[Lorenz Chaos] Verilog Implementation of Lorenz Chaos System Based on FPGA

1. Software version

matlab2021a ， quartusii

2. Theoretical knowledge of this algorithm

In 1963, when Lorenz was studying the phenomenon of atmospheric convection, he discovered the first three-dimensional autonomous chaotic system with a simple structure, which is the famous "butterfly effect" model. Its mathematical model is:

    

 

When r≤1, it holds for all x, y, z, and the equation holds only when r<1, x=y=z=0 and r=1, x=y, z=0.

When r>1, the system begins to show instability, and when r increases to rc, there is a subcritical Hopf bifurcation.

When r > rc, adjacent orbitals near the attractor are on average exponentially separated, so two orbits that start very close together quickly lose all correlation and appear chaotic.

3. Core code

% Dx=(25a+10)(y-x)
% Dy=(28-35a)x-xz+(29a-1)y
% Dz=xy-(a+8)z/3
% 当0<=a<0.8，广义Lorenz系统
% 当a=0.8，广义吕系统
% 当0.8<a<=1，广义陈系统
clear;clc


 x=1.2;y=1.3;z=1.6;
%x=1 ;y=1 ;z=1 ;

dt=0.005;
a=10.0;c=28.0;b=2.666667;
figure;
for i=1:10000
    newx=x+a*(y-x)*dt;
    newy=y+(c*x-y-x*z)*dt;
    newz=z+(x*y-b*z)*dt;
    if i>1000
 %   plot(x,y)
    % plot(x,z)
    % plot(y,z)
     plot3(x,y,z)
    hold on
    end
    x=newx;y=newy;z=newz;
end

figure;
for i=1:10000
    newx=x+a*(y-x)*dt;
    newy=y+(c*x-y-x*z)*dt;
    newz=z+(x*y-b*z)*dt;
    if i>1000
    plot(x,y)
    % plot(x,z)
    % plot(y,z)
     %plot3(x,y,z)
    hold on
    end
    x=newx;y=newy;z=newz;
end


figure;
for i=1:10000
    newx=x+a*(y-x)*dt;
    newy=y+(c*x-y-x*z)*dt;
    newz=z+(x*y-b*z)*dt;
    if i>1000
   % plot(x,y)
     plot(x,z)
    % plot(y,z)
     %plot3(x,y,z)
    hold on
    end
    x=newx;y=newy;z=newz;
end


figure;
for i=1:10000
    newx=x+a*(y-x)*dt;
    newy=y+(c*x-y-x*z)*dt;
    newz=z+(x*y-b*z)*dt;
    if i>1000
    %plot(x,y)
    % plot(x,z)
     plot(y,z)
     %plot3(x,y,z)
    hold on
    end
    x=newx;y=newy;z=newz;
end

 4. Operation steps and simulation conclusion

    Functional simulation of the system: the following results are obtained:

Figure 1. Overall simulation results of the system

Figure 2. Partial results of the overall simulation of the system

5. References

[01] Lorenz EN. The essence of chaos. Translated by Liu Shida et al. Meteorological Press, 1997

[02] Liu Bingzheng. Nonlinear Dynamics and Chaos Foundation. Changchun: Northeast Normal University Press, 1994

[03] Wu Xiangxing, Chen Zhong. Introduction to Chaos. Shanghai: Shanghai Science and Technology Press, 1996

[04] Li Xiaochun, Zhu Shuanghe, etc. Research on chaotic signal generation circuit. Journal of Air Force Engineering University, 2001

A07-04