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

The calculation and simulation results show that f (x, y, z) = sin (k(x(2) + y(2) + z(2))), f (x, y, z) = k (1 - (x(2) + y(2) + z(2)))e((-(x+y+z+u)2)), f (x, y, z) = k ((x(2) + y(2) + z(2))/3) (1 - (x(2) + y(2) + z(2))/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.