Dynamical behaviors in a system with the double Hopf bifurcation under slow-fast excitations and the enveloping slow-fast analysis method

The double Hopf system under slow-fast excitations appears frequently in the excited model of mechanical structure. It is of great significance to analyze the bursting behavior and predict the amplitude of such system, particularly for controlling the oscillation. Focusing on solving the problem with the enveloping slow-fast analysis method, we compute the equilibrium branch as well as bifurcation points in two limiting cases. Regarding a certain oscillator, we manage to reveal the bifurcation mechanism and predict the oscillation amplitude with the bifurcation line, the enveloping line and so on. By gradually amplifying the amplitude of the slow-varying excitation, the evolution rule of the oscillator is obtained. It is found that the oscillator is dominated by the limit cycle induced by Hopf bifurcations with single mode for the small excitation amplitude, while the control range as well as oscillation amplitude can be estimated by the limit cycle in limiting cases. The single mode oscillator may transform into the mixed mode oscillator with increasing excitation amplitude. Initially, the movement of the oscillator is composed by both single mode oscillation and mixed mode oscillation. The bifurcation line as well as the Hopf bifurcation may be accounted for the transformation. Since the amplitude of the slow-varying excitation grows large enough, the movement of the oscillator is entirely composed by mixed mode oscillation. The bifurcation line is used to explain the switch between jumping and the regular oscillation, of which the amplitude may also be evaluated by the bilateral enveloping line in limiting cases. Numerical computation also reveals that the error of the method in estimating amplitude of the limit cycle oscillation can be constrained within 10% of the truth or around 10% as predicted.