Global Dynamical Data Models and Rayleigh-Bénard Convection

Nonlinear problems have come increasingly into the forefront in different fields of natural science. Very often the underlying physical model of complex systems is not known. Much effort has already been made to construct global models for dynamical systems based on the information contained in measured scalar time series. Our aim is to develop a global modeling and a prediction method for systems belonging to a certain class of differential equations in the case that only one single variable time series is available. The tests of methods on known chaotic dynamical systems by comparing correlation and topological measures of re-constructed and original systems are promising. Thermal convection is the most common type of flow in liquids or gases. Moreover, convection plays an important role in various technological setups and physical applications, notably in geophysics. Rayleigh-Bénard convection represents a simple fluid system that shows a sequence of transitions leading from a two-dimensional laminar flow to more complicated three-dimensional cells and finally to turbulence. By applying a Fourier mode expansion on the Boussinesq equations governing thermal convection in a three-dimensional spatial domain, we examine the appearing regular and chaotic motion for different system parameters.