The The Van der Pol equation: qualitative and numerical study - Pinto, Vinícius Justen (2023)
This expositive paper aims at the study of nonlinear equations, focused on the van der Pol equation, including deduction, qualitative analysis, and numerical examples. The van der Pol equation is deduced using an electrical circuit as a physical model. The qualitative analysis is divided into two parts: the theoretical enunciation and its application. The main theorems used in this study are Poincaré-Bendixson’s and Lyapunov’s. The construction of a Lyapunov function is also performed. Finally, a series of numerical examples are graphically presented using computational tools such as Python and Octave. The phase portraits and temporal behavior of the van der Pol equation are exhibited, along with the basin of attraction obtained experimentally, compared with the basin of attraction yielded by the Lyapunov function. Therefore, the numerical study provides a visual representation of the results stated in the qualitative analysis