Using perturbative methods, this project will study properties of dynamical systems of very high dimension called Coupled Lattice Maps. These systems consist of many copies of a given simple and well-understood dynamical system, indexed by the points of a lattice, and interacting through a local coupling. The project will mainly focus on the statistical and geometric properties of solution trajectories. Particular attention will be devoted to global properties, including the fractal dimension of the attractor and the Lyapunov exponents and associated Lyapunov eigenspaces. Of particular interest is the possibility that the statistical properties of such systems may vary discontinuously under small modifications of the parameters that define the system (phase transitions). Finally, the results obtained for these maps will be extended, where possible, to coupled flows.
Coupled Lattice Maps are of interest as models in many different areas of research, including Statistical Mechanics, Fluid Dynamics, Neural Systems, Food Chains, and Microeconomics. In most of these fields the interest is in the statistical properties of the trajectories. These properties should explain the emergence of coherent macroscopic behavior, space-time chaos, and phase transitions. The most important characteristic of these models is the intrinsic dynamical instability (chaos) of the local dynamics. The theory of hyperbolic (chaotic) dynamical systems is well developed in systems with few degrees of freedom. Its extension to systems with many degrees of freedom presents new mathematical challenges and a strong potential for applications in the aforementioned fields of study.
