ETD

• chaos
• dynamical systems
• Lanford map
• maps
• non-linear dynamics

In the present thesis we investigate the dynamical properties of a discrete-time dynamical system, along with the corresponding dynamics arising when two identical copies of the original dynamical system are coupled. The one-dimensional dynamical system under investigation is the Lanford map $x_{k+1}=\alpha x_k+\frac{1}{2}x_k(1-x_k)\pmod1$, a piecewise continuous interval map, which, despite its simple analytical form, can generate a variety of interesting dynamical behaviors when changing the control parameter $\alpha$. In the current literature this map has received very limited attention, for this reason we first characterize its dynamical behaviors by studying its fixed points, bifurcation diagram, Lyapunov exponent and invariant measure as functions of $\alpha$. We then perform a similar analysis of two coupled Lanford maps and compare the results with the corresponding ones for the one-dimensional case. We consider three different coupling functions: diffusive, which exhibits a symmetry under variables' exchange; asymmetric, where the above symmetry is broken; and one-variable coupling. Here, in addition to $\alpha$, we also investigate the effects produced by varying the coupling strength. For each coupling function, we numerically determined the fixed points and their stability properties, bifurcation diagrams, Lyapunov exponents, invariant measures, ergodicity and mixing properties. The diffusive coupling essentially leaves the dynamics unchanged, until coupling becomes strong enough to destabilize fixed points and extend the region corresponding to chaotic behavior. By contrast, both asymmetric and one-variable couplings produce more evident differences, inducing even transient or intermittent chaotic regimes. The invariant measures corresponding to diffusive and asymmetric couplings reveal the formation of high-density clusters in phase space, while strong diffusive or one-variable couplings produce phase space partitioning into similar subdomains. Ergodicity is generally enhanced by coupling, though it may become weaker, or even be broken, where clusters are detected in the invariant measure or intermittent chaotic dynamics arise. Finally, mixing properties are also numerically investigated while varying $\alpha$ and the coupling strength by using Poincar{\'e} recurrence times. Such findings show unequivocally that even low-dimensional coupled maps can exhibit a very rich variety of dynamical behaviors, thus providing an extremely useful setting to investigate the relationships between deterministic chaos, invariant structures, ergodicity and mixing.