Explore chaotic dynamical systems in this 44-minute video lecture, focusing on their sensitive dependence on initial conditions and prevalence in natural and engineering systems. Discover how to recognize chaos and numerically integrate these systems, with examples ranging from planetary dynamics to double pendulums. Delve into concepts such as flow map Jacobian, Lyapunov exponents, and symplectic integration for chaotic Hamiltonian dynamics. Examine chaos in fluid turbulence and investigate the interplay between synchrony and order in dynamics. Gain insights into the complex behavior of systems found in turbulent fluids and celestial mechanics.