A scaling investigation for a Van der Pol circuit: normal form applied to a Hopf bifurcation

Scaling laws are generally associated with changes in the spatial structure of dynamical systems due to variations of control parameters. In the local bifurcation theory, when an equilibrium point changes stability from stable to unstable and a stable limit cycle shows up, we say the system has undergone a Hopf bifurcation. Some of the basic questions that remain to be explored about Hopf bifurcation are the regimes for which certain scaling laws exist and whether the exponents obtained for system obeying kinds of dynamics are valid for others. Based on this scenario, we explore the evolution towards the steady state at Hopf bifurcation in the Van der Pol circuit. The simplicity of Van der Pol circuit and the ability to generate a variety of behaviours motivate the choice of the system. Through the scaling analysis, we obtained the scaling properties and the critical exponents that characterise the bifurcation in study.