Periodic response and chaos in nonlinear systems with parametric excitation and time delay

In this paper, the periodic motions of a nonlinear system with quadratic, cubic, and parametrically excited stiffness terms and with time-delay terms are obtained by the incremental harmonic balance (IHB) method. The elements of the Jacobian matrix and residue vector arising in the IHB formulation are derived in closed form. A mechanism model representing the one-mode oscillation of beams and plates is considered as an example. A path-following algorithm with an arc-length parametric continuation procedure is used to obtain the response diagrams. The system also exhibits chaotic motion through a cascade of period-doubling bifurcations, which is characterized by phase planes, Poincaré sections and Lyapunov exponents. The interpolated cell mapping (ICM) procedure is used to obtain the initial condition map corresponding to multiple steady-state solutions.