【摘要】 简化Wiggins提出关于近Hamilton系统的Hopf分岔条件 ,并结合硬弹簧Duffing系统 ,研究该类系统的Hopf分岔行为 ,并用数值积分的方法验证结果的正确性。更多还原

【Abstract】 Hopf bifurcation conditions for a perturbed Hamiltonian system are studied by theoretical and numerical methods. Saddle node bifurcation of this system has been well studied, but Hopf bifurcation and torus motion of this system are not very clear yet. A series of concise Hopf bifurcation conditions via sub-Melnikov method and some mathematical skills are obtained. By the simplified formula deduced, the Hopf bifurcation curves can be easily found in the parameter space. Associated with the theoretical analysis, numerical simulations for a kind of Duffing equation are carried out. Numerical simulations verify that our theoretical results of odd and even number invariant circles for Hopf bifurcation correspond well to the KAM tori for the odd or even number order resonance in KAM theory. When the odd and even number order Hopf bifurcation conditions are satisfied at the same time, KAM tori for multi-resonance can be obtained, which may lead to further torus bifurcation of the system. This method may be useful to study how Hopf bifurcation connects with the KAM tori structure.更多还原