43,403 research outputs found
Another approach to Runge-Kutta methods
The condition equations are derived by the introduction of a system of equivalent differential equations, avoiding the usual formalism with trees and elementary differentials. Solutions to the condition equations are found by direct optimization, avoiding the necessity to introduce simplifying assumptions upon the Runge-Kutta coefficients. More favourable coefficients, in view of rounding errors, are found
On the solvability of the Yakubovich linear-quadratic infinite horizon minimization problem
The Yakubovich Frequency Theorem, in its periodic version and in its general
nonautonomous extension, establishes conditions which are equivalent to
the global solvability of a minimization problem of infinite horizon type,
given by the integral in the positive half-line of a quadratic functional
subject to a control system. It also provides the unique minimizing pair
\lq\lq solution, control\rq\rq~and
the value of the minimum. In this paper we establish less restrictive conditions
under which the problem is partially solvable, characterize the set of
initial data for which the minimum exists, and obtain its value as well a
minimizing pair. The occurrence of exponential dichotomy and the
null character of the rotation number for a nonautonomous
linear Hamiltonian system defined
from the minimization problem are fundamental in the analysis.Ministerio de Economía y Competitividad / FEDER, MTM2015-66330-PMinisterio de Ciencia, Innovación y Universidades, RTI2018-096523-B-I00European Commission, H2020-MSCA-ITN-2014INDAM -- GNAMPA Project 201
Shape of an elastica under growth restricted by friction
We investigate the quasi-static growth of elastic fibers in the presence of
dry or viscous friction. An unusual form of destabilization beyond a critical
length is described. In order to characterize this phenomenon, a new definition
of stability against infinitesimal perturbations over finite time intervals is
proposed and a semi-analytical method for the determination of the critical
length is developed. The post-critical behavior of the system is studied by
using an appropriate numerical scheme based on variational methods. We find
post-critical shapes for uniformly distributed as well as for concentrated
growth and demonstrate convergence to a figure-8 shape for large lengths when
self-crossing is allowed. Comparison with simple physical experiments yields
reasonable accuracy of the theoretical predictions
On Norm-Based Estimations for Domains of Attraction in Nonlinear Time-Delay Systems
For nonlinear time-delay systems, domains of attraction are rarely studied
despite their importance for technological applications. The present paper
provides methodological hints for the determination of an upper bound on the
radius of attraction by numerical means. Thereby, the respective Banach space
for initial functions has to be selected and primary initial functions have to
be chosen. The latter are used in time-forward simulations to determine a first
upper bound on the radius of attraction. Thereafter, this upper bound is
refined by secondary initial functions, which result a posteriori from the
preceding simulations. Additionally, a bifurcation analysis should be
undertaken. This analysis results in a possible improvement of the previous
estimation. An example of a time-delayed swing equation demonstrates the
various aspects.Comment: 33 pages, 8 figures, "This is a pre-print of an article published in
'Nonlinear Dynamics'. The final authenticated version is available online at
https://doi.org/10.1007/s11071-020-05620-8
Hysteresis in Adiabatic Dynamical Systems: an Introduction
We give a nontechnical description of the behaviour of dynamical systems
governed by two distinct time scales. We discuss in particular memory effects,
such as bifurcation delay and hysteresis, and comment the scaling behaviour of
hysteresis cycles. These properties are illustrated on a few simple examples.Comment: 28 pages, 10 ps figures, AMS-LaTeX. This is the introduction of my
Ph.D. dissertation, available at
http://dpwww.epfl.ch/instituts/ipt/berglund/these.htm
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