In this thesis I explore three new topics in Dynamical Systems. In Chapters 2 and 3 I investigate the dynamics of a family of asynchronous linear systems. These systems are of interest as models for asynchronous processes in economics and computer science and as novel ways to solve linear equations. I find a tight sandwich of bounds relating the Lyapunov exponents of these asynchronous systems to the eigenvalue of their synchronous counterparts. Using ideas from the theory of IFSs I show how the random behavior of these systems can be quickly sampled and go some way to characterizing the associated probability measures. In Chapter 4 I consider another family of random linear dynamical system but this time over the Max-plus semi-ring. These models provide a linear way to model essentially non-linear queueing systems. I show how the topology of the queue network impacts on the dynamics, in particular I relate an eigenvalue of the adjacency matrix to the throughput of the queue. In Chapter 5 I consider non-smooth systems which can be used to model a wide variety of physical systems in engineering as well as systems in control and computer science. I introduce the Moving Average Transformation which allows us to systematically 'smooth' these systems enabling us to apply standard techniques that rely on some smoothness, for example computing Lyapunov exponents from time series data.

這一篇論文是說明矩陣中的元素如果滿足熱帶格雷斯曼就可以將它用樹型圖表示。第二章簡述熱帶數學的基本運算和介紹超曲面，作為熱帶格雷斯曼的先備知識。第三章介紹演化樹，在第二節中用兩種方法重建演化樹，第一種方法UPGMA是要在特定的條件下所建造的演化樹才會正確，第二種方法是利用矩陣中兩個物種間的距離假設的樹型配置圖，解聯立方程組得到各段距離，第三節中先介紹代數幾何中的格雷斯曼再聯結到熱帶幾何中的格雷斯曼。第四章是將第二種方法寫成Python程式，輸入矩陣中的元素，如果滿足熱帶格雷斯曼，就可以顯示它可能的樹型圖以及算出距離。