1,416 research outputs found
On the Ionization of a Keplerian Binary System by Periodic Gravitational Radiation
The gravitational ionization of a Keplerian binary system via normally
incident periodic gravitational radiation of definite helicity is discussed.
The periodic orbits of the planar tidal equation are investigated on the basis
of degenerate continuation theory. The relevance of the Kolmogorov-Arnold-Moser
theory to the question of gravitational ionization is elucidated, and it is
conjectured that the process of ionization is closely related to the Arnold
diffusion of the perturbed system.Comment: 19 pages, REVTEX Style, To appear in JM
Symmetry breaking in periodic and solitary gravity-capillary waves on water of finite depth
A weakly nonlinear model is developed from the Hamiltonian formulation of water waves, to study the bifurcation structure of gravity-capillary waves on water of finite depth. It is found that, besides a very rich structure of symmetric solutions, non-symmetric Wilton's ripples exist. They appear via a spontaneous symmetry breaking bifurcation from symmetric solutions. The bifurcation tree is similar to that for gravity waves. The solitary wave with surface tension is studied with the same model close to a critical depth. It is found that the solution is not unique, and that further non-symmetric solitary waves are possible. The bifurcation tree has the same structure as for the case of periodic waves. The possibility of checking these results in low-gravity experiments is postulated
Weakly nonlinear non-symmetric gravity waves on water of finite depth
A weakly nonlinear Hamiltonian model for two-dimensional irrotational waves on water of finite depth is developed. The truncated model is used to study families of periodic travelling waves of permanent form. It is shown that non-symmetric periodic waves exist, which appear via spontaneous symmetry-breaking bifurcations from symmetric waves
Symmetry breaking perturbations and strange attractors
The asymmetrically forced, damped Duffing oscillator is introduced as a
prototype model for analyzing the homoclinic tangle of symmetric dissipative
systems with \textit{symmetry breaking} disturbances. Even a slight fixed
asymmetry in the perturbation may cause a substantial change in the asymptotic
behavior of the system, e.g. transitions from two sided to one sided strange
attractors as the other parameters are varied. Moreover, slight asymmetries may
cause substantial asymmetries in the relative size of the basins of attraction
of the unforced nearly symmetric attracting regions. These changes seems to be
associated with homoclinic bifurcations. Numerical evidence indicates that
\textit{strange attractors} appear near curves corresponding to specific
secondary homoclinic bifurcations. These curves are found using analytical
perturbational tools
Uniform approximations for pitchfork bifurcation sequences
In non-integrable Hamiltonian systems with mixed phase space and discrete
symmetries, sequences of pitchfork bifurcations of periodic orbits pave the way
from integrability to chaos. In extending the semiclassical trace formula for
the spectral density, we develop a uniform approximation for the combined
contribution of pitchfork bifurcation pairs. For a two-dimensional double-well
potential and the familiar H\'enon-Heiles potential, we obtain very good
agreement with exact quantum-mechanical calculations. We also consider the
integrable limit of the scenario which corresponds to the bifurcation of a
torus from an isolated periodic orbit. For the separable version of the
H\'enon-Heiles system we give an analytical uniform trace formula, which also
yields the correct harmonic-oscillator SU(2) limit at low energies, and obtain
excellent agreement with the slightly coarse-grained quantum-mechanical density
of states.Comment: LaTeX, 31 pp., 18 figs. Version (v3): correction of several misprint
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