Asymptotical behaviour of mean field coupled maps

Ms. Anastasyia Panchuk

Abstract.
   The behaviour over time of the dynamical system of the globally
coupled maps is considered in the report. As is known such systems
can be in several different states: coherent, partially coherent
(or clustered), and turbulent (or chaotic). Furthermore, in first
two states periodicity may take place. The system is investigated
in order to find out if there exist periodical clusters for various
values of parameters.
   As a result of the experiment different periodical clustered
states were found and represented graphically. The phenomenon of
partial synchronisation - or clustering - in a system of globally
coupled chaotic maps is analysed for small coupling parameter epsilon.
Perfectly synchronised states with equally populated m clusters
moving with period n, referred to as PmCn-states, are of main
interest. The domains in a-epsilon plane for differently clustered
states, which are found numerically, rise out of each n-periodic
window for one-dimensional logistic map. Furthermore, it is checked
that all the states found are transversally stable for the whole
range of parameters. It is also proved that for any period n and
number of clusters m the corresponding PmCn-state is transversally
stable.
   Examined problem can be made use of in different areas of science
either exact or technical or social and economical. It can be useful
as for modelling the behaviour of coupled oscillators as for
examining bound markets, noise stable transferring information, or
networking, or anywhere where there are elements of similar type with
mutual coupling.