114,887 research outputs found
Periodic Solutions for Circular Restricted 4-body Problems with Newtonian Potentials
We study the existence of non-collision periodic solutions with Newtonian
potentials for the following planar restricted 4-body problems: Assume that the
given positive masses in a Lagrange configuration move in
circular obits around their center of masses, the sufficiently small mass moves
around some body. Using variational minimizing methods, we prove the existence
of minimizers for the Lagrangian action on anti-T/2 symmetric loop spaces.
Moreover, we prove the minimizers are non-collision periodic solutions with
some fixed wingding numbers
Global Weinstein Type Theorem on Multiple Rotating Periodic Solutions for Hamiltonian Systems
This paper concerns the existence of multiple rotating periodic solutions for
dimensional convex Hamiltonian systems. For the symplectic orthogonal
matrix , the rotating periodic solution has the form of ,
which might be periodic, anti-periodic, subharmonic or quasi-periodic according
to the structure of . It is proved that there exist at least
geometrically distinct rotating periodic solutions on a given convex energy
surface under a pinched condition, so our result corresponds to the well known
Ekeland and Lasry's theorem on periodic solutions. It seems that this is the
first attempt to solve the symmetric quasi-periodic problem on the global
energy surface. In order to prove the result, we introduce a new index on
rotating periodic orbits.Comment: arXiv admin note: text overlap with arXiv:1812.0583
Periodic orbits in Hamiltonian systems with involutory symmetries
We study the existence of families of periodic solutions in a neighbourhood
of a symmetric equilibrium point in two classes of Hamiltonian systems with
involutory symmetries. In both classes, involutions reverse the sign of the
Hamiltonian function. In the first class we study a Hamiltonian system with a
reversing involution R acting symplectically. We first recover a result of
Buzzi and Lamb showing that the equilibrium point is contained in a three
dimensional conical subspace which consists of a two parameter family of
periodic solutions with symmetry R and there may or may not exist two families
of non-symmetric periodic solutions, depending on the coefficients of the
Hamiltonian. In the second problem we study an equivariant Hamiltonian system
with a symmetry S that acts anti-symplectically. Generically, there is no
S-symmetric solution in a neighbourhood of the equilibrium point. Moreover, we
prove the existence of at least 2 and at most 12 families of non-symmetric
periodic solutions. We conclude with a brief study of systems with both forms
of symmetry, showing they have very similar structure to the system with
symmetry R
Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
Coupled-mode systems are used in physical literature to simplify the
nonlinear Maxwell and Gross-Pitaevskii equations with a small periodic
potential and to approximate localized solutions called gap solitons by
analytical expressions involving hyperbolic functions. We justify the use of
the one-dimensional stationary coupled-mode system for a relevant elliptic
problem by employing the method of Lyapunov--Schmidt reductions in Fourier
space. In particular, existence of periodic/anti-periodic and decaying
solutions is proved and the error terms are controlled in suitable norms. The
use of multi-dimensional stationary coupled-mode systems is justified for
analysis of bifurcations of periodic/anti-periodic solutions in a small
multi-dimensional periodic potential.Comment: 18 pages, no figure
Synchronization of Huygens' clocks and the Poincare method
We study two models of connected pendulum clocks synchronizing their
oscillations, a phenomenon originally observed by Huygens. The oscillation
angles are assumed to be small so that the pendulums are modeled by harmonic
oscillators, clock escapements are modeled by the van der Pol terms. The mass
ratio of the pendulum bobs to their casings is taken as a small parameter.
Analytic conditions for existence and stability of synchronization regimes, and
analytic expressions for their stable amplitudes and period corrections are
derived using the Poincare theorem on existence of periodic solutions in
autonomous quasi-linear systems. The anti-phase regime always exists and is
stable under variation of the system parameters. The in-phase regime may exist
and be stable, exist and be unstable, or not exist at all depending on
parameter values. As the damping in the frame connecting the clocks is
increased the in-phase stable amplitude and period are decreasing until the
regime first destabilizes and then disappears. The results are most complete
for the traditional three degrees of freedom model, where the clock casings and
the frame are consolidated into a single mass.Comment: 23 pages, 8 figure
On periodic solutions of nonlinear wave equations, including Einstein equations with a negative cosmological constant
We construct periodic solutions of nonlinear wave equations using analytic
continuation. The construction applies in particular to Einstein equations,
leading to infinite-dimensional families of time-periodic solutions of the
vacuum, or of the Einstein-Maxwell-dilaton-scalar
fields-Yang-Mills-Higgs-Chern-Simons- equations, with a negative
cosmological constant.Comment: 15 pages, 1 figur
Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations
We prove the most general theorem about spectral stability of multi-site
breathers in the discrete Klein-Gordon equation with a small coupling constant.
In the anti-continuum limit, multi-site breathers represent excited
oscillations at different sites of the lattice separated by a number of "holes"
(sites at rest). The theorem describes how the stability or instability of a
multi-site breather depends on the phase difference and distance between the
excited oscillators. Previously, only multi-site breathers with adjacent
excited sites were considered within the first-order perturbation theory. We
show that the stability of multi-site breathers with one-site holes change for
large-amplitude oscillations in soft nonlinear potentials. We also discover and
study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site
breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure
Breathers in the weakly coupled topological discrete sine-Gordon system
Existence of breather (spatially localized, time periodic, oscillatory)
solutions of the topological discrete sine-Gordon (TDSG) system, in the regime
of weak coupling, is proved. The novelty of this result is that, unlike the
systems previously considered in studies of discrete breathers, the TDSG system
does not decouple into independent oscillator units in the weak coupling limit.
The results of a systematic numerical study of these breathers are presented,
including breather initial profiles and a portrait of their domain of existence
in the frequency-coupling parameter space. It is found that the breathers are
uniformly qualitatively different from those found in conventional spatially
discrete systems.Comment: 19 pages, 4 figures. Section 4 (numerical analysis) completely
rewritte
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