Qualitative analysis in a delayed Van der Pol oscillator.

The main purpose of this manuscript is to introduce and compare two methods for discussing the nature of Hopf bifurcation, namely, a research method combining the normal form theory and the center manifold theorem and the other method of multiple scales. Taking a delayed Van der Pol oscillator as an example, the local stability as well as the occurrence of Hopf bifurcation are explored via choosing time delay as the bifurcation parameter. On the basis of two different methods respectively, the characters of Hopf bifurcation which includes the direction of Hopf bifurcation as well as the stability of bifurcating periodic solutions are analyzed. Finally, numerical simulations supporting the theoretical findings are given, and it is observed that the results of two methods are consistent. • Propose a delayed Van der Pol oscillator. • Discuss local stability and occurrence of Hopf bifurcation. • Use Center-manifold reduction and multiple scales to study Hopf bifurcation. • Numerical simulations are in accord with theoretical analysis. • The results of two methods are consistent. [ABSTRACT FROM AUTHOR]