Explore the application of the Poincaré-Bendixson theorem in proving the existence of limit cycles through worked examples, including a biochemical oscillator model of glycolysis. Delve into the construction of trapping regions guided by nullclines, and examine the implications of the theorem for two-dimensional differential equation systems. Gain insights into analytical examples in polar coordinates, nullcline analysis, and the identification of parameter regions where stable limit cycles exist. Understand the significance of this theorem in ruling out chaotic behavior in 2D systems and its relevance to biological oscillations.