# Phase Portrait for the Lorenz System
> restart:
> with(plots):
> with(DEtools):
Warning, the name changecoords has been redefined
# Define the differential equation using parameters:
> LorenzDE := [diff(x(t),t) = sigma*(y(t)-x(t)),
>                diff(y(t),t) = r*x(t)-y(t) -x(t)*z(t),
>            diff(z(t),t) = x(t)*y(t) -b*z(t)];
            [ d                              
            [--- x(t) = sigma (y(t) - x(t)), 
            [ dt                             

               d                                    
              --- y(t) = r x(t) - y(t) - x(t) z(t), 
               dt                                   

               d                           ]
              --- z(t) = x(t) y(t) - b z(t)]
               dt                          ]
# The initial conditions are given in the first line and can be changed.
> IC:=[[x(0)= 0.1,y(0)=0.1, z(0)=0]];
              [[x(0) = 0.1, y(0) = 0.1, z(0) = 0]]
> sigma := 10; b := 8/3; r := 28;
                               10
                               8
                               -
                               3
                               28
# Plot x versus z
> DEplot(LorenzDE, [x(t),y(t),z(t)], t=0..20,  
> IC,linecolour=BLUE, x=-30..30, y=-30..30,z=0..50, stepsize=0.01,
> arrows=NONE,thickness=1, method=classical[rk4],scene=[x,z], 
> title=`Lorenz System: (x,z)`);

# Plot t versus x
> DEplot(LorenzDE, [x(t),y(t),z(t)], t=0..20,  
> IC,linecolour=BLUE,thickness=1, x=-30..30, y=-30..30,z=0..50, stepsize=0.01,
> arrows=NONE, method=classical[rk4],scene=[t,y], 
> title=`Lorenz System: (t,y)`);

# 3 Dimensional plot which can be rotated:
> DEplot3d(LorenzDE, [x(t),y(t),z(t)], t=0..20,  
> IC,linecolour=BLUE, x=-30..30, y=-30..30,z=0..50, stepsize=0.01,
> arrows=NONE, method=classical[rk4],scene=[x,y,z], 
> title=`Lorenz System: (x,y,z)`);

> 
# 
#