双参数平面上Duffing系统TLE的计算与分岔分析

摘要

The method of calculating the top Lyapunov exponent of a Duffing system in the multi-parameter space is given,and the distribution characteristic of the top Lyapunov exponent of the Duffing system in its two-parameter plane is calculated.Bifurcation characteristics of the Duffing system are discussed with the single parameter bifurcation diagram.The discussion results show that the Duffing system has various bifurcation curves including periodic jump,pitchfork bifurcation and period-doubling bifurcation in its two-parameter plane.The period-doubling bifurcation curve cycles constantly appear and nest each other,making the system finally evolve the chaotic state via period-doubling bifurcation sequences.These periodic bifurcation curves are truncated by the periodic jumping curve,making the system move into different periodic states via the cycle jump curve.The intersection of various bifurcation curves in the two-parameter plane makes the local bifurcation characteristic of the system become very complex.The calculation and analysis of the Duffing system prove that this method is effective and feasible in terms of computational chaos.%给出系统在参数空间最大Lyapunov指数的计算方法,计算Duffing系统在双参数平面上最大Lyapunov指数的分布特性.结合单参数分岔图讨论了Duffing系统在双参数平面上的分岔特性.结果表明系统在双参数平面上出现了周期跳跃、叉式分岔和倍周期分岔等各种分岔曲线,系统在倍周期分岔曲线环内不断嵌套新的倍周期分岔曲线环,使得系统最终经倍周期分岔序列进入混沌运动;这些倍周期分岔曲线环均被周期跳跃曲线截断,使得系统经过周期跳跃曲线后出现不同的周期运动.参数平面上各种分岔曲线的相交使得系统局部分岔特性变得极为复杂.通过对Duffing系统的计算与分析证明了本文方法在计算混沌问题方面的有效性与可行性.