Statistical Properties of High Dimensional Nonstationary Dynamical Systems

Classical dynamical systems involves the study of the long-time behavior of a fixed map or vector field. When dynamical instabilities are present, it is advantageous to study the dynamical system from a statistical perspective. It is important to move beyond the classical setup in order to model a more diverse array of physical and biochemical phenomena. The recent theory of nonstationary dynamical systems endeavors to do exactly that. In this theory, the dynamical model itself varies in time. This allows modelers to handle dynamical processes that evolve in time-varying environments, as well as systems with time-varying parameters. This dissertation formulates and solves two novel problems in nonstationary dynamical systems. The first project concerns what we call the quasistatic limit, an idea inspired by quasistatic processes in thermodynamics. We address the following question: If one assumes that the dynamical model itself varies sufficiently slowly, is it possible to recover a quasistatic ergodic theorem? We answer this question affirmatively for a class of quasistatic dynamical systems built from piecewise-smooth expanding maps in higher dimensions. The second project moves the theory of coupled map lattices (CMLs) into the nonstationary realm. CMLs have been used extensively to model phenomena in biology and physics. Classically, a CML consists of a lattice (or graph), a local dynamical system at each lattice site, and interactions between different lattice sites. Here, we allow the local dynamical model at each lattice site to vary in time, thereby producing nonstationary CMLs, a novel construct. For a certain class of nonstationary CMLs, we define a notion of statistical memory loss, an analog of decay of correlations. We then prove that memory is lost at an exponential rate. A common theme links the two parts of the dissertation: Dimension is high.