近哈密顿系统的Hopf分岔

HOPF BIFURCATION OF A PERTURBED HAMILTONIAN SYSTEM

简化了Ｗｉｇｇｉｎｓ提出的关于近哈密顿系统的Ｈｏｐｆ分岔条件，并结合硬弹簧Ｄｕｆｆｉｎｇ系统，研究了该类系统的Ｈｏｐｆ分岔行为，并用数值积分的方法验证了结果的正确性．

Hopf bifurcation conditions are studied for a perturbed Hamiltonian system in this paper by theoretical and numerical method. Saddle node bifurcation of such system have been well studied by now, but its Hopf bifurcation and torns motion are not very clear. This paper obtains a series of concise Hopf bifurcation conditions via sub-Melnikov method and some mathematical skills, these conditions were mentioned in some early resarches but they are very complicated and unpractical. Using the simplified formula deduced in this paper, we can find the Hopf bifurcation curves easily in the parameter space. Associated with the theoretical analysis, numerical simulations about a kind of Duffing equation are carried out. Numerical simulations show that our theory is correct because we get many odd number invariant circles and even number ones resulting from Hopf bifurcation separately(we call them odd number order or even number order Hopf bifurcation respectively) according to the parameters obtained by our theory analysis, and these invariant circles clearly correspond to the KAM ton 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, many interesting KAM ton corresponding multi-resonance can be obtained, which may be connected with further torns bifurcation of the system. So this paper's method may be very useful to study how Hopf bifurcation connects with the KAM ton structure, further work will be done.