Stability and bifurcation analysis of micro-electromechanical nonlinear coupling system with delay

In this paper, we study dynamics in delayed micro-electromechanical nonlinear coupling system, with particular attention focused on Hopf and Hopf-pitchfork bifurcations. Based on the distribution of eigenvalues, we prove that a sequence of Hopf and Hopf-pitchfork bifurcations occur at the trivial equilibrium as the delay increases and obtain the critical values of two types of bifurcations. Next, by applying the multiple time scales method, the normal forms near the Hopf and Hopf-pitchfork bifurcations critical points are derived. Finally, bifurcation analysis and numerical simulations are presented to demonstrate the application of the theoretical results. We show the regions near above bifurcation critical points in which the micro-electromechanical nonlinear coupling system exists stable fixed point or stable periodic solution. Detailed numerical analysis using MATLAB extends the local bifurcation analysis to a global picture, and stable windows are observed as we change control parameters. Namely, the stable fixed point and stable periodic solution can exist in large regions of unfolding parameters as the unfolding parameters increase away from the critical value.