Lyapunov's Theorem provides a powerful method for determining the stability of dynamical systems by using a scalar function known as the Lyapunov function. This theorem connects the concept of stability to energy-like properties, allowing for analysis of whether equilibrium points are stable or unstable without requiring explicit solutions to the system's equations. It plays a crucial role in understanding limit cycles and bifurcations in nonlinear control systems, as it helps characterize the behavior of trajectories in the state space around these points.
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