3,628 research outputs found
Pattern Selection and Super-patterns in the Bounded Confidence Model
We study pattern formation in the bounded confidence model of opinion
dynamics. In this random process, opinion is quantified by a single variable.
Two agents may interact and reach a fair compromise, but only if their
difference of opinion falls below a fixed threshold. Starting from a uniform
distribution of opinions with compact support, a traveling wave forms and it
propagates from the domain boundary into the unstable uniform state.
Consequently, the system reaches a steady state with isolated clusters that are
separated by distance larger than the interaction range. These clusters form a
quasi-periodic pattern where the sizes of the clusters and the separations
between them are nearly constant. We obtain analytically the average separation
between clusters L. Interestingly, there are also very small quasi-periodic
modulations in the size of the clusters. The spatial periods of these
modulations are a series of integers that follow from the continued fraction
representation of the irrational average separation L.Comment: 6 pages, 6 figure
High accuracy simulations of black hole binaries:spins anti-aligned with the orbital angular momentum
High-accuracy binary black hole simulations are presented for black holes
with spins anti-aligned with the orbital angular momentum. The particular case
studied represents an equal-mass binary with spins of equal magnitude
S/m^2=0.43757 \pm 0.00001. The system has initial orbital eccentricity ~4e-5,
and is evolved through 10.6 orbits plus merger and ringdown. The remnant mass
and spin are M_f=(0.961109 \pm 0.000003)M and S_f/M_f^2=0.54781 \pm 0.00001,
respectively, where M is the mass during early inspiral. The gravitational
waveforms have accumulated numerical phase errors of <~ 0.1 radians without any
time or phase shifts, and <~ 0.01 radians when the waveforms are aligned with
suitable time and phase shifts. The waveform is extrapolated to infinity using
a procedure accurate to <~ 0.01 radians in phase, and the extrapolated waveform
differs by up to 0.13 radians in phase and about one percent in amplitude from
the waveform extracted at finite radius r=350M. The simulations employ
different choices for the constraint damping parameters in the wave zone; this
greatly reduces the effects of junk radiation, allowing the extraction of a
clean gravitational wave signal even very early in the simulation.Comment: 14 pages, 15 figure
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
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