Local stability, Hopf and resonant codimension-Two bifurcation in a harmonic oscillator with Two Time delays

A harmonic oscillator with two discrete time delays is considered. The local stability of the zero solution of this equation is investigated by analyzing the corresponding transcendental characteristic equation of its linearized equation and employing the Nyquist criterion. Some general stability criteria involving the delays and the system parameters are derived. By choosing one of the delays as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. The direction and stability of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. Resonant codimension-two bifurcation is also found to occur in this model. A complete description is given to the location of points in the parameter space at which the transcendental characteristic equation possesses two pairs of pure imaginary roots, ±iω1, ±iω2 with ω1:ω2 = m:n, where m and n are positive integers. Some numerical examples are finally given for justifying the theoretical results.