Analytical period-3 motions to chaos in a hardening Duffing oscillator

作者：Luo Albert C J^*; Huang Jianzhe

来源：Nonlinear Dynamics, 2013, 73(3): 1905-1932.

DOI：10.1007/s11071-013-0913-9

• Generalized harmonic balance method
• Hardening Duffing oscillator
• Period-3 motions
• Bifurcation tree
• Nonlinear dynamical systems

In this paper, bifurcation trees of period-3 motions to chaos in the periodically forced, hardening Duffing oscillator are investigated analytically. Analytical solutions for period-3 and period-6 motions are used for the bifurcation trees of period-3 motions to chaos. Such bifurcation trees are based on the Hopf bifurcations of asymmetric period-3 motions. In addition, an independent symmetric period-3 motion without imbedding in chaos is discovered, and such a symmetric period-3 motion possesses saddle-node bifurcations only. The switching of symmetric to asymmetric period-3 motions is completed through saddle-node bifurcations, and the onset of asymmetric period-6 motions occurs at the Hopf bifurcations of asymmetric period-3 motions. Continuously, the onset of period-12 motions is at the Hopf bifurcation of asymmetric period-6 motions. With such bifurcation trees, the chaotic motions relative to asymmetric period-3 motions can be determined analytically. This investigation provides a systematic way to study analytical dynamics of chaos relative to period-m motions in nonlinear dynamical systems.

• 出版日期2013-8