截断误差诱导阵发混沌与拓展维

The intermittency and extended dimension induced by truncation error

进一步研究过焦系统,找到了截断误差诱导阵发混沌的直接实验证据.实验显示,过焦系统的数字解呈现开关阵发混沌的特征,分析解仅是一个简单极限序列;影子效应支配系统演化;截断误差的累计误差是演化轨道的影子距离,影子距离随系统逼近不动点增加,随系统远离不动点减少,增加量与减少量统计上相等;在非双曲不动点的邻域内,存在一个阈值,当累计误差超过该阈值时,影子效应失灵,系统从吸引相跳到排斥相.一种合理的猜测是在非双曲不动点局域截断误差可以诱导出新维度——拓展维.

We further studied the cross-focus system and found direct experimental evidence for the truncation error induced intermittency. The experiment demonstrated that, the numerical solution of the cross-focus system shows features of on-off intermittency, while the analytical solution of the cross-focus system is only a simple limited sequence, and the shadowability controls the system evolution. The accumulation of truncation errors imply the shadowing distances of evolving orbits, and the shadowing distance increases on approaching the fixed point and decreases on leaving the fixed point, and the increment equals the decrement in the cycles statistically. In the vicinity of nonhyperbolic fixed point, there is a threshold value, when the accumulation surpasses this threshold value, a tailure of the shadowability takes place, the system jumps from the attraction phase to the repulsion phase. It is reasonably conjectured that a new extended dimension may occur locally in the vicinity of nonhyperbolic fixed point due to truncation errors.