Computer Algebra Recipes

84 CHAPTER 2. PHASE-PLANE ANALYSIS
sol2 := [ [x =0;y=0;z=0]; [ x =6 p 2;y=6 p 2;z =27];
[ x = ¡6 p 2;y= ¡6 p 2;z =27]]
Thenumberof¯xedpointsisequaltothenumberN2 of operands in sol2 .
> N2:=nops(sol2);
N2 := 3
So the Lorenz system has three ¯xed points. The stability of these ¯xed points
will now be determined by extending the procedure used in the phase-plane
analysis. A functional operator f is introduced to substitute the coordinates
x + u, y + v, z + w of an ordinary point, near a ¯xed point (x; y; z), into a
dependent variable V ,whereV will be taken to be P , Q, andR.
> f:=V->simplify(subs(fx=x+u,y=y+v,z=z+wg,V)):
Assuming that u, v, andw are su±ciently small, only linear terms in these
variables will be retained in P , Q, andR. Since P is already linear, entering
f(P) produces the linear algebraic form shown in eq1 . However, on entering
f(Q) the nonlinear term uw will be present. This term can be removed with
the remove command, which is done in eq2 . Similarly, the nonlinear term uv
is removed from f(R) in eq3 .
> eq1:=f(P); eq2:=remove(has,f(Q),u*w);
eq3:=remove(has,f(R),u*v):
eq1 := 10 y +10v ¡ 10 x ¡ 10 u
eq2 := 28 x +28u ¡ y ¡ v ¡ xz¡ xw¡ uz
8 z 8 w
eq3 := xy+ xv+ uy¡ ¡
3 3
Let's take i = 1, so that assigning the ith operand in sol2 corresponds to taking
the ¯xed point coordinates to be x =0,y =0,z = 0. The stability of the other
two ¯xed points may be examined by choosing i =2ori =3.
> i:=1: assign(sol2[i]): #choose i
For i =1,eq1 , eq2 ,andeq3 reduce to the following linear forms:
> eq1:=eq1; eq2:=eq2; eq3:=eq3;
8 w
eq1 := 10 v ¡ 10 u eq2 := 28 u ¡ v eq3 := ¡
3
So, the linearized ODEs at an ordinary point near the ¯xed point at the origin
are
du
dv
dw 8 w
=10v ¡ 10 u; =28u ¡ v; = ¡
dt dt dt 3 :
The stability is determined by assuming that u; v; w ' e¸t . For the ODEs
to have a nontrivial solution, a determinantal equation must be satis¯ed that
when expanded produces a cubic equation in ¸ with three roots. To have a
stable ¯xed point, the real part of ¸ must be negative for all three roots. If any
root has a real part that is positive, the solution will grow with time, leading
to an unstable situation.