Chaos and Lyapunov Exponents: Properties and Dynamics

This lecture discusses the properties of chaos, focusing on Lyapunov exponents and their significance in chaotic systems. The instructor begins by reviewing the Jacobian matrix and its role in mapping neighborhoods of states along the flow. The concept of Lyapunov exponents is introduced, explaining how they characterize the expansion and compression of neighborhoods in chaotic dynamics. Positive Lyapunov exponents indicate sensitive dependence on initial conditions, while negative ones suggest convergence of nearby trajectories. The lecture further explores the implications of zero Lyapunov exponents and their relation to perturbations along the flow and continuous translations. The instructor then introduces the coupling York dimension as a measure of complexity in chaotic systems, linking it to the effective degrees of freedom involved in dynamics. The discussion includes examples of chaotic attractors, such as the Lorenz system, and emphasizes the importance of periodic orbits in chaotic dynamics. The lecture concludes with a definition of chaos based on Lyapunov exponents and the properties of chaotic systems.