Bifurcation dynamics of complex period-1 motions to chaos in an electromagnetically tuned duffing oscillator

In this paper, bifurcation dynamics of periodic motions in the electromagnetically tuned Duffing oscillator are studied through symmetric period-1 to asymmetric period-2 motions. On the bifurcation routes, there exist one branch of symmetric period-1 motions and 4 branches of asymmetric period-1 motions, one branch of the four asymmetric period-1 motion branches with asymmetric period-2 motion to chaos is obtained. The corresponding stability and bifurcations of periodic motions are determined. The frequency-amplitude characteristics for bifurcation routes of such periodi-1 motions to chaos are presented. Numerical illustrations are presented for complex symmetric period-1 motions, asymmetric period-1 to period-2 motions. For low frequency, the complex period-1 motions are obtained, and for the high frequency, the simple period-1 motions are observed. The asymmetric period-1 motions from the symmetric period-1 motion are obtained through the saddle-node bifurcation, and the asymmetric period-2 motions from the asymmetric period-1 motions are obtained from the saddle-node bifurcation. The study of periodic motions can help design the vibration reduction and energy harvesting system.