Bifurcations and chaos in three-well Duffing system with one external forcing

The dynamics of three-well Duffing system with one external forcing are investigated in detail. The conditions of existence and bifurcations for harmonics, subharmonics (second-order, third-order and m  -order) and superharmonics under small perturbations are given by using second-order averaging method and Melnikov’s method, the threshold values of chaotic motion under periodic perturbation are also given by Melnikov’s method. Moreover, the numerical simulations (including bifurcation curves, bifurcation diagrams in three-dimensional space and two-dimensional plane, phase portraits, leading Lyapunov Exponents and homoclinic and heteroclinic bifurcation surfaces in three-dimensional parametric space) not only show the consistence with the theoretical analyses but also exhibit more new complex dynamical behaviors, including cascades of period-doubling and reverse period doubling bifurcations, complex period windows, onset of chaos, symmetry-breaking, intermittent dynamics, different chaotic attractors, quasi-periodic orbits, and the system can leave the chaotic behavior to period-one orbit as parameters a, δ1δ1 and γ1γ1 increase. Combining the existing results of Li and Moon in 1990 with the new results reported in this paper, a more complete description of the system is now obtained.