Quasiperiodic Stability Diagram in a Nonlinear Delayed Self-Excited Oscillator Under Parametric Coupling

We determine analytical approximation of quasiperiodic response and its stability diagram in a two degree-of-freedom weakly nonlinear coupled self-excited oscillator. The coupling effect is implemented through a linear parametrically modulated stiffness and the delay is introduced in the position and velocity. The second-step perturbation method is applied on the slow flow of the coupled oscillators to derive the slow-slow flow near the first and the second naturel frequencies. Quasiperiodic solution corresponding to nontrivial equilibrium of the slow-slow flow as well as its modulation envelope are predicted analytically. The stability analysis is carried out and the stability charts of quasiperiodic solution are provided near the two naturel frequencies of the system. The effect of time delay parameters on the amplitude of responses is examined. It is shown that for appropriate values of delay parameters, the amplitude of quasiperiodic solution increases substantially and bistability phenomenon may appear in a very small region of the stability diagram. This bistability region ca be controlled by acting on the delay amplitude in the position or in the velocity.