此文档为旧博客搬运。——2020.05.11

Lorenz system is a simplified mathematical model to describe the atmospheric convection. The system consists of three ordinary differential equations given as:

\[\begin{aligned} \dfrac{\mathrm{d}x}{\mathrm{d}t} &= \sigma (y-x) \\\\ \dfrac{\mathrm{d}y}{\mathrm{d}t} &= x(\rho-z)-y \\\\ \dfrac{\mathrm{d}z}{\mathrm{d}t} &= xy -\beta z \end{aligned}\]

where $x$, $y$ and $z$ are the system state, $t$ is time, and $\sigma$, $\rho$, $\beta$ are the system parameters.

The above ODE system can be solved by the following code in Mathematica:


tend = 100;
eq = {x'[t] == \[Sigma] (y[t] - x[t]), 
   y'[t] == x[t] (\[Rho] - z[t]) - y[t], 
   z'[t] == x[t] y[t] - \[Beta] z[t]};
init1 = {x[0] == 10, y[0] == 10, z[0] == 10};
init2 = {x[0] == 10, y[0] == 10, z[0] == 10.1};
pars = {\[Sigma] -> 10, \[Rho] -> 28, \[Beta] -> 8/3};
{xs1, ys1, zs1} = 
  NDSolveValue[{eq /. pars, init1}, {x, y, z}, {t, 0, tend}];
{xs2, ys2, zs2} = 
  NDSolveValue[{eq /. pars, init2}, {x, y, z}, {t, 0, tend}];



ParametricPlot3D[{{xs1[t], ys1[t], zs1[t]}, {xs2[t], ys2[t], 
   zs2[t]}}, {t, 0, tend}, PlotStyle -> {Red, Blue}, 
 PlotPoints -> {1000, 1000}, Boxed -> False, Axes -> False]

A beautiful (well, the rendering color is not…) butterfly is traced out by the above code:

butterfly

For further description of the system and code, please refer to https://en.wikipedia.org/wiki/Lorenz_system

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