Juan Carlos Ponce Campuzano
Independent Mathematics Educator
Lorenz Attractor in GeoGebra
First of all
First of all
we need to know the equations
⎩⎨⎧dtdxdtdydtdz===σ(−x+y)−xz+ρx−yxy−βz
\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
σ(sigma),ρ(rho), and β(beta)are constants
\sigma\text{(sigma)},
\rho\text{(rho)},\text{ and }
\beta\text{(beta)} \;\;\;\\
\text{are constants}
t
t
t
t
t
t
x
x
y
y
z
z
x
x
x
x
x
x
y
y
y
y
y
y
z
z
z
z
x
x
Non-linear system of
ordinary differential equations (ODEs)
Lorenz system
σ(sigma),ρ(rho), and β(beta)are constants
\sigma\text{(sigma)},
\rho\text{(rho)},\text{ and }
\beta\text{(beta)} \;\;\;\\
\text{are constants}
⎩⎨⎧dtdxdtdydtdz===σ(−x+y)−xz+ρx−yxy−βz
\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
t
t
t
x
x
y
y
z
z
x
x
x
x
x
x
y
y
y
y
y
y
z
z
z
z
x
x
Lorenz system
dtdx=x′
⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
σ(sigma),ρ(rho), and β(beta)are constants
\sigma\text{(sigma)},
\rho\text{(rho)},\text{ and }
\beta\text{(beta)} \;\;\;\\
\text{are constants}
x
x
y
y
z
z
x
x
x
x
x
x
y
y
y
y
y
y
z
z
x
x
z
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
⎩⎨⎧x(t,x,y,z)y(t,x,y,z)z(t,x,y,z)
\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
⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\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
⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\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
⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\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
⎩⎨⎧x1′x2′x3′xn′===⋮=f1(t,x1,x2,…,xn)f2(t,x1,x2,…,xn)f3(t,x1,x2,…,xn)fn(t,x1,x2,…,xn)
\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
⎩⎨⎧x1′x2′x3′xn′===⋮=f1(t,x1,x2,…,xn)f2(t,x1,x2,…,xn)f3(t,x1,x2,…,xn)fn(t,x1,x2,…,xn)
\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
⎩⎨⎧x1′x2′x3′xn′===⋮=f1(t,x1,x2,…,xn)f2(t,x1,x2,…,xn)f3(t,x1,x2,…,xn)fn(t,x1,x2,…,xn)
\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()⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\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
t0
Initial conditions
(x0,y0,z0) at t=t0
Final time
tf
Lorenz system
(x0,y0,z0)
x
y
z
time
t
⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
t0
tf
Lorenz system
x
y
z
time
t
⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
t0
tf
(xf,yf,zf)
(x0,y0,z0)
Chaotic behaviour
⎩⎨⎧x′y′z′===σ(−x+y)−xz+ρx−yxy−βz
\displaystyle\left\{
\begin{array}{rcl}
x'&=&\sigma (-x + y)\\
\\
y'&=& -x z + \rho x - y \\
\\
z'&=& x y - \beta z
\end{array}
\right.
(u0,v0,w0)=(1,1.01,1)
(u_0,v_0,w_0)=(1,1.01,1)
(x0,y0,z0)=(1,1,1)
(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 in GeoGebra
Lorenz Attractor
By Juan Carlos Ponce Campuzano
