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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.