截面的几何形状决定三维函数的混沌特性

Chaotic characteristics of three-dimensional function determined by cross-section geometric shape

计算仿真发现,函数f(x,y,z)=sin(k(x2+y2+z2)),f(x,y,z)=k(1-(x2+y2+z2))e(-(x+y+z+u)2),f(x,y,z)=k((x2+y2+z2)/3)(1-(x2+y2+z2)/3)分别与另外两个随机产生的二次多项式函数均可组合成一个三维离散动力系统,当系数k,u在一定范围内取值时,系统出现混沌吸引子的概率可以大于90%.通过绘制分岔图、Lyapunov指数图等对上述系统的混沌特性进行了分析.分析发现,出现混沌概率高的原因是这3个函数的截面都是中间凸起或中间凹陷的曲面,在这样的截面条件下系统容易出现混沌.这普遍适用于三维函数,利用这些三维离散动力系统绘制出的大量吸引子图形具有使用价值和研究价值.

The calculation and simulation results show that f（x, y, z） = sin（k（x2＋y2＋z2））, f（x, y, z） = k（1-（x2＋y2＋z2）） e（-（x＋y＋z＋u）2）, f（x, y, z） = k（（x2 ＋y2 ＋z2）/3）（1-（x2 ＋y2 ＋z2）/3） can easily constructe a three-dimensional （3D） discrete dynamic system by combining other two polynomial functions generated randomly. Through calculating Lyapunov exponent and drawing the bifurcation diagram, the characteristics of chaos of the function are confirmed, and according to the bifurcation diagram of parameters and the Lyapunov exponent curve more chaotic mapping functions are found. Analysis shows that the cross-section geometric shape can determine the chaotic characteristics of 3D function, and the cross-sections are all the median convex or middle concave surfaces, which can constructe chaotic dynamic systems easily. In the future, the mathematical description model and some basic theorems are to be further investigated and their results will be used to solve practical problems such as turbulence.