基于Lorenz映射的混沌系统分支变换预报规律研究

Rules for predicting regime change in the Lorenz chaotic system based on the Lorenz map

尽管Lorenz系统具有混沌和非周期性质,但其分支变换是可预报的.本文以强迫Lorenz系统为数学模型,基于Lorenz映射,研究了混沌系统分支变换的预报规律,将原有关于分支开始变换条件和新分支持续时间的两条一般规律扩展到了3条,并首次分析了系统当前状态达到变换条件所需时间的预报规律,从而为预报混沌系统非周期演变提供了另一途径.结果表明:映射尖点位置为分支变换的临界值,当变量z超过相应临界值时,系统在当前分支的运动即将结束,下一循环将跳跃到另一分支运动;系统在同一分支循环的次数随极值zmax单调减小,zmax越小,达到变换条件需循环的次数越多;系统在新分支持续的时间是先前分支最大极值zM的单调增加函数,zM越大,持续时间增加的幅度也越大.此外,外强迫影响着混沌系统分支变换的预报规律,其不但使正负分支的变换条件出现差异,且与新分支持续时间的增加速率和达到变换条件所需时间的递减速率密切相关.

Corresponding to two strange Lorenz attractors, in the Lorenz model there exist two opposite regimes which can be called as positive and negative regimes. Despite the trajectory of the Lorenz system changing between the two regimes back and forth with an unfixed period, the regime change is predictable. In this paper, with the help of the Lorenz map, three rules for predicting regime change are obtained. In particular, besides two generic predictable rules for the condition of regime transition and duration in new regime, a new rule about length for reaching transition condition, which has not been reported in previous work, is also very important. It provides another approach to forecasting the evolution of the nonlinear dynamical system. The results show that the position for highest point in cusps is the critical value for regime change. When the value of variable z is greater than the corresponding critical value, the current regime is about to end, and the Lorenz model will move to other regime in the next cycle. The length for reaching transition condition in the current regime decreases monotonically with local maximum value Zmax, and the smaller Zm^x in current status implies the bigger length for reaching transition condition. The duration in new regime increases monotonically with the maximum value zM in the previous regime, and the bigger the value of zM, the larger the range for the duration increase is. In addition, the forcing is also associated with the prediction rules for regime change. It not only makes transition conditions for positive and negative regimes different, but also determines the speed of decrease in length for reaching transition condition and the range of increase for duration in new regime.