scholarly journals Diversified homotopic behavior of closed orbits of some -covered Anosov flows

2014 ◽  
Vol 36 (3) ◽  
pp. 767-780
Author(s):  
SÉRGIO R. FENLEY
Keyword(s):  
Periodic Orbits ◽  
Closed Orbit ◽  
The Other ◽  
Dehn Surgery ◽  
Geodesic Flows ◽  
Closed Orbits ◽  
Anosov Flows

We produce infinitely many examples of Anosov flows in closed $3$-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one other closed orbit. The examples are obtained by Dehn surgery on geodesic flows. The manifolds are toroidal and have Seifert pieces and atoroidal pieces in their torus decompositions.

PERIODIC ORBITS IN THE NEWTON–LEIPNIK SYSTEM

2002 ◽  
Vol 12 (03) ◽  
pp. 511-523 ◽  
Author(s):  
BENJAMIN A. MARLIN
Keyword(s):  
Periodic Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Closed Orbit ◽  
Numerical Results ◽  
Asymptotically Stable ◽  
System Of Differential Equations ◽  
Promising Candidate ◽  
Closed Orbits

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

scholarly journals Markov families for Anosov flows with an involutive action

1986 ◽  
Vol 104 ◽  
pp. 55-62 ◽  
Author(s):  
Toshiaki Adachi
Keyword(s):  
Dynamical Systems ◽  
Boundary Conditions ◽  
Periodic Orbits ◽  
Cross Sections ◽  
Negative Curvature ◽  
Finite Family ◽  
Geodesic Flows ◽  
Anosov Flows

The aim of this note is to construct “involutive” Markov families for geodesic flows of negative curvature. Roughly speaking, a Markov family for a flow is a finite family of local cross-sections to the flow with fine boundary conditions. They are basic tools in the study of dynamical systems. In 1973, R. Bowen [5] constructed Markov families for Axiom A flows. Using these families, he reduced the problem of counting periodic orbits of an Axiom A flow to the case of hyperbolic symbolic flows.

scholarly journals Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds

2014 ◽  
Vol 35 (6) ◽  
pp. 1681-1722 ◽  
Author(s):  
THIERRY BARBOT ◽  
SÉRGIO R. FENLEY
Keyword(s):  
Homotopy Class ◽  
Closed Orbit ◽  
Dehn Surgery ◽  
Anosov Flow ◽  
Free Homotopy Class ◽  
Graph Manifold ◽  
Finite Power ◽  
Graph Manifolds ◽  
Anosov Flows ◽  
Regular Fibers

In this article we analyze totally periodic pseudo-Anosov flows in graph 3-manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in Barbot and Fenley (Pseudo-Anosov flows in toroidal manifolds.Geom. Topol. 17(2013), 1877–1954). A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (i.e. there is a isotopy of$M$mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.

Error terms for closed orbits of hyperbolic flows

2001 ◽  
Vol 21 (2) ◽  
pp. 545-562 ◽  
Author(s):  
MARK POLLICOTT ◽  
RICHARD SHARP
Keyword(s):  
Error Term ◽  
Closed Orbit ◽  
Counting Function ◽  
Diophantine Condition ◽  
Weak Mixing ◽  
Closed Orbits ◽  
Hyperbolic Flows ◽  
Error Terms ◽  
Anosov Flows

In this paper we obtain a polynomial error term for the closed orbit counting function associated to certain hyperbolic flows. In the case of weak-mixing transitive Anosov flows no further conditions are required; for general weak-mixing hyperbolic flows a diophantine condition on the periods of the closed orbits is required.

scholarly journals Normal Form Near Orbit Segments of Convex Hamiltonian Systems

2021 ◽  
Author(s):  
Shahriar Aslani ◽  
Patrick Bernard
Keyword(s):  
Normal Form ◽  
Periodic Orbits ◽  
Hamiltonian Systems ◽  
Vector Fields ◽  
Energy Surface ◽  
Riemannian Metrics ◽  
Riemannian Metric ◽  
The Other ◽  
Finsler Metrics ◽  
The One

