The article explores new ways to analyze the stability of complex dynamical systems using non-monotonic Lyapunov functions. These functions don't always decrease steadily over time, unlike traditional ones. The researchers show that these non-monotonic functions can be more flexible and less conservative in predicting stability. They introduce the concept of multiple non-monotonic Lyapunov functions, which can vary with different system motions. The study establishes conditions for stability, boundedness, and different types of stability for discontinuous dynamical systems. The findings broaden the understanding of stability in diverse systems and provide practical examples to illustrate their applicability.