慢变参数系统的渐近解、分叉与混沌

研究了一类慢变参数振子系统,通过摄动方法得到其对称周期解的渐近展开式,并与数值解进行了比较.此外,通过系统解的相图、功率谱、倍周期分叉图和Lyapunov指数的计算,分析了系统的倍周期分叉至混沌的过程.结果表明,随着系统的小参数的变化,此系统的运动将经历与Lorenz模型极为类似的分叉而进入混沌状态.此外还可明显看出,此系统比起Lorenz模型相对说来容易处理一些,因为可得出系统的对称周期解的解析表达式.

Asymptotic Solutions, Bifurcations and Chaosof Slow-Varying System

A non-linear system with slow-varying parameters is dealt with.By using perturbation theory,the asymptotic expressions of periodic solutions are obtained and compared with the numerical results. By the phase portraits, power spectrum analysis, bifurcation diagram and computation of the largest Lyapunov exponent,the process from period-doubling bifurcations to chaos is studied. It is shown that, following the variation of the system's small parameter, the motion of the system becomes chaotic through a similar bifurcation as that in the Lorenz model. Moreover, it is not difficult to find that the system is more tractable than the Lorenz model and the analytic form of the symmetric periodic solutions can be got easily.