25,255 research outputs found
A minimization principle for the description of time-dependent modes associated with transient instabilities
We introduce a minimization formulation for the determination of a
finite-dimensional, time-dependent, orthonormal basis that captures directions
of the phase space associated with transient instabilities. While these
instabilities have finite lifetime they can play a crucial role by either
altering the system dynamics through the activation of other instabilities, or
by creating sudden nonlinear energy transfers that lead to extreme responses.
However, their essentially transient character makes their description a
particularly challenging task. We develop a minimization framework that focuses
on the optimal approximation of the system dynamics in the neighborhood of the
system state. This minimization formulation results in differential equations
that evolve a time-dependent basis so that it optimally approximates the most
unstable directions. We demonstrate the capability of the method for two
families of problems: i) linear systems including the advection-diffusion
operator in a strongly non-normal regime as well as the Orr-Sommerfeld/Squire
operator, and ii) nonlinear problems including a low-dimensional system with
transient instabilities and the vertical jet in crossflow. We demonstrate that
the time-dependent subspace captures the strongly transient non-normal energy
growth (in the short time regime), while for longer times the modes capture the
expected asymptotic behavior
Numerical Study of a Lyapunov Functional for the Complex Ginzburg-Landau Equation
We numerically study in the one-dimensional case the validity of the
functional calculated by Graham and coworkers as a Lyapunov potential for the
Complex Ginzburg-Landau equation. In non-chaotic regions of parameter space the
functional decreases monotonically in time towards the plane wave attractors,
as expected for a Lyapunov functional, provided that no phase singularities are
encountered. In the phase turbulence region the potential relaxes towards a
value characteristic of the phase turbulent attractor, and the dynamics there
approximately preserves a constant value. There are however very small but
systematic deviations from the theoretical predictions, that increase when
going deeper in the phase turbulence region. In more disordered chaotic regimes
characterized by the presence of phase singularities the functional is
ill-defined and then not a correct Lyapunov potential.Comment: 20 pages,LaTeX, Postcript version with figures included available at
http://formentor.uib.es/~montagne/textos/nep
Practical implementation of nonlinear time series methods: The TISEAN package
Nonlinear time series analysis is becoming a more and more reliable tool for
the study of complicated dynamics from measurements. The concept of
low-dimensional chaos has proven to be fruitful in the understanding of many
complex phenomena despite the fact that very few natural systems have actually
been found to be low dimensional deterministic in the sense of the theory. In
order to evaluate the long term usefulness of the nonlinear time series
approach as inspired by chaos theory, it will be important that the
corresponding methods become more widely accessible. This paper, while not a
proper review on nonlinear time series analysis, tries to make a contribution
to this process by describing the actual implementation of the algorithms, and
their proper usage. Most of the methods require the choice of certain
parameters for each specific time series application. We will try to give
guidance in this respect. The scope and selection of topics in this article, as
well as the implementational choices that have been made, correspond to the
contents of the software package TISEAN which is publicly available from
http://www.mpipks-dresden.mpg.de/~tisean . In fact, this paper can be seen as
an extended manual for the TISEAN programs. It fills the gap between the
technical documentation and the existing literature, providing the necessary
entry points for a more thorough study of the theoretical background.Comment: 27 pages, 21 figures, downloadable software at
http://www.mpipks-dresden.mpg.de/~tisea
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