38,902 research outputs found
The slow-flow method of identification in nonlinear structural dynamics
The Hilbert-Huang transform (HHT) has been shown to be effective for characterizing a wide range of nonstationary signals in terms of elemental components through what has been called the empirical mode decomposition. The HHT has been utilized extensively despite the absence of a serious analytical foundation, as it provides a concise basis for the analysis of strongly nonlinear systems. In this paper, we attempt to provide the missing link, showing the relationship between the EMD and the slow-flow equations of the system. The slow-flow model is established by performing a partition between slow and fast dynamics using the complexification-averaging technique, and a dynamical system described by slowly-varying amplitudes and phases is obtained. These variables can also be extracted directly from the experimental measurements using the Hilbert transform coupled with the EMD. The comparison between the experimental and analytical results forms the basis of a nonlinear system identification method, termed the slow-flowmodel identification method, which is demonstrated using numerical examples
Comment of Global dynamics of biological systems
In a recent study, (Grigorov, 2006) analyzed temporal gene expression
profiles (Arbeitman et al., 2002) generated in a Drosophila experiment using
SSA in conjunction with Monte-Carlo SSA. The author (Grigorov, 2006) makes
three important claims in his article, namely:
Claim1: A new method based on the theory of nonlinear time series analysis is
used to capture the global dynamics of the fruit-fly cycle temporal gene
expression profiles.
Claim 2: Flattening of a significant part of the eigen-spectrum confirms the
hypothesis about an underly-ing high-dimensional chaotic generating process.
Claim 3: Monte-Carlo SSA can be used to establish whether a given time series
is distinguishable from any well-defined process including deterministic chaos.
In this report we present fundamental concerns with respect to the above
claims (Grigorov, 2006) in a systematic manner with simple examples. The
discussion provided especially discourages the choice of SSA for inferring
nonlinear dynamical structure form time series obtained in any biological
paradigm.Comment: 6 pages, 2 figure
Inference of stochastic nonlinear oscillators with applications to physiological problems
A new method of inferencing of coupled stochastic nonlinear oscillators is
described. The technique does not require extensive global optimization,
provides optimal compensation for noise-induced errors and is robust in a broad
range of dynamical models. We illustrate the main ideas of the technique by
inferencing a model of five globally and locally coupled noisy oscillators.
Specific modifications of the technique for inferencing hidden degrees of
freedom of coupled nonlinear oscillators is discussed in the context of
physiological applications.Comment: 11 pages, 10 figures, 2 tables Fluctuations and Noise 2004, SPIE
Conference, 25-28 May 2004 Gran Hotel Costa Meloneras Maspalomas, Gran
Canaria, Spai
Nonlinear model order reduction via Dynamic Mode Decomposition
We propose a new technique for obtaining reduced order models for nonlinear
dynamical systems. Specifically, we advocate the use of the recently developed
Dynamic Mode Decomposition (DMD), an equation-free method, to approximate the
nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix
that correlates spatial features while simultaneously associating the activity
with periodic temporal behavior. With this decomposition, one can obtain a
fully reduced dimensional surrogate model and avoid the evaluation of the
nonlinear term in the online stage. This allows for an impressive speed up of
the computational cost, and, at the same time, accurate approximations of the
problem. We present a suite of numerical tests to illustrate our approach and
to show the effectiveness of the method in comparison to existing approaches
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