper belt dynamics
In the framework of the planar restricted three body problem we study a
considerable number of resonances associated to the Kuiper Belt dynamics and
located between 30 and 48 a.u. Our study is based on the computation of
resonant periodic orbits and their stability. Stable periodic orbits are
surrounded by regular librations in phase space and in such domains the capture
of trans-Neptunian object is possible. All the periodic orbits found are
symmetric and there is evidence for the existence of asymmetric ones only in
few cases. In the present work first, second and third order resonances are
under consideration. In the planar circular case we found that most of the
periodic orbits are stable. The families of periodic orbits are temporarily
interrupted by collisions but they continue up to relatively large values of
the Jacobi constant and highly eccentric regular motion exists for all cases.
In the elliptic problem and for a particular eccentricity value of the primary
bodies the periodic orbits are isolated. The corresponding families, where they
belong to, bifurcate from specific periodic orbits of the circular problem and
seem to continue up to the rectilinear problem. Both stable and unstable orbits
are obtained for each case. In the elliptic problem the unstable orbits found
are associated with narrow chaotic domains in phase space. The evolution of the
orbits, which are located in such chaotic domains, seems to be practically
regular and bounded for long time intervals.Comment: preprint, 20 pages, 10 figure
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