134 research outputs found
Dynamics beyond dynamic jam; unfolding the Painlev\'e paradox singularity
This paper analyses in detail the dynamics in a neighbourhood of a
G\'enot-Brogliato point, colloquially termed the G-spot, which physically
represents so-called dynamic jam in rigid body mechanics with unilateral
contact and Coulomb friction. Such singular points arise in planar rigid body
problems with slipping point contacts at the intersection between the
conditions for onset of lift-off and for the Painlev\'e paradox. The G-spot can
be approached in finite time by an open set of initial conditions in a general
class of problems. The key question addressed is what happens next. In
principle trajectories could, at least instantaneously, lift off, continue in
slip, or undergo a so-called impact without collision. Such impacts are
non-local in momentum space and depend on properties evaluated away from the
G-spot. The results are illustrated on a particular physical example, namely
the a frictional impact oscillator first studied by Leine et al.
The answer is obtained via an analysis that involves a consistent contact
regularisation with a stiffness proportional to . Taking a
singular limit as , one finds an inner and an outer
asymptotic zone in the neighbourhood of the G-spot. Two distinct cases are
found according to whether the contact force becomes infinite or remains finite
as the G-spot is approached. In the former case it is argued that there can be
no such canards and so an impact without collision must occur. In the latter
case, the canard trajectory acts as a dividing surface between trajectories
that momentarily lift off and those that do not before taking the impact. The
orientation of the initial condition set leading to each eventuality is shown
to change each time a certain positive parameter passes through an
integer
Moving Embedded Solitons
The first theoretical results are reported predicting {\em moving} solitons
residing inside ({\it embedded} into) the continuous spectrum of radiation
modes. The model taken is a Bragg-grating medium with Kerr nonlinearity and
additional second-derivative (wave) terms. The moving embedded solitons (ESs)
are doubly isolated (of codimension 2), but, nevertheless, structurally stable.
Like quiescent ESs, moving ESs are argued to be stable to linear approximation,
and {\it semi}-stable nonlinearly. Estimates show that moving ESs may be
experimentally observed as 10 fs pulses with velocity th that
of light.Comment: 9 pages 2 figure
Embedded Solitons in a Three-Wave System
We report a rich spectrum of isolated solitons residing inside ({\it embedded
} into) the continuous radiation spectrum in a simple model of three-wave
spatial interaction in a second-harmonic-generating planar optical waveguide
equipped with a quasi-one-dimensional Bragg grating. An infinite sequence of
fundamental embedded solitons are found, which differ by the number of internal
oscillations. Branches of these zero-walkoff spatial solitons give rise,
through bifurcations, to several secondary branches of walking solitons. The
structure of the bifurcating branches suggests a multistable configuration of
spatial optical solitons, which may find straightforward applications for
all-optical switching.Comment: 5 pages 5 figures. To appear in Phys Rev
Thirring Solitons in the presence of dispersion
The effect of dispersion or diffraction on zero-velocity solitons is studied
for the generalized massive Thirring model describing a nonlinear optical fiber
with grating or parallel-coupled planar waveguides with misaligned axes. The
Thirring solitons existing at zero dispersion/diffraction are shown numerically
to be separated by a finite gap from three isolated soliton branches. Inside
the gap, there is an infinity of multi-soliton branches. Thus, the Thirring
solitons are structurally unstable. In another parameter region (far from the
Thirring limit), solitons exist everywhere.Comment: 12 pages, Latex. To appear in Phys. Rev. Let
Bifurcation of Limit Cycles from Boundary Equilibria in Impacting Hybrid Systems
A semianalytical method is derived for finding the existence and stability of single-impact periodicorbits born in a boundary equilibrium bifurcation in a generaln-dimensional impacting hybridsystem. Known results are reproduced for planar systems and general formulae derived for three-dimensional (3D) systems. A numerical implementation of the method is illustrated for several 3Dexamples and for an 8D wing-flap model that shows coexistence of attractors. It is shown how themethod can easily be embedded within numerical continuation, and some remarks are made aboutnecessary and sufficient conditions in arbitrary dimensional system
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