Abstract In the study of Hamiltonian systems on cotangent bundles, it is natural to perturb Hamiltonians by adding potentials (functions depending only on the base point). This led to the definition of Mañé genericity [ 8]: a property is generic if, given a Hamiltonian $H$, the set of potentials $g$ such that $H+g$ satisfies the property is generic. This notion is mostly used in the context of Hamiltonians that are convex in $p$, in the sense that $\partial ^2_{pp} H$ is positive definite at each point. We will also restrict our study to this situation. There is a close relation between perturbations of Hamiltonians by a small additive potential and perturbations by a positive factor close to one. Indeed, the Hamiltonians $H+g$ and $H/(1-g)$ have the same level one energy surface, hence their dynamics on this energy surface are reparametrisation of each other, this is the Maupertuis principle. This remark is particularly relevant when $H$ is homogeneous in the fibers (which corresponds to Finsler metrics) or even fiberwise quadratic (which corresponds to Riemannian metrics). In these cases, perturbations by potentials of the Hamiltonian correspond, up to parametrisation, to conformal perturbations of the metric. One of the widely studied aspects is to understand to what extent the return map associated to a periodic orbit can be modified by a small perturbation. This kind of question depends strongly on the context in which they are posed. Some of the most studied contexts are, in increasing order of difficulty, perturbations of general vector fields, perturbations of Hamiltonian systems inside the class of Hamiltonian systems, perturbations of Riemannian metrics inside the class of Riemannian metrics, and Mañé perturbations of convex Hamiltonians. It is for example well known that each vector field can be perturbed to a vector field with only hyperbolic periodic orbits, this is part of the Kupka–Smale Theorem, see [ 5, 13] (the other part of the Kupka–Smale Theorem states that the stable and unstable manifolds intersect transversally; it has also been studied in the various settings mentioned above but will not be discussed here). In the context of Hamiltonian vector fields, the statement has to be weakened, but it remains true that each Hamiltonian can be perturbed to a Hamiltonian with only non-degenerate periodic orbits (including the iterated ones), see [ 11, 12]. The same result is true in the context of Riemannian metrics: every Riemannian metric can be perturbed to a Riemannian metric with only non-degenerate closed geodesics, this is the bumpy metric theorem, see [ 1, 2, 4]. The question was investigated only much more recently in the context of Mañé perturbations of convex Hamiltonians, see [ 9, 10]. It is proved in [ 10] that the same result holds: if $H$ is a convex Hamiltonian and $a$ is a regular value of $H$, then there exist arbitrarily small potentials $g$ such that all periodic orbits (including iterated ones) of $H+g$ at energy $a$ are non-degenerate. The proof given in [ 10] is actually rather similar to the ones given in papers on the perturbations of Riemannian metrics. In all these proofs, it is very useful to work in appropriate coordinates around an orbit segment. In the Riemannian case, one can use the so-called Fermi coordinates. In the Hamiltonian case, appropriate coordinates are considered in [ 10,Lemma 3.1] itself taken from [ 3, Lemma C.1]. However, as we shall detail below, the proof of this Lemma in [ 3], Appendix C, is incomplete, and the statement itself is actually wrong. Our goal in the present paper is to state and prove a corrected version of this normal form Lemma. Our proof is different from the one outlined in [ 3], Appendix C. In particular, it is purely Hamiltonian and does not rest on the results of [ 7] on Finsler metrics, as [ 3] did. Although our normal form is weaker than the one claimed in [ 10], it is actually sufficient to prove the main results of [ 6, 10], as we shall explain after the statement of Theorem 1, and probably also of the other works using [ 3, Lemma C.1].

scholarly journals Periodic Orbits and Warps

1983 ◽  
Vol 100 ◽  
pp. 189-190
Author(s):  
W. A. Mulder
Keyword(s):  
Density Distribution ◽  
Periodic Orbits ◽  
Spiral Galaxies ◽  
Closed Orbits ◽  
The Galaxy

Orbit calculations were done in a rotating triaxial system with a density distribution in accordance with recent observations of spiral galaxies. A search was made for simple closed orbits which are tilted with respect to the plane of the galaxy. A family of stable prograde tilted orbits was found which can explain warps as stationary phenomena.

scholarly journals A Complete Family of Periodic Solutions of the Planar Three-Body Problem and their Stability

1977 ◽  
Vol 33 ◽  
pp. 159-159
Author(s):  
M. Hénon
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Body Problem ◽  
The Other ◽  
Three Body Problem ◽  
Complete Family ◽  
First Order ◽  
Hierarchical Arrangement ◽  
The Family ◽  
Three Body

AbstractWe give a complete description of a one-parameter family of periodic orbits in the planar problem of three bodies with equal masses. This family begins with a rectilinear orbit, computed by Schubart in 1956. It ends in retrograde revolution, i.e., a hierarchy of two binaries rotating in opposite directions. The first-order stability of the orbits in the plane is also computed. Orbits of the retrograde revolution type are stable; more unexpectedly, orbits of the “interplay” type at the other end of the family are also stable. This indicates the possible existence of triple stars with a motion entirely different from the usual hierarchical arrangement.

scholarly journals Hopf bifurcation and the existence and stability of closed orbits in three-sector models of optimal endogenous growth

2019 ◽  
Vol 23 (4) ◽  
Author(s):  
Kazuo Nishimura ◽  
Tadashi Shigoka
Keyword(s):  
Hopf Bifurcation ◽  
Endogenous Growth ◽  
Bifurcation Analysis ◽  
Stable Manifold ◽  
Closed Orbit ◽  
One Dimensional ◽  
Convex Structure ◽  
Equilibrium Dynamics ◽  
Closed Orbits ◽  
Existence And Stability

Abstract The present paper constructs a family of three-sector models of optimal endogenous growth, and conducts exact bifurcation analysis. In so doing, original six-dimensional equilibrium dynamics is decomposed into five-dimensional stationary autonomous dynamics and one-dimensional endogenously growing component. It is shown that the stationary dynamics thus decomposed undergoes supercritical Hopf bifurcation. It is inferred from the convex structure of our model that the dimension of a stable manifold of each closed orbit thus bifurcated in this five-dimensional dynamics should be two.

An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints

2010 ◽  
Vol 6 (3) ◽  
Author(s):  
Harry Dankowicz ◽  
Frank Schilder
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Bifurcation Analysis ◽  
The Other ◽  
Extended Formulation ◽  
Hybrid Dynamical System ◽  
Families Of Periodic Orbits ◽  
Continuation Problem ◽  
The One ◽  
Additional Constraints

This paper presents an extended formulation of the basic continuation problem for implicitly defined, embedded manifolds in Rn. The formulation is chosen so as to allow for the arbitrary imposition of additional constraints during continuation and the restriction to selective parametrizations of the corresponding higher-codimension solution manifolds. In particular, the formalism is demonstrated to clearly separate between the essential functionality required of core routines in application-oriented continuation packages, on the one hand, and the functionality provided by auxiliary toolboxes that encode classes of continuation problems and user definitions that narrowly focus on a particular problem implementation, on the other hand. Several examples are chosen to illustrate the formalism and its implementation in the recently developed continuation core package COCO and auxiliary toolboxes, including the continuation of families of periodic orbits in a hybrid dynamical system with impacts and friction as well as the detection and constrained continuation of selected degeneracies characteristic of such systems, such as grazing and switching-sliding bifurcations.

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