Synchronization in chaotic systems: Coupling of chaotic maps, data assimilation and weather forecasting
Аннотации
The theme of this thesis is the synchronization of coupled chaotic systems.
Background and introductory material are presented in Chapter 1.
In Chapter 2,
we study the transition to coherence of ensembles of globally
coupled chaotic maps allowing for ensembles of non-identical maps and
for noise. The transition coupling strength
is determined from a transfer function of the perturbation evolution.
Analytical results are presented and tested using numerical
experiments.
One of our examples suggests that the validity of the perturbation
theory approach can be problematic for an ensemble of noiseless
identical `nonhyperbolic' maps, but can be restored by noise and/or parameter spread.
The problem of estimating the state of a large
evolving spatiotemporally chaotic system from noisy observations
and a model of the system dynamics
is studied in Chapters 3 - 5.
This problem, refered to as `data assimilation', can
be thought of as a synchorization problem
where one attempts to synchronize the model state to the system
state by using incoming data to correct
synchronization error.
In Chapter 3,
using a simple data assimilation technique, we show the possible
occurrence of temporally and spatially localized bursts
in the estimation error. We discuss the similarity of these bursts
to those occurring at the `bubbling transition' in
the synchronization of low dimensional chaotic systems.
In general, the model used for state estimation
is imperfect and does not exactly represent the system dynamics.
In Chapter 4 we modify an ensemble Kalman filter scheme
to incorporate the effect of model bias
for large chaotic systems based on augmentation of the system
state by the bias estimates,
and we consider different ways of parameterizing
the model bias.
The experimental results highlight the critical role played by the selection
of a good parameterization model for representing the form of
the possible bias in the model.
In Chapter 5 we further test the method developed in
Chapter 4 via numerical experiments employing previously
developed codes for global weather forecasting. The results suggest
that our method can be effective for obtaining improved forecasting
results when using an ensemble Kalman filter scheme.