Chaos and Lyapunov Exponents: Analyzing Predictability

This lecture introduces the concept of Lyapunov exponents as a measure of chaos in dynamical systems. The instructor explains that for a system to be classified as chaotic, the largest Lyapunov exponent must be positive. The lecture details how to compute the spectrum of Lyapunov exponents, emphasizing the significance of the largest exponent, referred to as Kaivan, which indicates the degree of chaos. The instructor discusses the relationship between Lyapunov time and predictability, noting that a larger Kaivan corresponds to a more chaotic system. The lecture also covers the mathematical framework for analyzing perturbation vectors and their evolution over time, including the use of Jacobians and the concept of volume in the context of perturbations. The instructor illustrates how to construct orthogonal sets of vectors using the Gram-Schmidt process and explains the QR decomposition method for evaluating the Lyapunov exponents. The session concludes with practical considerations for implementing these concepts in computational algorithms.