1,392 research outputs found
Forward and Adjoint Sensitivity Computation of Chaotic Dynamical Systems
This paper describes a forward algorithm and an adjoint algorithm for
computing sensitivity derivatives in chaotic dynamical systems, such as the
Lorenz attractor. The algorithms compute the derivative of long time averaged
"statistical" quantities to infinitesimal perturbations of the system
parameters. The algorithms are demonstrated on the Lorenz attractor. We show
that sensitivity derivatives of statistical quantities can be accurately
estimated using a single, short trajectory (over a time interval of 20) on the
Lorenz attractor.Comment: 3/7/2012: applied chain rule to Equation (7), so that Equations (15),
(16) and (17) go through w.o. differentiable Lyapunov vectors. = 4/10/2012:
Windowing scheme for forward sensitivity. Reduced heavy-tailedness in
computed derivatives. Updated Figs 8 and 9. = 9/7/2012: Minor revision,
accepted by JC
Probability density adjoint for sensitivity analysis of the Mean of Chaos
Sensitivity analysis, especially adjoint based sensitivity analysis, is a
powerful tool for engineering design which allows for the efficient computation
of sensitivities with respect to many parameters. However, these methods break
down when used to compute sensitivities of long-time averaged quantities in
chaotic dynamical systems.
The following paper presents a new method for sensitivity analysis of {\em
ergodic} chaotic dynamical systems, the density adjoint method. The method
involves solving the governing equations for the system's invariant measure and
its adjoint on the system's attractor manifold rather than in phase-space. This
new approach is derived for and demonstrated on one-dimensional chaotic maps
and the three-dimensional Lorenz system. It is found that the density adjoint
computes very finely detailed adjoint distributions and accurate sensitivities,
but suffers from large computational costs.Comment: 29 pages, 27 figure
Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations
The adjoint method, among other sensitivity analysis methods, can fail in
chaotic dynamical systems. The result from these methods can be too large,
often by orders of magnitude, when the result is the derivative of a long time
averaged quantity. This failure is known to be caused by ill-conditioned
initial value problems. This paper overcomes this failure by replacing the
initial value problem with the well-conditioned "least squares shadowing (LSS)
problem". The LSS problem is then linearized in our sensitivity analysis
algorithm, which computes a derivative that converges to the derivative of the
infinitely long time average. We demonstrate our algorithm in several dynamical
systems exhibiting both periodic and chaotic oscillations.Comment: submitted to JCP in revised for
Theory and computation of covariant Lyapunov vectors
Lyapunov exponents are well-known characteristic numbers that describe growth
rates of perturbations applied to a trajectory of a dynamical system in
different state space directions. Covariant (or characteristic) Lyapunov
vectors indicate these directions. Though the concept of these vectors has been
known for a long time, they became practically computable only recently due to
algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and
by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in
covariant Lyapunov vectors and their wide range of potential applications, in
this article we summarize the available information related to Lyapunov vectors
and provide a detailed explanation of both the theoretical basics and numerical
algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The
angles between these vectors and the original covariant vectors are
norm-independent and can be considered as characteristic numbers. Moreover, we
present and study in detail an improved approach for computing covariant
Lyapunov vectors. Also we describe, how one can test for hyperbolicity of
chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure
Least Squares Shadowing Sensitivity Analysis of a Modified Kuramoto-Sivashinsky Equation
Computational methods for sensitivity analysis are invaluable tools for
scientists and engineers investigating a wide range of physical phenomena.
However, many of these methods fail when applied to chaotic systems, such as
the Kuramoto-Sivashinsky (K-S) equation, which models a number of different
chaotic systems found in nature. The following paper discusses the application
of a new sensitivity analysis method developed by the authors to a modified K-S
equation. We find that least squares shadowing sensitivity analysis computes
accurate gradients for solutions corresponding to a wide range of system
parameters.Comment: 23 pages, 14 figures. Submitted to Chaos, Solitons and Fractals, in
revie
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