8,510 research outputs found
Editorial: The pre-history of Chaos—An Interdisciplinary Journal of Nonlinear Science
Published versio
On Born's conjecture about optimal distribution of charges for an infinite ionic crystal
We study the problem for the optimal charge distribution on the sites of a
fixed Bravais lattice. In particular, we prove Born's conjecture about the
optimality of the rock-salt alternate distribution of charges on a cubic
lattice (and more generally on a d-dimensional orthorhombic lattice).
Furthermore, we study this problem on the two-dimensional triangular lattice
and we prove the optimality of a two-component honeycomb distribution of
charges. The results holds for a class of completely monotone interaction
potentials which includes Coulomb type interactions. In a more general setting,
we derive a connection between the optimal charge problem and a minimization
problem for the translated lattice theta function.Comment: 32 pages. 3 Figures. To appear in Journal of Nonlinear Science. DOI
:10.1007/s00332-018-9460-
Local stable and unstable manifolds and their control in nonautonomous finite-time flows
It is well-known that stable and unstable manifolds strongly influence fluid
motion in unsteady flows. These emanate from hyperbolic trajectories, with the
structures moving nonautonomously in time. The local directions of emanation at
each instance in time is the focus of this article. Within a nearly autonomous
setting, it is shown that these time-varying directions can be characterised
through the accumulated effect of velocity shear. Connections to Oseledets
spaces and projection operators in exponential dichotomies are established.
Availability of data for both infinite and finite time-intervals is considered.
With microfluidic flow control in mind, a methodology for manipulating these
directions in any prescribed time-varying fashion by applying a local velocity
shear is developed. The results are verified for both smoothly and
discontinuously time-varying directions using finite-time Lyapunov exponent
fields, and excellent agreement is obtained.Comment: Under consideration for publication in the Journal of Nonlinear
Science
Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems
We consider a dynamical system with state space , a smooth, compact subset
of some , and evolution given by , , ;
is invertible and the time may be discrete, , , or continuous, . Here we show that starting with a
continuous positive initial probability density , with respect
to , the smooth volume measure induced on by Lebesgue measure on , the expectation value of , with respect to any
stationary (i.e. time invariant) measure , is linear in , . depends only on and vanishes
when is absolutely continuous wrt .Comment: 7 pages, plain TeX; [email protected],
[email protected], [email protected], to appear in Chaos: An
Interdisciplinary Journal of Nonlinear Science, Volume 8, Issue
Chaos at the border of criticality
The present paper points out to a novel scenario for formation of chaotic
attractors in a class of models of excitable cell membranes near an
Andronov-Hopf bifurcation (AHB). The mechanism underlying chaotic dynamics
admits a simple and visual description in terms of the families of
one-dimensional first-return maps, which are constructed using the combination
of asymptotic and numerical techniques. The bifurcation structure of the
continuous system (specifically, the proximity to a degenerate AHB) endows the
Poincare map with distinct qualitative features such as unimodality and the
presence of the boundary layer, where the map is strongly expanding. This
structure of the map in turn explains the bifurcation scenarios in the
continuous system including chaotic mixed-mode oscillations near the border
between the regions of sub- and supercritical AHB. The proposed mechanism
yields the statistical properties of the mixed-mode oscillations in this
regime. The statistics predicted by the analysis of the Poincare map and those
observed in the numerical experiments of the continuous system show a very good
agreement.Comment: Chaos: An Interdisciplinary Journal of Nonlinear Science
(tentatively, Sept 2008
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