Multiple scales scheme for bifurcation in a delayed extended van der Pol oscillator

The van der Pol(VDP) oscillator is an essential nonlinear oscillator, which has been extensively studied. However, the results of delayed generalized VDP oscillator are relatively less. The issue of bifurcation control for a delayed extended VDP oscillator is fully considered in this paper. Firstly, a nonlinear time-delayed feedback controller is carefully designed to control the dynamics for the proposed oscillator, and the results of stability of such oscillator are obtained by means of stability switches theory. Secondly, a general explicit formula is derived to establish the properties of Hopf bifurcation by utilizing the method of multiple scales. Moreover, the impact of feedback gain on the bifurcation point for the controlled oscillator is demonstrated numerically. Finally, the effectiveness of theoretical results is verified via a numerical example. (C) 2017 Elsevier B.V. All rights reserved.