2,397 research outputs found
Resonance bifurcations of robust heteroclinic networks
Robust heteroclinic cycles are known to change stability in resonance
bifurcations, which occur when an algebraic condition on the eigenvalues of the
system is satisfied and which typically result in the creation or destruction
of a long-period periodic orbit. Resonance bifurcations for heteroclinic
networks are more complicated because different subcycles in the network can
undergo resonance at different parameter values, but have, until now, not been
systematically studied. In this article we present the first investigation of
resonance bifurcations in heteroclinic networks. Specifically, we study two
heteroclinic networks in and consider the dynamics that occurs as
various subcycles in each network change stability. The two cases are
distinguished by whether or not one of the equilibria in the network has real
or complex contracting eigenvalues. We construct two-dimensional Poincare
return maps and use these to investigate the dynamics of trajectories near the
network. At least one equilibrium solution in each network has a
two-dimensional unstable manifold, and we use the technique developed in [18]
to keep track of all trajectories within these manifolds. In the case with real
eigenvalues, we show that the asymptotically stable network loses stability
first when one of two distinguished cycles in the network goes through
resonance and two or six periodic orbits appear. In the complex case, we show
that an infinite number of stable and unstable periodic orbits are created at
resonance, and these may coexist with a chaotic attractor. There is a further
resonance, for which the eigenvalue combination is a property of the entire
network, after which the periodic orbits which originated from the individual
resonances may interact. We illustrate some of our results with a numerical
example.Comment: 46 pages, 20 figures. Supplementary material (two animated gifs) can
be found on
http://www.maths.leeds.ac.uk/~alastair/papers/KPR_res_net_abs.htm
Testing the no-hair theorem with GW150914
We analyze gravitational-wave data from the first LIGO detection of a binary
black-hole merger (GW150914) in search of the ringdown of the remnant black
hole. Using observations beginning at the peak of the signal, we find evidence
of the fundamental quasinormal mode and at least one overtone, both associated
with the dominant angular mode (), with confidence. A
ringdown model including overtones allows us to measure the final mass and spin
magnitude of the remnant exclusively from postinspiral data, obtaining an
estimate in agreement with the values inferred from the full signal. The mass
and spin values we measure from the ringdown agree with those obtained using
solely the fundamental mode at a later time, but have smaller uncertainties.
Agreement between the postinspiral measurements of mass and spin and those
using the full waveform supports the hypothesis that the GW150914 merger
produced a Kerr black hole, as predicted by general relativity, and provides a
test of the no-hair theorem at the level. An independent
measurement of the frequency of the first overtone yields agreement with the
no-hair hypothesis at the level. As the detector sensitivity
improves and the detected population of black hole mergers grows, we can expect
that using overtones will provide even stronger tests.Comment: v2: journal versio
The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit
The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic
cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known
to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various
phase space variables, but details of the mechanisms underlying the complicated dynamics have
not previously been investigated. We identify global bifurcations that induce the onset of chaotic
dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis
of approximate return maps, locate the global bifurcations in parameter space. We find there is a
threshold in the size of certain symmetry-breaking terms below which there can be no persistent
switching. Our results are illustrated by a numerical example
Treating instabilities in a hyperbolic formulation of Einstein's equations
We have recently constructed a numerical code that evolves a spherically
symmetric spacetime using a hyperbolic formulation of Einstein's equations. For
the case of a Schwarzschild black hole, this code works well at early times,
but quickly becomes inaccurate on a time scale of 10-100 M, where M is the mass
of the hole. We present an analytic method that facilitates the detection of
instabilities. Using this method, we identify a term in the evolution equations
that leads to a rapidly-growing mode in the solution. After eliminating this
term from the evolution equations by means of algebraic constraints, we can
achieve free evolution for times exceeding 10000M. We discuss the implications
for three-dimensional simulations.Comment: 13 pages, 9 figures. To appear in Phys. Rev.
Numerical simulations of neutron star-black hole binaries in the near-equal-mass regime
Simulations of neutron star-black hole (NSBH) binaries generally consider
black holes with masses in the range , where we expect to find
most stellar mass black holes. The existence of lower mass black holes,
however, cannot be theoretically ruled out. Low-mass black holes in binary
systems with a neutron star companion could mimic neutron star-neutron (NSNS)
binaries, as they power similar gravitational wave (GW) and electromagnetic
(EM) signals. To understand the differences and similarities between NSNS
mergers and low-mass NSBH mergers, numerical simulations are required. Here, we
perform a set of simulations of low-mass NSBH mergers, including systems
compatible with GW170817. Our simulations use a composition and temperature
dependent equation of state (DD2) and approximate neutrino transport, but no
magnetic fields. We find that low-mass NSBH mergers produce remnant disks
significantly less massive than previously expected, and consistent with the
post-merger outflow mass inferred from GW170817 for moderately asymmetric mass
ratio. The dynamical ejecta produced by systems compatible with GW170817 is
negligible except if the mass ratio and black hole spin are at the edge of the
allowed parameter space. That dynamical ejecta is cold, neutron-rich, and
surprisingly slow for ejecta produced during the tidal disruption of a neutron
star : . We also find that the final mass of the remnant
black hole is consistent with existing analytical predictions, while the final
spin of that black hole is noticeably larger than expected -- up to for our equal mass case
Casimir forces from a loop integral formulation
We reformulate the Casimir force in the presence of a non-trivial background.
The force may be written in terms of loop variables, the loop being a curve
around the scattering sites. A natural path ordering of exponentials take place
when a particular representation of the scattering centres is given. The basic
object to be evaluated is a reduced (or abbreviated) classical pseudo-action
that can be operator valued.Comment: references added, text clarified in place
On Toroidal Horizons in Binary Black Hole Inspirals
We examine the structure of the event horizon for numerical simulations of
two black holes that begin in a quasicircular orbit, inspiral, and finally
merge. We find that the spatial cross section of the merged event horizon has
spherical topology (to the limit of our resolution), despite the expectation
that generic binary black hole mergers in the absence of symmetries should
result in an event horizon that briefly has a toroidal cross section. Using
insight gained from our numerical simulations, we investigate how the choice of
time slicing affects both the spatial cross section of the event horizon and
the locus of points at which generators of the event horizon cross. To ensure
the robustness of our conclusions, our results are checked at multiple
numerical resolutions. 3D visualization data for these resolutions are
available for public access online. We find that the structure of the horizon
generators in our simulations is consistent with expectations, and the lack of
toroidal horizons in our simulations is due to our choice of time slicing.Comment: Submitted to Phys. Rev.
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