332 research outputs found
New Solutions for Slow Moving Kinks in a Forced Frenkel-Kontorova Chain
We construct new traveling wave solutions of moving kink type for a modified, driven, dynamic Frenkel-Kontorova model, representing dislocation motion under stress. Formal solutions known so far are inadmissible for velocities below a thresh- old value. The new solutions fill the gap left by this loss of admissibility. Analytical and numerical evidence is presented for their existence; however, dynamic simula- tions suggest that they are probably unstable
Schelling's Segregation Model: Parameters, scaling, and aggregation
Thomas Schelling proposed a simple spatial model to illustrate how, even with relatively mild assumptions on each individual's nearest neighbor preferences, an integrated city would likely unravel to a segregated city, even if all individuals prefer integration. This agent based lattice model has become quite influential amongst social scientists, demographers, and economists. Aggregation relates to individuals coming together to form groups and Schelling equated global aggregation with segregation. Many authors assumed that the segregation which Schelling observed in simulations on very small cities persists for larger, realistic size cities. We describe how different measures could be used to quantify the segregation and unlock its dependence on city size, disparate neighbor comfortability threshold, and population density. We identify distinct scales of global aggregation, and show that the striking global aggregation Schelling observed is strictly a small city phenomenon. We also discover several scaling laws for the aggregation measures. Along the way we prove that as the Schelling model evolves, the total perimeter of the interface between the different agents decreases, which provides a useful analytical tool to study the evolution.clusters, segregation, simulation, statistics
Using Resonances to Control Chaotic Mixing within a Translating and Rotating Droplet
Enhancing and controlling chaotic advection or chaotic mixing within liquid
droplets is crucial for a variety of applications including digital
microfluidic devices which use microscopic ``discrete'' fluid volumes
(droplets) as microreactors. In this work, we consider the Stokes flow of a
translating spherical liquid droplet which we perturb by imposing a
time-periodic rigid-body rotation. Using the tools of dynamical systems, we
have shown in previous work that the rotation not only leads to one or more
three-dimensional chaotic mixing regions, in which mixing occurs through the
stretching and folding of material lines, but also offers the possibility of
controlling both the size and the location of chaotic mixing within the drop.
Such a control was achieved through appropriate tuning of the amplitude and
frequency of the rotation in order to use resonances between the natural
frequencies of the system and those of the external forcing. In this paper, we
study the influence of the orientation of the rotation axis on the chaotic
mixing zones as a third parameter, as well as propose an experimental set up to
implement the techniques discussed.Comment: 15 pages, 6 figure
A Unifying Perspective: Solitary Traveling Waves As Discrete Breathers And Energy Criteria For Their Stability
In this work, we provide two complementary perspectives for the (spectral)
stability of solitary traveling waves in Hamiltonian nonlinear dynamical
lattices, of which the Fermi-Pasta-Ulam and the Toda lattice are prototypical
examples. One is as an eigenvalue problem for a stationary solution in a
co-traveling frame, while the other is as a periodic orbit modulo shifts. We
connect the eigenvalues of the former with the Floquet multipliers of the
latter and based on this formulation derive an energy-based spectral stability
criterion. It states that a sufficient (but not necessary) condition for a
change in the wave stability occurs when the functional dependence of the
energy (Hamiltonian) of the model on the wave velocity changes its
monotonicity. Moreover, near the critical velocity where the change of
stability occurs, we provide explicit leading-order computation of the unstable
eigenvalues, based on the second derivative of the Hamiltonian
evaluated at the critical velocity . We corroborate this conclusion with a
series of analytically and numerically tractable examples and discuss its
parallels with a recent energy-based criterion for the stability of discrete
breathers
An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices
In this work, we revisit a criterion, originally proposed in [Nonlinearity
{\bf 17}, 207 (2004)], for the stability of solitary traveling waves in
Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the
implications of this criterion from the point of view of stability theory, both
at the level of the spectral analysis of the advance-delay differential
equations in the co-traveling frame, as well as at that of the Floquet problem
arising when considering the traveling wave as a periodic orbit modulo a shift.
We establish the correspondence of these perspectives for the pertinent
eigenvalue and Floquet multiplier and provide explicit expressions for their
dependence on the velocity of the traveling wave in the vicinity of the
critical point. Numerical results are used to corroborate the relevant
predictions in two different models, where the stability may change twice. Some
extensions, generalizations and future directions of this investigation are
also discussed
Quasiadiabatic description of nonlinear particle dynamics in typical magnetotail configurations
International audienceIn the present paper we discuss the motion of charged particles in three different regions of the Earth magnetotail: in the region with magnetic field reversal and in the vicinities of neutral line of X- and O-types. The presence of small parameters (ratio of characteristic length scales in and perpendicular to the equatorial plane and the smallness of the electric field) allows us to introduce a hierarchy of motions and use methods of perturbation theory. We propose a parameter that plays the role of a measure of mixing in the system
Controlling Mixing Inside a Droplet by Time Dependent Rigid-body Rotation
The use of microscopic discrete fluid volumes (i.e., droplets) as
microreactors for digital microfluidic applications often requires mixing
enhancement and control within droplets. In this work, we consider a
translating spherical liquid droplet to which we impose a time periodic
rigid-body rotation which we model using the superposition of a Hill vortex and
an unsteady rigid body rotation. This perturbation in the form of a rotation
not only creates a three-dimensional chaotic mixing region, which operates
through the stretching and folding of material lines, but also offers the
possibility of controlling both the size and the location of the mixing. Such a
control is achieved by judiciously adjusting the three parameters that
characterize the rotation, i.e., the rotation amplitude, frequency and
orientation of the rotation. As the size of the mixing region is increased,
complete mixing within the drop is obtained.Comment: 6 pages, 6 figure
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