Three dimensional periodic orbits in three dipole problem

1996 ◽  
Vol 241 (2) ◽  
pp. 249-262
Author(s):  
P. G. Kazantzis ◽  
C. D. Desiniotis
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional

On the integrability of intermediate distributions for Anosov diffeomorphisms

1995 ◽  
Vol 15 (2) ◽  
pp. 317-331 ◽  
Author(s):  
M. Jiang ◽  
Ya B. Pesin ◽  
R. de la Llave
Keyword(s):  
Finite Number ◽  
Lyapunov Exponents ◽  
Periodic Orbits ◽  
Three Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphism ◽  
Anosov Diffeomorphisms ◽  
Stable Foliation

AbstractWe study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.

Topology of ambient manifolds of non-singular Morse – Smale flows with three periodic orbits

2021 ◽  
Vol 29 (6) ◽  
pp. 863-868
Author(s):  
Danila Shubin ◽  
◽  
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional ◽  
Solid Torus ◽  
Lens Space ◽  
Tubular Neighborhood ◽  
Lens Spaces ◽  
Ambient Manifold ◽  
Orientable Manifold ◽  
Smale Flows ◽  
Published Result

The purpose of this study is to establish the topological properties of three-dimensional manifolds which admit Morse – Smale flows without fixed points (non-singular or NMS-flows) and give examples of such manifolds that are not lens spaces. Despite the fact that it is known that any such manifold is a union of circular handles, their topology can be investigated additionally and refined in the case of a small number of orbits. For example, in the case of a flow with two non-twisted (having a tubular neighborhood homeomorphic to a solid torus) orbits, the topology of such manifolds is established exactly: any ambient manifold of an NMS-flow with two orbits is a lens space. Previously, it was believed that all prime manifolds admitting NMS-flows with at most three non-twisted orbits have the same topology. Methods. In this paper, we consider suspensions over Morse – Smale diffeomorphisms with three periodic orbits. These suspensions, in turn, are NMS-flows with three periodic trajectories. Universal coverings of the ambient manifolds of these flows and lens spaces are considered. Results. In this paper, we present a countable set of pairwise distinct simple 3-manifolds admitting NMS-flows with exactly three non-twisted orbits. Conclusion. From the results of this paper it follows that there is a countable set of pairwise distinct three-dimensional manifolds other than lens spaces, which refutes the previously published result that any simple orientable manifold admitting an NMS-flow with at most three orbits is lens space.

scholarly journals The Mechanism of Branching of Three-Dimensional Periodic Orbits from the Plane

1983 ◽  
Vol 74 ◽  
pp. 213-224
Author(s):  
I.A. Robin ◽  
V.V. Markellos
Keyword(s):  
Recent Work ◽  
Periodic Orbits ◽  
Restricted Problem ◽  
Three Dimensional ◽  
General Situation ◽  
Orbital Symmetry ◽  
Resonant Orbits ◽  
Symmetry Properties ◽  
Elliptic Restricted Problem ◽  
Symmetric Periodic Orbits

AbstractA linearised treatment is presented of vertical bifurcations of symmetric periodic orbits(bifurcations of plane with three-dimensional orbits) in the circular restricted problem. Recent work on bifurcations from vertical-critical orbits (av = ±1) is extended to deal with the v more general situation of bifurcations from vertical self-resonant orbits (av = cos(2Πn/m) for integer m,n) and it is shown that in this more general case bifurcating families of three-dimensional orbits always occur in pairs, the orbital symmetry properties being governed by the evenness or oddness of the integer m. The applicability of the theory to the elliptic restricted problem is discussed.

scholarly journals Reeb periodic orbits after a bypass attachment

2013 ◽  
Vol 35 (2) ◽  
pp. 615-672
Author(s):  
ANNE VAUGON
Keyword(s):  
Periodic Orbits ◽  
Contact Structure ◽  
Convex Surface ◽  
Three Dimensional ◽  
Contact Manifold ◽  
Contact Structures ◽  
Contact Homology ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Manifold With Boundary

AbstractOn a three-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.

Exact Torus Knot Periodic Orbits and Homoclinic Orbits in a Class of Three-Dimensional Flows Generated by a Planar Cubic System

2017 ◽  
Vol 27 (13) ◽  
pp. 1750205 ◽  
Author(s):  
Tonghua Zhang ◽  
Jibin Li
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Numerical Examples ◽  
Torus Knot ◽  
Cubic System ◽  
Theoretical Results

This paper considers a class of three-dimensional systems constructed by a rotating planar symmetric cubic vector field. To study its periodic orbits including homoclinic orbits, which may be knotted in space, we classify the types of periodic orbits and then calculate their exact parametric representations. Our study shows that this class of systems has infinitely many distinct types of knotted periodic orbits, which lie on three families of invariant tori. Numerical examples of [Formula: see text]-torus knot periodic orbits have also been provided to illustrate our theoretical results.

scholarly journals Periodic orbits relevant for the formation of inner rings in barred galaxies

1985 ◽  
Vol 106 ◽  
pp. 543-544
Author(s):  
M. Michalodimitrakis ◽  
Ch. Terzides
Keyword(s):  
Periodic Orbits ◽  
Three Dimensional ◽  
Particle Trapping ◽  
Barred Galaxies ◽  
Future Paper ◽  
Stability Of Periodic Orbits ◽  
Wide Range ◽  
The Galaxy ◽  
Stable Periodic ◽  
The Stability

The study of orbits of a test particle in the gravitational field of a model barred galaxy is a first step toward the understanding of the origin of the morphological characterstics observed in real barred galaxies. In this paper we confine our attention to the inner rings. Inner rings are a very common characteristic of barred galaxies. They are narrow, round or slightly elongated along the bar (with typical axial ratios from 0.7 to near 1.0), and of the same size as the bar. A first step to test the old hypothesis that inner rings consist of stars trapped near stable periodic orbits would be a study of particle trapping around periodic orbits encircling the bar. Such a study is contained in the work of several authors (Danby 1965, de Vaucouleurs and Freeman 1972, Michalodimitrakis 1975, Contopoulos and Papayannopoulos 1980, Athanassoula et al. 1983). In the above works the stability of periodic orbits was studied with respect to perturbations which lie on the plane of motion z = 0 (planar stability). To ensure the possibility of formation of rings, a study of stability with respect to perturbations perpendicular to the plane of motion (vertical stability) is necessary. In this paper we investigate the properties of periodic orbits which we believe to be relevant for the inner-ring problem using a sufficiently general model for the galaxy and sets of values for the parameters which cover a wide range of different possible cases. We also study the stability, planar and vertical, with respect to large perturbations in order to estimate the extent of particle trapping. A detailed numerical investigation of three-dimensional periodic orbits will be given in a future paper.

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