843 research outputs found
Center conditions for a simple class of quintic systems
We obtain center conditions for a -symmetric system of degree 5 for which
the origin is a uniformly isochronous singular point. In the revised paper some
misprints are corrected in the reference list.Comment: 9 pages, 0 figures, LaTeX 2.0
Regular and Singular Pulse and Front Solutions and Possible Isochronous Behavior in the Short-Pulse Equation: Phase-Plane, Multi-Infinite Series and Variational Approaches
In this paper we employ three recent analytical approaches to investigate the
possible classes of traveling wave solutions of some members of a family of
so-called short-pulse equations (SPE). A recent, novel application of
phase-plane analysis is first employed to show the existence of breaking kink
wave solutions in certain parameter regimes. Secondly, smooth traveling waves
are derived using a recent technique to derive convergent multi-infinite series
solutions for the homoclinic (heteroclinic) orbits of the traveling-wave
equations for the SPE equation, as well as for its generalized version with
arbitrary coefficients. These correspond to pulse (kink or shock) solutions
respectively of the original PDEs.
Unlike the majority of unaccelerated convergent series, high accuracy is
attained with relatively few terms. And finally, variational methods are
employed to generate families of both regular and embedded solitary wave
solutions for the SPE PDE. The technique for obtaining the embedded solitons
incorporates several recent generalizations of the usual variational technique
and it is thus topical in itself. One unusual feature of the solitary waves
derived here is that we are able to obtain them in analytical form (within the
assumed ansatz for the trial functions). Thus, a direct error analysis is
performed, showing the accuracy of the resulting solitary waves. Given the
importance of solitary wave solutions in wave dynamics and information
propagation in nonlinear PDEs, as well as the fact that not much is known about
solutions of the family of generalized SPE equations considered here, the
results obtained are both new and timely.Comment: accepted for publication in Communications in Nonlinear Science and
Numerical Simulatio
Two novel classes of solvable many-body problems of goldfish type with constraints
Two novel classes of many-body models with nonlinear interactions "of
goldfish type" are introduced. They are solvable provided the initial data
satisfy a single constraint (in one case; in the other, two constraints): i.
e., for such initial data the solution of their initial-value problem can be
achieved via algebraic operations, such as finding the eigenvalues of given
matrices or equivalently the zeros of known polynomials. Entirely isochronous
versions of some of these models are also exhibited: i.e., versions of these
models whose nonsingular solutions are all completely periodic with the same
period.Comment: 30 pages, 2 figure
Isochronous island bifurcations driven by resonant magnetic perturbations in Tokamaks
Recent evidences show that heteroclinic bifurcations in magnetic islands may
be caused by the amplitude variation of resonant magnetic perturbations in
tokamaks. To investigate the onset of these bifurcations, we consider a large
aspect ratio tokamak with an ergodic limiter composed of two pairs of rings
that create external primary perturbations with two sets of wave numbers. An
individual pair produces hyperbolic and elliptic periodic points, and its
associated islands, that are consistent with the Poincar\'e-Birkhoff fixed
point theorem. However, for two pairs producing external perturbations resonant
on the same rational surface, we show that different configurations of
isochronous island chains may appear on phase space according to the amplitude
of the electric currents in each pair of the ergodic limiter. When one of the
electric currents increases, isochronous bifurcations take place and new
islands are created with the same winding number as the preceding islands. We
present examples of bifurcation sequences displaying (a) direct transitions
from the island chain configuration generated by one of the pairs to the
configuration produced by the other pair, and (b) transitions with intermediate
configurations produced by the limiter pairs coupling. Furthermore, we identify
shearless bifurcations inside some isochronous islands, originating
nonmonotonic local winding number profiles with associated shearless invariant
curves
A Two-Process Model for Control of Legato Articulation Across a Wide Range of Tempos During Piano Performance
Prior reports indicated a non-linear increase in key overlap times (KOTs) as tempo slows for scales/arpeggios performed at internote intervals (INIs) of I00-1000 ms. Simulations illustrate that this function can be explained by a two-process model. An oscillating neural network based on dynamics of the vector-integration-to-endpoint model for central generation of voluntary actions, allows performers to compute an estimate of the time remaining before the oscillator's next cycle onset. At fixed successive threshold values of this estimate they first launch keystroke n+l and then lift keystroke n. As tempo slows, time required to pass between threshold crossings elongates, and KOT increases. If only this process prevailed, performers would produce longer than observed KOTs at the slowest tempo. The full data set is explicable if subjects lift keystroke n whenever they cross the second threshold or receive sensory feedback from stroke n+l, whichever comes earlier.Fulbright grant; Office of Naval Research (N00014-92-J-1309, N0014-95-1-0409
Traffic Abstractions of Nonlinear Homogeneous Event-Triggered Control Systems
In previous work, linear time-invariant event-triggered control (ETC) systems
were abstracted to finite-state systems that capture the original systems'
sampling behaviour. It was shown that these abstractions can be employed for
scheduling of communication traffic in networks of ETC loops. In this paper, we
extend this framework to the class of nonlinear homogeneous systems, however
adopting a different approach in a number of steps. Finally, we discuss how the
proposed methodology could be extended to general nonlinear systems
Traffic Abstractions of Nonlinear Homogeneous Event-Triggered Control Systems
In previous work, linear time-invariant event-triggered control (ETC) systems
were abstracted to finite-state systems that capture the original systems'
sampling behaviour. It was shown that these abstractions can be employed for
scheduling of communication traffic in networks of ETC loops. In this paper, we
extend this framework to the class of nonlinear homogeneous systems, however
adopting a different approach in a number of steps. Finally, we discuss how the
proposed methodology could be extended to general nonlinear systems
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