1,226 research outputs found
A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
The Koopman operator is a linear but infinite dimensional operator that
governs the evolution of scalar observables defined on the state space of an
autonomous dynamical system, and is a powerful tool for the analysis and
decomposition of nonlinear dynamical systems. In this manuscript, we present a
data driven method for approximating the leading eigenvalues, eigenfunctions,
and modes of the Koopman operator. The method requires a data set of snapshot
pairs and a dictionary of scalar observables, but does not require explicit
governing equations or interaction with a "black box" integrator. We will show
that this approach is, in effect, an extension of Dynamic Mode Decomposition
(DMD), which has been used to approximate the Koopman eigenvalues and modes.
Furthermore, if the data provided to the method are generated by a Markov
process instead of a deterministic dynamical system, the algorithm approximates
the eigenfunctions of the Kolmogorov backward equation, which could be
considered as the "stochastic Koopman operator" [1]. Finally, four illustrative
examples are presented: two that highlight the quantitative performance of the
method when presented with either deterministic or stochastic data, and two
that show potential applications of the Koopman eigenfunctions
Arnold maps with noise: Differentiability and non-monotonicity of the rotation number
Arnold's standard circle maps are widely used to study the quasi-periodic
route to chaos and other phenomena associated with nonlinear dynamics in the
presence of two rationally unrelated periodicities. In particular, the El
Nino-Southern Oscillation (ENSO) phenomenon is a crucial component of climate
variability on interannual time scales and it is dominated by the seasonal
cycle, on the one hand, and an intrinsic oscillatory instability with a period
of a few years, on the other. The role of meteorological phenomena on much
shorter time scales, such as westerly wind bursts, has also been recognized and
modeled as additive noise. We consider herein Arnold maps with additive,
uniformly distributed noise. When the map's nonlinear term, scaled by the
parameter , is sufficiently small, i.e. , the map is
known to be a diffeomorphism and the rotation number is a
differentiable function of the driving frequency . We concentrate on
the rotation number's behavior as the nonlinearity becomes large, and show
rigorously that is a differentiable function of ,
even for , at every point at which the noise-perturbed map is
mixing. We also provide a formula for the derivative of the rotation number.
The reasoning relies on linear-response theory and a computer-aided proof. In
the diffeomorphism case of , the rotation number
behaves monotonically with respect to . We show, using again a
computer-aided proof, that this is not the case when and the
map is not a diffeomorphism.Comment: Electronic copy of final peer-reviewed manuscript accepted for
publication in the Journal of Statistical Physic
Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
Transfer operators such as the Perron--Frobenius or Koopman operator play an
important role in the global analysis of complex dynamical systems. The
eigenfunctions of these operators can be used to detect metastable sets, to
project the dynamics onto the dominant slow processes, or to separate
superimposed signals. We extend transfer operator theory to reproducing kernel
Hilbert spaces and show that these operators are related to Hilbert space
representations of conditional distributions, known as conditional mean
embeddings in the machine learning community. Moreover, numerical methods to
compute empirical estimates of these embeddings are akin to data-driven methods
for the approximation of transfer operators such as extended dynamic mode
decomposition and its variants. One main benefit of the presented kernel-based
approaches is that these methods can be applied to any domain where a
similarity measure given by a kernel is available. We illustrate the results
with the aid of guiding examples and highlight potential applications in
molecular dynamics as well as video and text data analysis
Dynamics of Oscillators Coupled by a Medium with Adaptive Impact
In this article we study the dynamics of coupled oscillators. We use
mechanical metronomes that are placed over a rigid base. The base moves by a
motor in a one-dimensional direction and the movements of the base follow some
functions of the phases of the metronomes (in other words, it is controlled to
move according to a provided function). Because of the motor and the feedback,
the phases of the metronomes affect the movements of the base while on the
other hand, when the base moves, it affects the phases of the metronomes in
return.
For a simple function for the base movement (such as in which is the velocity of the base,
is a multiplier, is a proportion and and
are phases of the metronomes), we show the effects on the dynamics of the
oscillators. Then we study how this function changes in time when its
parameters adapt by a feedback. By numerical simulations and experimental
tests, we show that the dynamic of the set of oscillators and the base tends to
evolve towards a certain region. This region is close to a transition in
dynamics of the oscillators; where more frequencies start to appear in the
frequency spectra of the phases of the metronomes
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