scholarly journals Dynamical convexity of the Euler problem of two fixed centers

2017 ◽  
Vol 165 (2) ◽  
pp. 359-384 ◽  
Author(s):  
SEONGCHAN KIM
Keyword(s):  
Periodic Orbits ◽  
Universal Cover ◽  
Explicit Formulas ◽  
Energy Hypersurface ◽  
Collision Orbits ◽  
Euler Problem

AbstractWe give thorough analysis for the rotation functions of the critical orbits from which one can understand bifurcations of periodic orbits. Moreover, we give explicit formulas of the Conley–Zehnder indices of the interior and exterior collision orbits and show that the universal cover of the regularised energy hypersurface of the Euler problem is dynamically convex for energies below the critical Jacobi energy.

scholarly journals The Conley–Zehnder indices of the rotating Kepler problem

2012 ◽  
Vol 154 (2) ◽  
pp. 243-260 ◽  
Author(s):  
PETER ALBERS ◽  
JOEL W. FISH ◽  
URS FRAUENFELDER ◽  
OTTO VAN KOERT
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Circular Orbit ◽  
Kepler Problem ◽  
Universal Cover ◽  
Bounded Component ◽  
Energy Hypersurface

AbstractWe determine the Conley–Zehnder indices of all periodic orbits of the rotating Kepler problem for energies below the critical Jacobi energy. Consequently, we show the universal cover of the bounded component of the regularized energy hypersurface is dynamically convex. Moreover, in the universal cover there is always precisely one periodic orbit with Conley–Zehnder index 3, namely the lift of the doubly covered retrograde circular orbit.

scholarly journals Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

2019 ◽  
Vol 23 (01) ◽  
pp. 1950084
Author(s):  
Wenmin Gong
Keyword(s):  
Magnetic Field ◽  
Periodic Orbits ◽  
Homotopy Class ◽  
Linear Growth ◽  
Floer Homology ◽  
Hamiltonian Function ◽  
Universal Cover ◽  
Zero Section ◽  
Free Homotopy Class ◽  
The Given

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

A Higher-Order Period Function and Its Application

2015 ◽  
Vol 25 (10) ◽  
pp. 1550140 ◽  
Author(s):  
Linping Peng ◽  
Lianghaolong Lu ◽  
Zhaosheng Feng
Keyword(s):  
Periodic Orbits ◽  
Upper Bound ◽  
Critical Period ◽  
Higher Order ◽  
Quadratic System ◽  
Critical Periods ◽  
Explicit Formulas ◽  
Period Function ◽  
Isochronous System ◽  
Isochronous Centers

This paper derives explicit formulas of the q th period bifurcation function for any perturbed isochronous system with a center, which improve and generalize the corresponding results in the literature. Based on these formulas to the perturbed quadratic and quintic rigidly isochronous centers, we prove that under any small homogeneous perturbations, for ε in any order, at most one critical period bifurcates from the periodic orbits of the unperturbed quadratic system. For ε in order of 1, 2, 3, 4 and 5, at most three critical periods bifurcate from the periodic orbits of the unperturbed quintic system. Moreover, in each case, the upper bound is sharp. Finally, a family of perturbed quintic rigidly isochronous centers is shown, which has three, for ε in any order, as the exact upper bound of the number of critical periods.

scholarly journals Topology of tropical moduli of weighted stable curves

2020 ◽  
Vol 20 (4) ◽  
pp. 445-462
Author(s):  
Alois Cerbu ◽  
Steffen Marcus ◽  
Luke Peilen ◽  
Dhruv Ranganathan ◽  
Andrew Salmon
Keyword(s):  
Fundamental Group ◽  
Moduli Space ◽  
Betti Numbers ◽  
Weight Vector ◽  
Universal Cover ◽  
Explicit Formulas ◽  
Structural Result ◽  
Stable Curves ◽  
Dual Complex ◽  
Stark Contrast

AbstractThe moduli space Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett’s spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0, w is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space Δ0,w is disconnected and for which the fundamental group of Δ0,w has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri–Hampe–Markwig–Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces Δ1,w.

Adaptive Control of Transient Dynamics to Periodic Orbits

2014 ◽  
Vol 2 ◽  
pp. 82-85
Author(s):  
Hiroyasu Ando ◽  
Kazuyuki Aihara
Keyword(s):  
Adaptive Control ◽  
Periodic Orbits ◽  
Transient Dynamics

New family of Whitney numbers

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 309-320 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad Cakic ◽  
F.A. Shiha
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Stirling Numbers ◽  
Explicit Formulas ◽  
Basic Properties ◽  
New Family ◽  
Whitney Numbers ◽  
Special Cases

In this paper we give a new family of numbers, called ??-Whitney numbers, which gives generalization of many types of Whitney numbers and Stirling numbers. Some basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Finally many interesting special cases are derived.

On a generalization of the Rényi–Srivastava characterization of the Poisson law

2021 ◽  
Vol 58 (1) ◽  
pp. 68-82
Author(s):  
Jean-Renaud Pycke
Keyword(s):  
Life Insurance ◽  
Collective Model ◽  
New Method ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Compound Distributions ◽  
Poisson Law ◽  
Pierre Loti

AbstractWe give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

Sign in / Sign up

Export Citation Format

Share Document