We study some topological properties of dynamical systems. In particular the rela-
tionship between spatio-temporal chaotic and Li-Yorke sensitive dynamical systems
establishing that for minimal dynamical systems those properties are equivalent. In
the same direction we show that being a Li-Yorke sensitive dynamical system implies
that the system is also Li-Yorke chaotic. On the other hand we survey the possibility
of lifting some topological properties from a given dynamical system (Y, S) to an-
other (X, T). After studying some basic facts about topological dynamical systems,
we move to the particular case of interval maps. We know that through the knowl-
edge of interval maps, f : I → I, precious information about the chaotic behavior
of general nonlinear dynamical systems can be obtained. It is also well known that
the analysis of the spectrum of time series encloses important material related to the
signal itself. In this work we look for possible connections between chaotic dynamical
systems and the behavior of its Fourier coefficients. We have found that a natural
bridge between these two concepts is given by the total variation of a function and
its connection with the topological entropy associated to the n-th iteration, fn(x), of
the map. Working in a natural way using the Sobolev spaces Wp,q(I) we show how
the Fourier coefficients are related to the chaoticity of interval maps.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-1810 |
| Date | 02 June 2009 |
| Creators | Roque Sol, Marco A. |
| Contributors | Chen, Goong |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
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