3,089 research outputs found
Helping Students Master Concepts in Mechanics by Graphing with Spreadsheets
An example of a curricular activity to help students master concepts in mechanics is presented. Students measure positions and times of movements using calculators, and construct graphs using spreadsheets. Students learn to connect concepts in mechanics and reinforce them following a spiral approach of increasing complexity. Comments from students about the activity are also presented
Mean field theory of assortative networks of phase oscillators
Employing the Kuramoto model as an illustrative example, we show how the use
of the mean field approximation can be applied to large networks of phase
oscillators with assortativity. We then use the ansatz of Ott and Antonsen
[Chaos 19, 037113 (2008)] to reduce the mean field kinetic equations to a
system of ordinary differential equations. The resulting formulation is
illustrated by application to a network Kuramoto problem with degree
assortativity and correlation between the node degrees and the natural
oscillation frequencies. Good agreement is found between the solutions of the
reduced set of ordinary differential equations obtained from our theory and
full simulations of the system. These results highlight the ability of our
method to capture all the phase transitions (bifurcations) and system
attractors. One interesting result is that degree assortativity can induce
transitions from a steady macroscopic state to a temporally oscillating
macroscopic state through both (presumed) Hopf and SNIPER (saddle-node,
infinite period) bifurcations. Possible use of these techniques to a broad
class of phase oscillator network problems is discussed.Comment: 8 pages, 7 figure
Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans
The spatiotemporal dynamics of cardiac tissue is an active area of research
for biologists, physicists, and mathematicians. Of particular interest is the
study of period-doubling bifurcations and chaos due to their link with cardiac
arrhythmogenesis. In this paper we study the spatiotemporal dynamics of a
recently developed model for calcium-driven alternans in a one dimensional
cable of tissue. In particular, we observe in the cable coexistence of regions
with chaotic and multi-periodic dynamics over wide ranges of parameters. We
study these dynamics using global and local Lyapunov exponents and spatial
trajectory correlations. Interestingly, near nodes -- or phase reversals --
low-periodic dynamics prevail, while away from the nodes the dynamics tend to
be higher-periodic and eventually chaotic. Finally, we show that similar
coexisting multi-periodic and chaotic dynamics can also be observed in a
detailed ionic model
Synchronization in large directed networks of coupled phase oscillators
We extend recent theoretical approximations describing the transition to
synchronization in large undirected networks of coupled phase oscillators to
the case of directed networks. We also consider extensions to networks with
mixed positive/negative coupling strengths. We compare our theory with
numerical simulations and find good agreement
Negative-energy perturbations in cylindrical equilibria with a radial electric field
The impact of an equilibrium radial electric field on negative-energy
perturbations (NEPs) (which are potentially dangerous because they can lead to
either linear or nonlinear explosive instabilities) in cylindrical equilibria
of magnetically confined plasmas is investigated within the framework of
Maxwell-drift kinetic theory. It turns out that for wave vectors with a
non-vanishing component parallel to the magnetic field the conditions for the
existence of NEPs in equilibria with E=0 [G. N. Throumoulopoulos and D.
Pfirsch, Phys. Rev. E 53, 2767 (1996)] remain valid, while the condition for
the existence of perpendicular NEPs, which are found to be the most important
perturbations, is modified. For ( is the
electrostatic potential) and ( is
the total plasma pressure), a case which is of operational interest in magnetic
confinement systems, the existence of perpendicular NEPs depends on ,
where is the charge of the particle species . In this case the
electric field can reduce the NEPs activity in the edge region of tokamaklike
and stellaratorlike equilibria with identical parabolic pressure profiles, the
reduction of electron NEPs being more pronounced than that of ion NEPs.Comment: 30 pages, late
- …