Abstract
The complex behaviors of Duffing equation with periodic damping and external excitations are investigated. The existence conditions and bifurcations of periodic orbits with three different frequencies resonant conditions are concerned by the second-order averaging method and the Melnikov method. The rich dynamical behaviors are so distinct when different periodic damping excitations are added, including more complicated averaged equations, bifurcation curves, bifurcation conditions, and even chaos. The numerical simulations show the consistence with the theoretical analysis and reveal new complex phenomena which cannot be given by theoretical analysis.
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