86 research outputs found
Focusing Revisited: an MN-dynamics Approach
The nonlinear Schr{\"o}dinger (NLS) equation is a ubiquitous example of an
envelope wave equation for conservative, dispersive systems. We revisit here
the problem of self-similar focusing of waves in the case of the focusing NLS
equation through the prism of a dynamic renormalization technique (MN dynamics)
that factors out self-similarity and yields a bifurcation view of the onset of
focusing. As a result, identifying the focusing self-similar solution becomes a
steady state problem. The discretized steady states are subsequently obtained
and their linear stability is numerically examined. The calculations are
performed in the setting of variable index of refraction, in which the onset of
focusing appears as a supercritical bifurcation of a novel type of mixed
Hamiltonian-dissipative dynamical system (reminiscent, to some extent, of a
pitchfork bifurcation).Comment: 6 pages, 2 figure
Microscopic/stochastic timesteppers and coarse control: a kinetic Monte Carlo example
Coarse timesteppers provide a bridge between microscopic / stochastic system
descriptions and macroscopic tasks such as coarse stability/bifurcation
computations. Exploiting this computational enabling technology, we present a
framework for designing observers and controllers based on microscopic
simulations, that can be used for their coarse control. The proposed
methodology provides a bridge between traditional numerical analysis and
control theory on the one hand and microscopic simulation on the other
Coarse Brownian Dynamics for Nematic Liquid Crystals: Bifurcation Diagrams via Stochastic Simulation
We demonstrate how time-integration of stochastic differential equations
(i.e. Brownian dynamics simulations) can be combined with continuum numerical
bifurcation analysis techniques to analyze the dynamics of liquid crystalline
polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the
approach analyzes the (unavailable in closed form) coarse macroscopic
equations, estimating the necessary quantities through appropriately
initialized, short bursts of Brownian dynamics simulation. Through this
approach, both stable and unstable branches of the equilibrium bifurcation
diagram are obtained for the Doi model of LCPs and their coarse stability is
estimated. Additional macroscopic computational tasks enabled through this
approach, such as coarse projective integration and coarse stabilizing
controller design, are also demonstrated
Coarse-graining the dynamics of network evolution: the rise and fall of a networked society
We explore a systematic approach to studying the dynamics of evolving
networks at a coarse-grained, system level. We emphasize the importance of
finding good observables (network properties) in terms of which coarse grained
models can be developed. We illustrate our approach through a particular social
network model: the "rise and fall" of a networked society [1]: we implement our
low-dimensional description computationally using the equation-free approach
and show how it can be used to (a) accelerate simulations and (b) extract
system-level stability/bifurcation information from the detailed dynamic model.
We discuss other system-level tasks that can be enabled through such a
computer-assisted coarse graining approach.Comment: 18 pages, 11 figure
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