In this paper, a van der Pol-type oscillator with an additional external periodic force and a white Gaussian noise term is reported. Four possible scenarios with respect to the external periodic force and the white Gaussian noise term are discussed for three different oscillators, i.e. the classical van der Pol model and generalized models with supercritical and subcritical Hopf bifurcations, respectively. Then, a complete classification of phase portraits without a periodic force and a white Gaussian noise term is presented. Further, a complete dynamical analysis is performed for all three considered oscillators with additional external periodic force and without a white Gaussian noise term, respectively. In the meantime, a stiffness term is added and dynamical analysis is performed by utilizing different tools having the perturbed term with or without noise, respectively. For a deeper analysis, the Ott–Grebogi–Yorke (OGY) method is utilized with the noise term. The significance of stochastic effects in modeling nonlinear systems is highlighted, which show rich behavior such as quasi-periodicity, mild chaos, and fractal basin boundaries.
Wali et al. (Sat,) studied this question.