Complex dynamics of Physical, Biological and Socio-Economical systems

Eguíluz, Víctor M. (Directors E. Hernandez-Garcia and O. Piro)
PhD Thesis (1999)

In this Thesis, I study different aspects of Nonlinear Science, focusing on applications
to other felds parallel to Physics. The common thread through this work is the type
of bifurcations that appear in each studied system.
In Chapter 2, I study the dynamical behavior of the hair cells in the cochlea (in-
ner ear). These cells are responsible of the transduction of sound pressure waves into
nervous impulses. I investigate the universal properties of oscillators in the vicinity
of a Hopf bifurcation to explain the behavior observed experimentally. In particu-
lar, I show that the amplitude-response curves of periodically forced oscillators have
the same characteristics as the sensitivity curves of our auditory system. In Chap-
ter 3, I present a mechanism that explains via a Turing bifurcation the formation of
a `sausage-string\' pattern that appears when increasing the arterial pressure in small
blood vessels. This structure appears as an instability due to the nonlinear stress-
strain relation of the blood vessel walls. In the context of extended dynamical systems
with some kind of disordered behavior, Chapter 4 is dedicated to the formation of
disordered structures in space but stationary in time, the so called spatial chaos. Al-
though frozen spatial chaos has been previously observed in other contexts, our work
is the first to show an example where its appearance is a consequence of the shape
of the domain and the boundary conditions. In Chapter 5, I study spatio-temporally
chaotic systems in bounded domains. A generic system displaying a chaotic regime
in a domain with boundary conditions other than periodic gives rise to a structured
time-averaged pattern similar to the ones experimentally observed. Changing the
boundary conditions, I fnd that the average also changes adjusting to the global
symmetry of the problem, including both the evolution equations and the boundary
conditions. Chapter 6 is dedicated to the complex Ginzburg-Landau equation that
is a model equation of extended dynamical system with a great wealth of dynamical
regimes. Studying this system in square, circular and stadium-like shaped domains
there appear solutions like targets, that are difficult to obtain without these contours.
Finally, Chapter 7 is dedicated to the study of dynamical systems with many interact-
ing components. In particular, I propose a model for the formation of opinion groups
in a financial market. The model displays several qualitative properties empirically
found in real markets.

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