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| Titel |
Parameter estimation of nonlinear dynamical systems using synchronization |
| VerfasserIn |
Naoya Fujiwara, Jürgen Kurths |
| Konferenz |
EGU General Assembly 2011
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 13 (2011) |
| Datensatznummer |
250046870
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| Zusammenfassung |
| It is very important for climate science to model nonlinear dynamical systems, because
geosystem is governed by highly nonlinear equation of motion with huge degrees of freedom.
Even if we assume a model, it is a difficult task to estimate the parameters in the model. In
this study, we propose a new method which estimates parameter values of nonlinear
dynamical equations from measured data. We assume that the data x obeys an equation of
motion
-N
xË= cifi(x),
i=1
(1)
where x is a vector and fi’s are vector functions. Our goal is to estimate unknown
parameters ci. Here we employ the concept of synchronization between data and
model; we run the numerical simulation of a model equation which contains the
coupling term in order to achieve synchronization between them. We can expect that a
good model has a good synchronizability to the data, although it is impossible to
completely synchronize the model with data because we do not know the correct
value of ci. However, we found that if we introduce the nonlinear coupling between
them in an appropriate way, synchronization error is proportional to the difference
between the correct parameter value and one used in the model. This means that the
information of synchronization error can be converted to the information of the
parameter value of the system. If there are many unknown parameters (N > 1), we can
decompose the synchronization error into the ones originated from fi(x) term. This
decomposition enables us to estimate all of those unknown parameter values ci.
In the absence of noise, we prove that we can exactly achieve them by using the
proposed method. Our method is applicable if we have no a priori knowledge about the
system. In such a case we assume that the governing equation can be expanded in
Taylor series and determine all coefficients with our method. Note that our method
works even when the system shows the complex behavior, e.g., turbulent or chaotic
motion.
Moreover, our method works even in the presence of noise. We numerically verified that
parameters can be well estimated if there is additive noise term in the governing equation and
in the measurement, if the noise is not very strong. We will study the dependence of the
accuracy of the estimation on the noise intensity.
In many cases, it is impossible to measure all variables. Therefore we need to develop a
method to achieve a good estimation with a few observables. We propose a method to apply
the embedding technique which is widely used in nonlinear dynamics; we estimate the
parameters for the evolution equation in the embedded space.
In the presentation, we describe the method and show that we can estimate all parameters
exactly in an ideal case that all variables can be measured and there is no noise. We discuss
detailed properties of the method such as the robustness of the method, extension to
spatially dependent systems with large degrees of freedom, and cost for computation. |
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