Lorenz Attractor in GeoGebra
First of all
First of all
we need to know the equations
\displaystyle\left\{
\begin{array}{rcl}
\dfrac{dx}{dt}&=&\sigma (-x + y)\\
\\
\dfrac{dy}{dt}&=& -x z + \rho x - y \\
\\
\dfrac{dz}{dt}&=& x y - \beta z
\end{array}
\right.
System of differential equations
Lorenz system
\sigma\text{(sigma)},
\rho\text{(rho)},\text{ and }
\beta\text{(beta)} \;\;\;\\
\text{are constants}
t
t
t
x
y
z
x
x
x
y
y
y
z
z
x
Non-linear system of
ordinary differential equations (ODEs)
Lorenz system
\sigma\text{(sigma)},
\rho\text{(rho)},\text{ and }
\beta\text{(beta)} \;\;\;\\
\text{are constants}
\displaystyle\left\{
\begin{array}{rcl}
\dfrac{dx}{dt}&=&\sigma (-x + y)\\
\\
\dfrac{dy}{dt}&=& -x z + \rho x - y \\
\\
\dfrac{dz}{dt}&=& x y - \beta z
\end{array}
\right.
t
t
t
x
y
z
x
x
x
y
y
y
z
z
x
Lorenz system
\(\dfrac{dx}{dt}=x'\)
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
\sigma\text{(sigma)},
\rho\text{(rho)},\text{ and }
\beta\text{(beta)} \;\;\;\\
\text{are constants}
x
y
z
x
x
x
y
y
y
z
x
z
Lorenz system
Lorenz E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences. 20(2): 130–141.
It was introduced in the 1960s
by Edward Norton Lorenz
Lorenz system
Lorenz E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences. 20(2): 130–141.
as a simplified mathematical model
for the atmospheric convection.
It was introduced in the 1960s
by Edward Norton Lorenz
Lorenz system
\displaystyle\left\{
\begin{array}{r}
x(t,x,y,z)\\
\\
y(t,x,y,z)\\
\\
z(t,x,y,z)
\end{array}
\right.
We are looking for the functions
such that
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
What does it mean to solve a system?
Lorenz system
\((x,y,z)\)
\(x\)
\(y\)
\(z\)
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
The solution gives you the position
If the system represents the velocity of this particle
time
\(t\)
Lorenz system
\((x,y,z)\)
\(x\)
\(y\)
\(z\)
time
\(t\)
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
The solution gives you the position
at any time \(t\)
General systems
\displaystyle\left\{
\begin{array}{rcl}
x'_1&=&f_1(t,x_1,x_2,\ldots,x_n)\\
x'_2&=&f_2(t,x_1,x_2,\ldots,x_n)\\
x'_3&=&f_3(t,x_1,x_2,\ldots,x_n)\\
&\vdots& \\
x'_n&=&f_n(t,x_1,x_2,\ldots,x_n)
\end{array}
\right.
In general, solving systems of ODEs is incredibly difficult,
sometimes even impossible!
The good news are that we still can solve them numerically!
General systems
\displaystyle\left\{
\begin{array}{rcl}
x'_1&=&f_1(t,x_1,x_2,\ldots,x_n)\\
x'_2&=&f_2(t,x_1,x_2,\ldots,x_n)\\
x'_3&=&f_3(t,x_1,x_2,\ldots,x_n)\\
&\vdots& \\
x'_n&=&f_n(t,x_1,x_2,\ldots,x_n)
\end{array}
\right.
General systems
In GeoGebra we can use the command
NSolveODE()Here is where computers are quite useful
\displaystyle\left\{
\begin{array}{rcl}
x'_1&=&f_1(t,x_1,x_2,\ldots,x_n)\\
x'_2&=&f_2(t,x_1,x_2,\ldots,x_n)\\
x'_3&=&f_3(t,x_1,x_2,\ldots,x_n)\\
&\vdots& \\
x'_n&=&f_n(t,x_1,x_2,\ldots,x_n)
\end{array}
\right.
NSolveODE(
List of Derivatives,
Initial x-coordinate,
List of Initial y-coordinates,
Final x-coordinate
)Command
NSolveODE()\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
Initial time
\(t_0\)
Initial conditions
\((x_0,y_0,z_0)\) at \(t=t_0\)
Final time
\(t_f\)
Lorenz system
\((x_0,y_0,z_0)\)
\(x\)
\(y\)
\(z\)
time
\(t\)
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
\(t_0\)
\(t_f\)
Lorenz system
\(x\)
\(y\)
\(z\)
time
\(t\)
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
\(t_0\)
\(t_f\)
\((x_f,y_f,z_f)\)
\((x_0,y_0,z_0)\)
Chaotic behaviour
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
(u_0,v_0,w_0)=(1,1.01,1)
(x_0,y_0,z_0)=(1,1,1)
d = 10
b = 8/3
p = 28
x'(t,x,y,z) = d * (y - x)
y'(t,x,y,z) = x * (p - z) - y
z'(t,x,y,z) = x * y - b * z
x0 = 1
y0 = 1
z0 = 1
NSolveODE({x', y', z'}, 0, {x0, y0, z0}, 20)
len = Length(numericalIntegral1)
L_1 = Sequence( (y(Point(numericalIntegral1, i)), y(Point(numericalIntegral2, i)), y(Point(numericalIntegral3, i))), i, 0, 1, 1 / len )
f = Polyline(L_1)Script
The algorithm behind this command is based on
Runge-Kutta numeric methods
Command
NSolveODE()NSolveODE(
List of Derivatives,
Initial x-coordinate,
List of Initial y-coordinates,
Final x-coordinate
)Lorenz Attractor
By Juan Carlos Ponce Campuzano
