2019, 40(10): 1122-1134.
doi: 10.21656/1000-0887.390234
Abstract:
Some Duffing systems with external excitation and quintic nonlinear restoring forces were considered, the amplitude-frequency response equation for the system was obtained with the multi-scale method, and the amplitude-frequency characteristic curves and their changing rules under different parameter changes were given. At the same time, the singularity theory was applied to get the transition sets and the corresponding topological structures of the system in 3 cases. Second, the fixed point of the system was determined, and the Hamiltonian function was used to get the heteroclinic orbit of the system, so the threshold of chaos in the Smale horseshoe sense was obtained with the Melnikov method. Then, the dynamic bifurcation and chaotic behavior of the system under external excitation and quintic nonlinear coefficients were given through numerical simulation. It is found that there are nonlinear phenomena such as periodic motion, period doubling motion, quasi periodic motion and chaos. The correctness of the theory was verified with nonlinear methods such as the Lyapunov exponent, the phase diagram and the Poincaré sections. The work provides a theoretical reference for further understanding of the nonlinear characteristics of Duffing systems and their evolution laws.
