2,920 research outputs found
Capturing correlations in chaotic diffusion by approximation methods
We investigate three different methods for systematically approximating the
diffusion coefficient of a deterministic random walk on the line which contains
dynamical correlations that change irregularly under parameter variation.
Capturing these correlations by incorporating higher order terms, all schemes
converge to the analytically exact result. Two of these methods are based on
expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method
approximates Markov partitions and transition matrices by using the escape rate
theory of chaotic diffusion. We check the practicability of the different
methods by working them out analytically and numerically for a simple
one-dimensional map, study their convergence and critically discuss their
usefulness in identifying a possible fractal instability of parameter-dependent
diffusion, in case of dynamics where exact results for the diffusion
coefficient are not available.Comment: 11 pages, 5 figure
Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns
Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is
described by means of an infinite hierarchy of its unstable spatiotemporally
periodic solutions. An intrinsic parametrization of the corresponding invariant
set serves as accurate guide to the high-dimensional dynamics, and the periodic
orbit theory yields several global averages characterizing the chaotic
dynamics.Comment: Latex, ioplppt.sty and iopl10.sty, 18 pages, 11 PS-figures,
compressed and encoded with uufiles, 170 k
Invariant manifolds, discrete mechanics, and trajectory design for a mission to Titan
With an environment comparable to that of primordial Earth, a surface strewn with liquid hydrocarbon lakes, and an atmosphere denser than that of any other moon in the solar system, Saturn's largest moon Titan is a treasure trove of potential scientific discovery and is the target of a proposed NASA mission scheduled for launch in roughly one decade. A chief consideration associated with the design of any such mission is the constraint imposed by fuel limitations that accompany the spacecraft's journey between celestial bodies. In this study, we explore the use of patched three-body models in conjunction with a discrete mechanical optimization algorithm for the design of a fuel-efficient Saturnian moon tour focusing on Titan. In contrast to the use of traditional models for trajectory design such as the patched conic approximation, we exploit subtleties of the three-body problem, a classic problem from celestial mechanics that asks for the motion of three masses in space under mutual gravitational interaction, in order to slash fuel costs. In the process, we demonstrate the aptitude of the DMOC (Discrete Mechanics and Optimal Control) optimization algorithm in handling celestial mechanical trajectory optimization problems
On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular
The main part of this contribution to the special issue of EJM-B/Fluids
dedicated to Patrick Huerre outlines the problem of the subcritical transition
to turbulence in wall-bounded flows in its historical perspective with emphasis
on plane Couette flow, the flow generated between counter-translating parallel
planes. Subcritical here means discontinuous and direct, with strong
hysteresis. This is due to the existence of nontrivial flow regimes between the
global stability threshold Re_g, the upper bound for unconditional return to
the base flow, and the linear instability threshold Re_c characterized by
unconditional departure from the base flow. The transitional range around Re_g
is first discussed from an empirical viewpoint ({\S}1). The recent
determination of Re_g for pipe flow by Avila et al. (2011) is recalled. Plane
Couette flow is next examined. In laboratory conditions, its transitional range
displays an oblique pattern made of alternately laminar and turbulent bands, up
to a third threshold Re_t beyond which turbulence is uniform. Our current
theoretical understanding of the problem is next reviewed ({\S}2): linear
theory and non-normal amplification of perturbations; nonlinear approaches and
dynamical systems, basin boundaries and chaotic transients in minimal flow
units; spatiotemporal chaos in extended systems and the use of concepts from
statistical physics, spatiotemporal intermittency and directed percolation,
large deviations and extreme values. Two appendices present some recent
personal results obtained in plane Couette flow about patterning from numerical
simulations and modeling attempts.Comment: 35 pages, 7 figures, to appear in Eur. J. Mech B/Fluid
Accurate modelling of the low-order secondary resonances in the spin-orbit problem
We provide an analytical approximation to the dynamics in each of the three
most important low order secondary resonances (1:1, 2:1, and 3:1) bifurcating
from the synchronous primary resonance in the gravitational spin-orbit problem.
To this end we extend the perturbative approach introduced in Gkolias et. al.
(2016), based on normal form series computations. This allows to recover
analytically all non-trivial features of the phase space topology and
bifurcations associated with these resonances. Applications include the
characterization of spin states of irregular planetary satellites or double
systems of minor bodies with irregular shapes. The key ingredients of our
method are: i) the use of a detuning parameter measuring the distance from the
exact resonance, and ii) an efficient scheme to `book-keep' the series terms,
which allows to simultaneously treat all small parameters entering the problem.
Explicit formulas are provided for each secondary resonance, yielding i) the
time evolution of the spin state, ii) the form of phase portraits, iii) initial
conditions and stability for periodic solutions, and iv) bifurcation diagrams
associated with the periodic orbits. We give also error estimates of the
method, based on analyzing the asymptotic behavior of the remainder of the
normal form series.Comment: Accepted for publication in Communications in Nonlinear Science and
Numerical Simulatio
High-order adaptive methods for computing invariant manifolds of maps
The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps
- âŚ